IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS
MAHDI ASGARI AND FREYDOON SHAHIDI

Abstract. We give a complete description of the image of the endoscopic functorialtransfer of generic automorphic representations from the quasi-split general spin groups
to general linear groups over arbitrary number fields. This result is not covered by
the recent work of Arthur on endoscopic classification of automorphic representations ofclassical groups. The image is expected to be the same for the whole tempered spectrum,
whether generic or not, once the transfer for all tempered representations is proved. We
give a number of applications including estimates toward the Ramanujan conjecture forthe groups involved and the characterization of automorphic representations of GL(6)which are exterior square transfers from GL(4), among others. More applications to
reducibility questions for the local induced representations of p-adic groups will also
follow.

1. Introduction
The purpose of this article is to completely determine the image of the transfer of globally
generic, automorphic representations from the quasi-split general spin groups to the generallinear groups. We prove the image is an isobaric, automorphic representation of a generallinear group. Moreover, we prove that the isobaric summands of this representation havethe property that their twisted symmetric square or twisted exterior square L-function has
a pole at s = 1, depending on whether the transfer is from an even or an odd general spingroup. We are also able to determine whether the image is a transfer from a split group, ora quasi-split, non-split group.
Arthur’s recent book on endoscopic classification of representations of classical groups[Ar] establishes similar results for automorphic representations of orthogonal and symplectic
groups, generic or not, but does not cover the case of the general spin groups we consider
here.
To describe our results in more detail, let k be a number field and let A = Ak denote
its ring of adèles. Let G be the split group GSpin(2n + 1), GSpin(2n), or one of its quasisplit non-split forms GSpin∗ (2n) associated with a quadratic extension K/k (cf. Section 2).
There is a natural embeddingι : L G −→ GL(2n, C) × Γk

(1.1)

of the L-group of G, as a group over k, into that of GL(2n) described in Section 3. Let
π be a globally generic, (unitary) cuspidal, automorphic representation of G = G(A). For
almost all places v of k the local representation πv is parameterized by a homomorphism
φv : Wv −→ L Gv ,
L

(1.2)

where Wv is the local Weil group of kv and Gv is the L-group of G as a group over
kv . Langlands Functoriality then predicts that there is an automorphic representation Π ofGL(2n, A) such that for almost all v, the local representation Πv is parameterized by ι ◦ φv .
Our main result is to prove that the transferred representation Π of GL(2n), from either
an even or an odd general spin group, is indeed an isobaric, automorphic representation
1

2

MAHDI ASGARI AND FREYDOON SHAHIDI(cf. Theorem 4.20 and its corollary). We then give a complete description of the image ofthese transfers in terms of the twisted symmetric square or exterior square L-functions (cf.
Theorem 4.26). We refer to [L1] for the notion of isobaric representations.
To prove that the image is isobaric, one needs to know some of the analytic propertiesof the Rankin-Selberg type L-functions L(s, π × τ ), where τ is a cuspidal representation ofGL(m, A) and π is a generic representation of G(A). In particular, one needs to know that
the L-functions for G × GL(m) for m up to 2n are holomorphic for <(s) > 1 and to know
under what conditions these L-function have poles at s = 1. These facts are established in
Section 4. Here, we use a result on possible poles of the Rankin-Selberg L-functions for
G × GL(m) (Proposition 4.9) which we establish in [ACS1] as part of a more extensive
project. Consequently, we are able to prove that the transferred representation Π is unique;
it is an isobaric sum of pairwise inequivalent, unitary, cuspidal, automorphic representationsΠ = Π1  · · ·  Πt ,(1.3)

and each Πi satisfies the condition that its twisted symmetric square or twisted exteriorsquare L-function has a pole at s = 1, depending on whether we are transferring from evene ⊗ ω (andor odd general spin groups (cf. Theorem 4.26). In particular, we prove that Π ∼=Πe denotes the contragredient of Π.
not just nearly equivalent as we showed in [AS1]). Here, ΠThe automorphic representations Π of GL(2n, A) which are transfers from representations
π of general spin groups satisfye ⊗ ω,
Π∼(1.4)=Π
as predicted by the theory of twisted endoscopy [KoSh]. In fact, these representations
comprise precisely the image of the transfer. While we prove half of this statement, we notethat the other half of this, i.e., the fact that any representation of GL(2n, A) satisfying (1.4)
is a transfer from a representation of a general spin group has now also been proved thanks
to the work of J. Hundley and E. Sayag [HS1, HS2], extending the descent theory resultsof Ginzburg, Rallis, and Soudry [GRS] from the case of classical groups (ω = 1) to generalspin groups .
If a representation Π of GL(2n, A) satisfies (1.4), then
LT s, Π × (Π ⊗ ω −1 ) = LT (s, Π, Sym2 ⊗ ω −1 )LT (s, Π, ∧2 ⊗ ω −1 ),(1.5)
where T is a sufficiently large finite set of places of k and LT denotes the product over v 6∈ Tof the local L-functions. The L-function on the left hand side of (1.5) has a pole at s = 1,
which implies that one, and only one, of the two L-functions on the right hand side of (1.5)
has a pole at s = 1. If the twisted exterior square L-function has a pole at s = 1, then Π isa transfer from an odd general spin group and if the twisted symmetric square L-function
has a pole at s = 1, then Π is a transfer from an even general spin group (which may be
split or quasi-split non-split).
To tell the split and quasi-split non-split cases apart, note that from (1.4) we have2ωΠ
= ω 2n .
−n(1.6)

In other words, µ = ωΠ ω
is a quadratic idèle class character. If µ is the trivial character,then Π is the transfer of a generic representation of the even split general spin group and ifµ is a non-trivial quadratic character, then Π is a transfer from a generic representation ofa quasi-split group associated with the quadratic extension of k determined by µ throughclass field theory.
Our results here along with those of Hundley and Sayag [HS1, HS2] give a complete
description of the image of the transfer for the generic representations of the general spin

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

3

groups. It remains to study the transfers of non-generic, cuspidal, automorphic representations of the general spin groups, which our current method cannot handle. However, theimage of the generic spectrum is conjecturally the full image of the tempered spectrum,
generic or not, of the general spin groups since each tempered L-packet is expected to contain a generic member. We refer to [Sh6] for more details on this conjecture. We point out
that Arthur’s upcoming book [Ar] would answer this question in the case of classical groups.
However, his book does not cover the case of general spin groups.
While we characterize the automorphic representation Π in the image, we should remarkthat the existence of Π in the case of the split groups GSpin(2n) and GSpin(2n + 1) wasestablished in [AS1]. However, we were not able to prove the quasi-split case then. Oneof our results, therefore, is also to establish the transfer of globally generic, automorphic
representations from the quasi-split non-split even general spin group, GSpin∗ (2n, Ak ), toGL(2n, Ak ) (cf. Theorem 3.5). The quasi-split case had to wait because the local technicaltools of “stability of γ-factors” (cf. Proposition 3.15) and a result on local L-functions
and normalized intertwining operators (Proposition 3.13) were not available in the quasisplit non-split case. The local result is now available in our cases thanks to the thesis of
Wook Kim [WKim], and more generally the work of Heiermann and Opdam [HO], and thestability of γ-factors is available in appropriate generality thanks to a recent work of Cogdell,
Piatetski-Shapiro and Shahidi [CPSS1].
As in the split case, the method of proving the existence of an automorphic representation
Π is to use converse theorems. This requires knowledge of the analytic properties of the
L-functions for GL(m) × GL(2n) for m ≤ 2n − 1. The two local tools allow us to relate the
L-functions for G×GL from the Langlands-Shahidi method to those required in the converse
theorems in the following way. Due to the lack of the local Langlands correspondence ingeneral, there is no natural choice for the local components of our candidate representation
Π at the finite number of exceptional places of k where some of our data may be ramified.
This means that we have to pick these local representations essentially arbitrarily. However,
we show that the local γ-factors appearing will become independent of the representation,
depending only on the central character, if we twist by a highly ramified character. Globally we can afford to twist our original representation by an idèle class character which ishighly ramified at a finite number of places. With this technique we succeed in applying
an appropriate version of the converse theorems. The conclusion is to have an automorphic
representation Π of GL(2n, A) which is locally the transfer of π associated with ι outside
a finite number of finite places. Moreover, if ω = ωπ is the central character of π, thenωΠ = ω n µ, where µ is a quadratic idèle class character, only non-trivial in the quasi-split
non-split case. In that case it determines the defining quadratic extension. This settles theexistence of Π as an automorphic representation in all cases, whose complete description we
give, as we explained above.
We summarize the results on Π being isobaric and its description in terms of twisted
symmetric or exterior square L-functions in Theorem 4.26.
These results allow us to give a number of applications. As a first application, we are able
to describe the local component of the transferred representation at the ramified places. In
particular, we show that these local components are generic (cf. Proposition 5.1).
Another application is to prove estimates toward the Ramanujan conjecture for thegeneric spectrum of the general spin groups. We do this by using the best estimates currently known for the general linear groups [LRS]. In particular, our estimates show that
if we know the Ramanujan conjecture for GL(m) for m up to 2n, then the Ramanujan
conjecture for the generic spectrum of GSpin(2n + 1) and GSpin(2n) follows. We note that

4

MAHDI ASGARI AND FREYDOON SHAHIDI

the Ramanujan conjecture is expected not to hold for certain non-generic representations of
the general spin groups.
Yet another application of our main results is to give more information about H. Kim’s
exterior square transfer from GL(4) to GL(6) with the help of some recent work of J. Hundleyand E. Sayag. We prove that a cuspidal representation Π of GL(6) is in the image of Kim’s
transfer if and only if the (partial) twisted symmetric square L-function of Π has a pole ats = 1 (cf. Proposition 5.10).
We can apply our results in this paper, along with those of Cogdell, Kim, Krishnamurthy,
Piatetski-Shapiro, and Shahidi for the classical and unitary groups, to give some uniform
results on reducibility of local induced representations of non-exceptional p-adic groups. We
will address this question along with other local applications of generic functoriality in aforthcoming paper [ACS2].
We would like to thank J. Cogdell for many helpful discussions on the material in thisarticle. We are also grateful to the referee for comments that helped improve the presentationand precision.
Parts of this work were done while the authors were visiting The Erwin Schrödinger
Institute for Mathematical Physics (ESI) and the Institute for Research in Fundamental
Sciences (IPM). We would like to thank both institutes for their hospitality and great
working environment. The first author was partially supported by NSA grants H-98230–
09–1–0049 and H-98230–11–1–0149, an Alexander von Humboldt Fellowship, and grant #245422 from the Simons Foundation. The second author was partially supported by NSF
grants DMS–0700280 and DMS–1162299.
Contents
1. Introduction
2. The Preliminaries
2.1. Structure of GSpin Groups
3. Weak Transfer for the Quasi-split GSpin(2n)3.1. The Converse Theorem
3.2. L-functions for GL(m) × GSpin∗ (2n)3.3. Stability of γ-factors
3.4. The Candidate Transfer
3.5. Proof of Theorem 3.5
4. The Transferred Representation
4.1. Low Rank Cases
4.2. The Global Transfer
4.3. Description of the Image of Transfer
5. Applications
5.1. Local Representations at the Ramified Places
5.2. Ramanujan Estimates
5.3. Image of Kim’s exterior square
References

1
4
5
6
7
8
9
9
13
14
14
15
18
20
20
21
22
23

2. The Preliminaries
Let k be a number field and let A = Ak be the ring of adèles of k. Let n ≥ 1 be an
integer. We consider the general spin groups. The group GSpin(2n+1) is a split, connected,

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

5

reductive group of type Bn defined over k whose dual group is GSp(2n, C). Similarly, thesplit, connected, reductive group GSpin(2n) over k is of type Dn and its dual is isomorphic
to GSO(2n, C), the connected component of the group GO(2n, C). There are also quasisplit non-split groups GSpin∗ (2n) in the even case. They are of type 2 Dn and correspondto quadratic extensions of k. A more precise description is given below. We also refer to[CPSS2, §7] for a review of the generalities about these quasi-split groups.
We fix a Borel subgroup B and a Cartan subgroup T ⊂ B. The associated based root
datum to (B, T) will be denoted by (X, ∆, X ∨ , ∆∨ ), which we further explicate below. Our
choice of the notation for the root data below is consistent with the Bourbaki notation [Bou].
2.1. Structure of GSpin Groups. We describe the odd and even GSpin groups by introducing a based root datum for each as in [Spr, §7.4.1]. More detailed descriptions can also
be found in [AS1, §2] and [HS2, §4]. We use these data as our tools to work with the groups
in question due to the lack of a convenient matrix representation.
2.1.1. The root datum of GSpin(2n + 1). The root datum of GSpin(2n + 1) is given by(X, R, X ∨ , R∨ ), where X and X ∨ are Z-modules generated by generators e0 , e1 , · · · , en ande∗0 , e∗1 , · · · , e∗n , respectively. The pairing
h , i : X × X ∨ −→ Z

(2.1)

is the standard pairing. Moreover, the roots and coroots are given by
R = R2n+1
R∨∨= R2n+1= {±(ei ± ej ) : 1 ≤ i < j ≤ n} ∪ {±ei | 1 ≤ i ≤ n}
(2.2)

∗
∗
=
±(ei − ej ) : 1 ≤ i < j ≤ n ∪
(2.3)

∗
∗
∗
∗
∗
±(ei + ej − e0 ) : 1 ≤ i < j ≤ n ∪ {±(2ei − e0 ) | 1 ≤ i ≤ n}along with the bijection R −→ R∨ given by(±(ei − ej ))∨=±(e∗i − e∗j )(2.4)(±(ei + ej ))

∨

=

(2.5)

(±ei )∨=±(e∗i + e∗j − e∗0 )±(2e∗i − e∗0 ).

(2.6)

It is easy to verify that the conditions (RD 1) and (RD 2) of [Spr, §7.4.1] hold. Moreover,
we fix the following choice of simple roots and coroots:
∆

=

∆∨

=

{e1 − e2 , e2 − e3 , · · · , en−1 − en , en } , ∗e1 − e∗2 , e∗2 − e∗3 , · · · , e∗n−1 − e∗n , 2e∗n − e∗0 .

(2.7)
(2.8)

This datum determines the group GSpin(2n + 1) uniquely, equipped with a Borel subgroupcontaining a maximal torus.
2.1.2. The root datum of GSpin(2n). Next, we give a similar description for the even case.
The root datum of GSpin(2n) is given by (X, R, X ∨ , R∨ ) where X and X ∨ and the pairing
is as above and the roots and coroots are given by
R = R2n∨R∨ = R2n= {±(ei ± ej ) : 1 ≤ i < j ≤ n}=±(e∗i − e∗j ) : 1 ≤ i < j ≤ n ∪±(e∗i + e∗j − e∗0 ) : 1 ≤ i < j ≤ n(2.9)(2.10)

along with the bijection R −→ R∨ given by(±(ei − ej ))∨(±(ei + ej ))∨= ±(e∗i − e∗j )=±(e∗i + e∗j − e∗0 ).

(2.11)(2.12)

6

MAHDI ASGARI AND FREYDOON SHAHIDI

It is easy again to verify that the conditions (RD 1) and (RD 2) of [Spr, §7.4.1] hold. Similar
to the odd case we fix the following choice of simple roots and coroots:∆∆∨{e1 − e2 , e2 − e3 , · · · , en−1 − en , en−1 + en } , ∗=e1 − e∗2 , e∗2 − e∗3 , · · · , e∗n−1 − e∗n , e∗n−1 + e∗n − e∗0 .

=(2.13)(2.14)

This based root datum determines the split group GSpin(2n) uniquely, equipped with a
Borel subgroup containing a maximal torus.
2.1.3. The quasi-split forms of GSpin(2n). In the even case, quasi-split non-split forms also
exist. We fix a splitting (B, T, {xα }α∈∆ ), where {xα } is a collection of root vectors, one
for each simple root of T in B. The quasi-split forms of GSpin(2n) over k are in oneone correspondence with homomorphisms from Gal(k̄/k) to the group of automorphisms ofthe character lattice preserving ∆. This group has two elements: the trivial and the one
switching en−1 − en and en−1 + en while keeping all other simple roots fixed (cf. [HS2, §4.3],
and [CPSS2, §7.1] for the quasi-split forms of SO(2n)).
By Class Field Theory such homomorphisms correspond to quadratic charactersµ : k × \A×k −→ {±1}.

(2.15)

When µ is non-trivial we denote the associated quasi-split non-split group with GSpinµ (2n)or simply GSpin∗ (2n) when the particular µ is unimportant. We will also denote the quadratic extension of k associated with µ by K µ /k or simply K/k.
3. Weak Transfer for the Quasi-split GSpin(2n)In this section, n ≥ 1 will be an integer and G = GSpin∗ (2n) will denote one of thequasi-split non-split forms of GSpin(2n) as in 2.1.3. We will denote the associated quadratic
extension by K/k and A = Ak will continue to denote the ring of adèles of k. Also, G is
associated with a non-trivial quadratic character µ : k × \A×k −→ {±1}.
L 0
The connected component of the L-group of G is G = GSO(2n, C) and the L-groupcan be written as
L
G = GSO(2n, C) o Wk ,(3.1)
where the Weil group acts through the quotient∼ Gal(K/k).
Wk /WK =(3.2)
The L-group of GL(2n) is GL(2n, C) × Wk , a direct product because GL(2n) is split. These
are the Weil forms of the L-group, or we can equivalently use the Galois forms of theL-groups.
We define a map
ι : GSO(2n, C) o Γk −→ GL(2n, C) × Γk((g, γ)
if γ|K = 1,(g, γ) 7→(hgh−1 , γ) if γ|K 6= 1,
where γ ∈ Γk , g ∈ GSO(2n, C) ⊂ GL(2n, C), and

In−1

0
−1
h=h =

1

(3.3)


1
0


.


(3.4)

In−1
(We refer to [CPSS2, §7.1] for more details.) The map ι is an L-homomorphism. We also
have a compatible family of local L-homomorphisms ιv : L Gv −→ GL(2n, C) × Wv . Our

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

7

purpose in this section is to prove the existence of a weak transfer of globally generic, cuspidal, automorphic representations of G = G(A) to automorphic representations of GL(2n, A)
associated with ι.
Theorem 3.5. Assume that n ≥ 1 is an integer. Let K/k be a quadratic extension of
number fields and let G = GSpin∗ (2n) be as above. Let ψ be a non-trivial continuous
additive character of k\Ak . The choice of ψ and the splitting above defines a non-degenerate
additive character of U(k)\U(A), again denoted by ψ.
Let π = ⊗v πv be an irreducible, globally ψ-generic, cuspidal, automorphic representationof G = G(Ak ). Write ψ = ⊗v ψv . Let S be a non-empty finite set of non-archimedean places
v of k such that for every non-archimedean v 6∈ S both πv and ψv , as well as Kw /kv for w|v,
are unramified. Then there exists an automorphic representation Π = ⊗v Πv of GL(2n, Ak )such that for all v 6∈ S the homomorphism parameterizing the local representation Πv is
given byΦv = ιv ◦ φv : Wkv → GL(2n, C),
where Wkv denotes the local Weil group of kv and φv : Wkv −→ L G is the homomorphism
parameterizing πv . Moreover, if ωΠ and ωπ denote the central characters of Π and π,
respectively, then ωΠ = ωπn µ, where µ is the non-trivial quadratic idèle class charactere ⊗ ωπ arecorresponding to K/k, the quadratic extension defining G. Furthermore, Π and Π
nearly equivalent.
Remark 3.6. We proved an analogous result for the split groups GSpin(2n+1) and GSpin(2n)in [AS1].
To prove the theorem we will use a suitable version of the converse theorems of Cogdell
and Piatetski-Shapiro [CPS1, CPS2]. The exact version we need can be found in [CPSS2, §2]
which we quickly review below. Next we introduce an irreducible, admissible representation
Π of GL(2n, A) as a candidate for the transfer of π. We then prove that Π satisfies therequired conditions of the converse theorem and hence is automorphic. Along the way wealso verify the remaining properties of Π stated in Theorem 3.5.
3.1. The Converse Theorem. Let k be a number field and fix a non-empty finite set S
of non-archimedean places of k. For each integer m let
A0 (m) = {τ |τ is a cuspidal representation of GL(m, Ak )}

(3.7)

and
AS0 (m) = {τ ∈ A0 (m)|τv is unramified for all v ∈ S} .
Also, for a positive integer N let
N
−1
a

N
−1
a

AS0 (m)(3.9)

and for η a continuous character of k × \A×
k let

S
T (S; η) = T (N − 1) ⊗ η = τ = τ 0 ⊗ η|τ 0 ∈ T S (N − 1) .

(3.10)

T (N − 1) =A0 (m)and T S (N − 1) =

(3.8)

m=1

m=1

For our purposes we will apply the following theorem with N = 2n.
Theorem 3.11. (Converse theorem of Cogdell and Piatetski-Shapiro) Let Π = ⊗Πv be anirreducible, admissible representation of GL(N, Ak ) whose central character ωΠ is invariant
under k × and whose L-functionY
L(s, Π) =L(s, Πv )v

8

MAHDI ASGARI AND FREYDOON SHAHIDI

is absolutely convergent in some right half plane. Let S be a finite set of non-archimedean
places of k and let η be a continuous character of k × \A× . Suppose that for every τ ∈ T (S; η)
the L-function L(s, τ × Π) is nice, i.e., it satisfies the following three conditions:e extend to entire functions of s ∈ C.
(1) L(s, τ × Π) and L(s, τe × Π)
e are bounded in vertical strips.
(2) L(s, τ × Π) and L(s, τe × Π)e holds.
(3) The functional equation L(s, τ × Π) = (s, τ × Π)L(s, τe × Π)0
∼ Π0 for
Then there exists an automorphic representation Π of GL(N, Ak ) such that Πv =v
all v 6∈ S.
2
The twisted L- and -factors in the statement are those in [CPS1]. In particular, they
are Artin factors and known to be the same as the ones coming from the Langlands-Shahidi
method at all places.
3.2. L-functions for GL(m) × GSpin∗ (2n). Let π be an irreducible, admissible, globally
generic representation of GSpin∗ (2n, Ak ) and let τ be a cuspidal representation of GL(m, Ak )with m ≥ 1. The group GSpin∗ (2(m+n)) has a standard maximal Levi GL(m)×GSpin∗ (2n)and we have the completed L-functions
Y
Y
L(s, τ × π) =L(s, τv × πv ) =L(s, τv ⊗ πv , ι0v ⊗ ιv ) = L(s, τ ⊗ π, ι0 ⊗ ι),(3.12)
v

v

with similar - and γ-factors, defined via the Langlands-Shahidi method in [Sh3]. Here, ι isthe representation of the L-group of GSpin∗ (2n) we described before and ι0 is the projection
map onto the first factor in the L-group L GL(m) = GL(m, C) × Wk .
Proposition 3.13. Let S be a non-empty finite set of finite places of k and let η be a2
character of k × \A×k such that, for some v ∈ S, ηv is ramified. Then for all τ ∈ T (S; η) the
L-function L(s, τ × π) is nice, i.e., it satisfies the following three conditions:(1) L(s, τ × π) and L(s, τe × πe) extend to entire functions of s ∈ C.
(2) L(s, τ × π) and L(s, τe × πe) are bounded in vertical strips.
(3) The functional equation L(s, τ × π) = (s, τ × π)L(s, τe × πe) holds.
Proof. Twisting by η is necessary for conditions (1) and (2). Both (2) and (3) hold in wide
generality.
Condition (2) follows from [GS, Cor. 4.5] and is valid for all τ ∈ T (N − 1), provided
that one removes neighborhoods of the finite number of possible poles of the L-function.
Condition (3) is a consequence of [Sh3, Thm. 7.7] and is valid for all τ ∈ T (N − 1).
Condition (1) follows from a more general result, [KS1, Prop. 2.1]. Note that this result
rests on Assumption 1.1 of [KS1], sometimes called Assumption A [K1], on certain normalized intertwining operators being holomorphic and non-zero. Fortunately the assumption
has been verified in our cases. The assumption requires two ingredients: the so-called “standard modules conjecture” and the “tempered L-functions conjecture”. Both of these havebeen verified in our cases in Wook Kim’s thesis [WKim]. For results proving various cases ofthis assumption we refer to [Sh3, CSh, MuSh, Mu, A, K3, Hei, KK]. Recently V. Heiermann
and E. Opdam have proved the assumption in full generality in [HO].
The key now is to relate the L-functions L(s, τ × π), defined via the Langlands-Shahidi
method, to the L-functions L(s, Π × τ ) in the converse theorem. We note that, when we
introduce our candidate for Π, for archimedean places and those non-archimedean places atwhich all data are unramified we know the equality of the local L-functions. However, we
do not know this to be the case for ramified places. We get around this problem throughIMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

9

stability of γ-factors, which basically makes the choice of local components of Π at theramified places irrelevant as long as we can twist by highly ramified characters.
3.3. Stability of γ-factors. In this subsection let F denote a non-archimedean local fieldof characteristic zero. Let G = GSpin∗ (2n, F ), where the quasi-split non-split group isassociated with a quadratic extension E/F .
Fix a non-trivial additive character ψ of F . Let π be an irreducible, admissible, ψ-genericrepresentation of G and let η denote a continuous character of GL(1, F ). Let γ(s, η × π, ψ)be the associated γ-factor defined via the Langlands-Shahidi method [Sh3, Theorem 3.5].
We have(s, η × π, ψ)L(1 − s, η −1 × πe).
(3.14)
γ(s, η × π, ψ) =L(s, η × π)
Proposition 3.15. Let π1 and π2 be two irreducible, admissible, ψ-generic representationsof G having the same central characters. Then for a suitably highly ramified character η ofGL(1, F ) we haveγ(s, η × π1 , ψv ) = γ(s, η × π2 , ψv ).
Proof. This is a special case of a more general theorem which is the main result of [CPSS1].
We note that in our case one has to apply that theorem to the self-associate maximal Levi
subgroup GL(1) × GSpin∗ (2n) in GSpin∗ (2n + 2) which does satisfy the assumptions of thattheorem.

3.4. The Candidate Transfer. We construct now a candidate global transfer Π = ⊗v Πv
as a restricted tensor product of its local components Πv , which are irreducible, admissible
representations of GL(2n, kv ). There are three cases to consider: (i) archimedean v, (ii)
non-archimedean unramified v, (iii) non-archimedean ramified v.
3.4.1. The archimedean transfer. If v is an archimedean place of k, then by the local Langlands correspondence [L2, Bor] the representation πv is parameterized by an admissible
homomorphism φv and we choose Πv to be the irreducible, admissible representation ofGL(2n, kv ) parameterized by Φv as in the statement of Theorem 3.5. We then haveL(s, πv ) = L(s, ιv ◦ φv ) = L(s, Πv )(3.16)

(s, πv , ψv ) = (s, ιv ◦ φv , ψv ) = (s, Πv , ψv ),(3.17)

andwhere the middle factors are the local Artin-Weil L- and -factors attached to representationsof the Weil group as in [T]. The other L- and -factors are defined via the Langlands-Shahidi
method which, in the archimedean case, are known to be the same as the Artin factors
defined through the arithmetic Langlands classification [Sh1].
If τv is an irreducible, admissible representation of GL(m, kv ), then it is parameterized
by an admissible homomorphism φ0v : Wkv −→ GL(m, C) and the tensor product homomorphism (ιv ◦ φv ) ⊗ φ0v : Wkv −→ GL(2mn, C) is another admissible homomorphism and weagain haveL(s, πv × τv ) = L(s, (ιv ◦ φv ) ⊗ φ0v ) = L(s, Πv × τv )(3.18)
and(s, πv × τv , ψv ) = (s, (ιv ◦ φv ) ⊗ φ0v , ψv ) = (s, Πv × τv , ψv ).
Hence, we get the following matching of the twisted local L- and -factors.

(3.19)

10

MAHDI ASGARI AND FREYDOON SHAHIDI

Proposition 3.20. Let v be an archimedean place of k and let πv be an irreducible, admissible, generic representation of GSpin∗ (2n, kv ), Πv its local functorial transfer to GL(2n, kv ),and τv an irreducible, admissible, generic representation of GL(m, kv ). Then
L(s, πv × τv ) = L(s, Πv × τv ),e v × τev )L(s, πev × τev ) = L(s, Πand(s, πv × τv , ψv ) = (s, Πv × τv , ψv ).
2
3.4.2. The non-archimedean unramified transfer. If v is a non-archimedean place of k suchthat πv , as well as all Kw /kv for w|v, are unramified, then by the arithmetic Langlands
classification or the Satake classification [Bor, Sat], the representation πv is parameterized
by an unramified admissible homomorphism φv : Wkv −→ L Gv . Again we take Φv as in thestatement of the theorem. It defines an irreducible, admissible, unramified representation
Πv of GL(2n, kv ) [HT, H1].
Given that πv is unramified, its parameter φv factors through an unramified homomorphism into the maximal torus L Tv ,→ L Gv . Then Φv has its image in a torus of GL(2n, C)and Πv is the corresponding unramified representation. We haveL(s, πv ) = L(s, Πv )(3.21)

(s, πv , ψv ) = (s, Πv , ψv )(3.22)

andand the factors on either side of the above equations can be expressed as products of onedimensional abelian Artin factors by multiplicativity of the local factors.
Let τv be an irreducible, admissible, generic, unramified representation of GL(m, kv ).
Again appealing to the general multiplicativity of local factors [JPSS, Sh3, Sh4] we haveL(s, πv × τv ) = L(s, Πv × τv )(3.23)

(s, πv × τv , ψv ) = (s, Πv × τv , ψv ).

(3.24)

andHence, we again get the following matching of the twisted local L- and -factors.
Proposition 3.25. Let v be a non-archimedean place of k and let πv be an irreducible, admissible, generic, unramified representation of GSpin∗ (2n, kv ), Πv its local functorial transfer to GL(2n, kv ), and τv an irreducible, admissible, generic representation of GL(m, kv ).
Thene v × τev )L(s, πv × τv ) = L(s, Πv × τv ), L(s, πev × τev ) = L(s, Πand(s, πv × τv , ψv ) = (s, Πv × τv , ψv ).
2
Proposition 3.25 is what is needed for the application of the converse theorem, establishingthat the global representation Π is automorphic. However, to prove further properties of Π,
we also make the local transfer in this case explicit. The analysis is similar to that of thequasi-split SO(2n) carried out in [CPSS2, §7.2], which we refer to for more detail.
Let G = GSpin∗ (2n), associated with the quadratic extension K/k of number fields. Letµ be the associated quadratic idèle class character of k as in (2.15). Assume that v is a
non-archimedean place of k where all the data is unramified. There are two cases: either v

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

11

splits as two places in K or v is inert, i.e., there is a single place in K above the place v ink.
First, consider the case that v is inert in K and let w be the single place of K above v.
The maximal torus Tv in Gv is isomorphic to GL(1)n−1 × T0 , where T0 ∼= GSpin∗ (2) is a
× n−1
×
∼
non-split torus in Gv . We have Tv (kv ) = (kv )× Kw and the center of Gv (kv ) sits as a×copy of kv× in Kw. The unramified representation πv is then determined by n − 1 unramified×characters χ1 , . . . , χn−1 of kv× along with a character χ̃n of Kw
satisfying
χ̃n |kv× = ωπv .

(3.26)

Note that the χi ’s are characters of kv× even though no v appears; they are not necessarily
local components of global idèle class characters.
The representation Πv of GL(2n, kv ) is induced from




W
−1
χ1 , . . . , χn−1 , π IndWkKvw χ̃n , χ−1
ω
,
.
.
.
,
χ
ω
(3.27)πv ,n−1 πv1WW
where π IndWkKvw χ̃n is the Weil representation of GL(2, kv ) defined by IndWkKvw χ̃n . (For aquick review of its definition and properties, see [GL, Appendix
B], for example.)W

Being a Weil representation, the central character of π IndWkKvw χ̃n is equal toµv · χ˜n |kv× = µv · ωπv ,(3.28)

×where µv , the component of the global µ at v, is the generating character of kv× /NKw /kv (Kw),××
i.e., the non-trivial character of kv which is trivial on norm elements, NKw /kv (Kw ). Hence,the central character of Πv is equal to ωπnv µv .


W

en
Moreover, by properties of the Weil representation, the contragredient of π IndWkKvw χW

is the Weil representation defined by IndWkKvw χ
e−1
n and




W
W
π
e IndWkKvw χen ⊗ ωπv ∼
e−1
⊗ ωπv
= π IndWkKvw χ
n


W
∼
e−1
= π IndWkKvw χn · ωπv ◦ NKw /kv


W
∼
en .
= π IndWkKvw χ(3.29)The last equivalence holds becauseχ
en−1 · ωπv ◦ NKw /kv = χ
en γ ,(3.30)

ωπv ◦ NKw /kv = χ
en · χ
en γ ,(3.31)

or equivalently,
which immediately follows from (3.26). Here, γ denotes the non-trivial element of the Galoise v ⊗ ωπ .
group Gal(Kw /kv ). Hence, Πv ∼=Πv
Furthermore, we note that for the transferred representation Πv to be a principal series,
the character χen must factor through the norm. Write χ
en = χn ◦NKw /kv with χn a character×2
of kv . Then, (3.26) implies that χn = ωπv and the representation Πv is a constituent of therepresentation of GL(2n, kv ) induced from−1−1
χ1 , . . . , χn−1 , χn µv , χ−1(3.32)n ωπv , χn−1 ωπv , . . . , χ1 ωπv .
Next, consider the case that the place v splits in K. We haveGv (kv ) ∼= GSpin(2n, kv ).

(3.33)

12

MAHDI ASGARI AND FREYDOON SHAHIDI

The unramified representation πv is given by n unramified characters χ1 , . . . , χn of kv× alongwith the central character ωπv .
The transferred representation Πv of GL(2n, kv ) is now the same as in the case of the
split GSpin(2n) we treated in [AS1, Eq. (64)]. More precisely, the representation Πv is aconstituent of the representation of GL(2n, kv ) induced from−1−1
χ1 , . . . , χn−1 , χn µv , χ−1(3.34)n ωπv , χn−1 ωπv , . . . , χ1 ωπv ,where µv , the component at v of the quadratic idèle class character µ associated withe v ⊗ ωv andthe quadratic extension K/k is now trivial. Hence, we again have Πv ∼= Πn
ωΠv = ωπv µv .
Therefore, we have proved the following.
Proposition 3.35. Let v be a non-archimedean place of k and let πv be an irreducible,
admissible, generic, unramified representation of GSpin∗ (2n, kv ) with Πv its local functorialtransfer to GL(2n, kv ) defined above. ThenωΠv = ωπnv µv
ande v ⊗ ωπ .
Πv ∼
=Π
v
Here, µv is a quadratic character of kv× associated with the quadratic extension defining GSpin∗ (2n), i.e., the non-trivial character of kv× which is trivial on norm elements,
×).
2
NKw /kv (Kw
3.4.3. The non-archimedean ramified transfer. For v a non-archimedean ramified place of k
we take Πv to be an arbitrary, irreducible, admissible representation of GL(2n, kv ) whose
central character satisfiesωΠv = ωπnv µv ,(3.36)
where µv is the v-component of the quadratic idèle class character of k associated with thequadratic extension K/k.
We can no longer expect equality of L- and -factors as in the previous cases. However,
we do still get equality if we twist by a highly ramified character thanks to stability ofγ-factors.
Proposition 3.37. Let v be a non-archimedean place of k such that the irreducible, admissible, generic representation πv of GSpin∗ (2n, kv ) is ramified. Let Πv be an irreducible,
admissible representation of GL(2n, kv ) as above. If τv = τv0 ⊗ ηv is an irreducible, admissible, generic representation of GL(m, kv ) with τv0 unramified and ηv a sufficiently ramified
character of GL(1, kv ), thenL(s, πv × τv ) = L(s, Πv × τv ),e v × τev )L(s, πev × τev ) = L(s, Πand(s, πv × τv , ψv ) = (s, Πv × τv , ψv ).
Proof. The representation τv0 can be written as a full induced principal series. Hence,τv = Ind ν b1 ⊗ · · · ⊗ ν bm ⊗ ηv = Ind ηv ν b1 ⊗ · · · ⊗ ηv ν bm ,(3.38)
where ν(·) = | · |v . By multiplicativity of the L- and -factors we haveL(s, πv × τv ) =

m
Y
i=1

L(s + bi , πv × ηv )(3.39)

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

13

and(s, πv × τv , ψv ) =mY

L(s + bi , πv × ηv , ψv ).

(3.40)

i=1

Similarly,mY

L(s, Πv × τv ) =L(s + bi , Πv × ηv )(3.41)L(s + bi , Πv × ηv , ψv ).

(3.42)i=1and
(s, Πv × τv , ψv ) =

m
Y
i=1

This reduces the proof to the case of m = 1.
Next, note that because ηv is sufficiently ramified (depending on πv ) the L-functions
stabilize to one and we haveL(s, πv × ηv ) ≡ 1(3.43)
and(s, πv × ηv , ψv ) = γ(s, πv × ηv , ψv ).

(3.44)On the other hand, by stability of gamma factors, Proposition 3.15, we may replace πv withanother representation with the same central character. Hence, for n arbitrary charactersχ1 , χ2 , . . . , χn , χ0 = ωπv and the quadratic character µv we have!
n−1Yγ(s, πv × ηv , ψv ) =γ(s, ηv χi , ψv )γ(s, ηv χ−1
i χ0 , ψv )i=1·γ(s, ηv χn µ, ψv )γ(s, ηv χ−1
n χ0 , ψv ).

We refer to [AS1, §6] for more details in the split case. The calculations in the quasi-split
case are similar, the only difference being the appearance of the quadratic character µv .
We have similar relations also for the GL case. More precisely, because ωΠv = χn0 µ, by[JS3, Proposition 2.2] we haveL(s, Πv × ηv ) ≡ 1(3.45)
and(s, Πv × ηv , ψv )

=

n−1
Y

!
γ(s, ηv χi , ψv )γ(s, ηv χ−1
i χ0 , ψv )i=1·γ(s, ηv χn µ, ψv )γ(s, ηv χ−1
n χ0 , ψv ).

(3.46)

Note that this is a special case of the multiplicativity of the local factors. This gives theequalities for the case of m = 1 and hence completes the proof.

3.5. Proof of Theorem 3.5. Let ω denote the central character ωπ of π and let S be as
in the statement of Theorem 3.5. We let Π = ⊗v Πv with Πv the candidates we constructed
in 3.4.1–3.4.3. Also, let µ = ⊗v µv be a quadratic idèle class character associated, by classfield theory, with the quadratic extension K/k, where K is the quadratic extension of k overwhich GSpin∗ (2n) is split.
Choose an idèle class character η of k which is sufficiently ramified at places v ∈ S sothat the requirements of Propositions 3.13 and 3.15 are satisfied. We apply Theorem 3.11
to the representation Π and T (S; η) with S and η as above.

14

MAHDI ASGARI AND FREYDOON SHAHIDI

By construction the central character ωΠ of Π is equal to ω n µ. Therefore, it is invariant
under k × . Moreover, by (3.16) and (3.21) we haveYY
LS (s, Π) =L(s, Πv ) =L(s, πv ) = LS (s, π).
(3.47)
v6∈S

v6∈S

Q

This implies that L(s, Π) = v L(s, Πv ) is absolutely convergent in some right half plane.
Furthermore, by Propositions 3.20, 3.25, and 3.37, we may verify the remaining propertiesof being nice for the L-functions L(s, Π×τ ) and the corresponding -factors, for τ ∈ T (S; η),by establishing the similar properties for the L-functions L(s, π × τ ) and their corresponding
-factors. Now the converse theorem can be applied thanks to Proposition 3.13 to conclude
that there exists an automorphic representation of GL(2n, Ak ) whose local components at
v 6∈ S agree with those of Π. This automorphic representation is what we are calling Π inthe statement of the theorem.
Finally, by Proposition 3.35, outside the finite set S ∪ {v : v|∞}, the central character of
the automorphic representation Π agrees with the idèle class character ω n µ, which impliese v ⊗ ωπ for v 6∈
that it is equal to ω n µ. The same proposition also gives that Πv ∼= Πv
S ∪ {v : v|∞}. This completes the proof.
2
4. The Transferred Representation
In this section k will continue to denote an arbitrary number field and A = Ak will denote
its ring of adèles.
Fix n ≥ 1 and let G = G(n) denote GSpin(2n + 1), the split GSpin(2n), or one of thenon-split quasi-split groups GSpin∗ (2n) associated with a non-trivial quadratic extension Kof the number field k. We will refer to the case of G(n) = GSpin(2n + 1) as the odd caseand the remaining cases as the even case.
4.1. Low Rank Cases. Assume that n = 1. In these low rank cases, the transfers are
well-known. We quickly review them below before assuming n ≥ 2 in what follows.
(i) Odd General Spin Groups. The group GSpin(3) is isomorphic to GL(2) and thetransfer from GSpin(3) to GL(2) that we are dealing with is simply the identity.
Obviously, the L-functions are preserved in this case: L(s, π) = L(s, Π).
(ii) Even General Spin Groups. In the split case, the group GSpin(2) is isomorphic to a
two-dimensional split torus. A cuspidal automorphic representation π of GSpin(2, Ak )is given by two unitary idèle class characters χ and ω. The dual of GSpin(2) embeds
in the diagonal of GL(2, C) with local unramified Satake parameters appearing asχv ($v )
0
,
(4.1)
0
ωv ($v )χ−1v ($v )
where $v denotes a uniformizer at v. The transfer Π of π = πω,χ to GL(2, Ak ) is
the appropriate constituent ofInd χ ⊗ ωχ−1 ,(4.2)
namely, the isobaric sum Π = χ  ωχ−1 , which takes the local Langlands quotientat each place when there is reducibility.
Next, consider the quasi-split non-split case with n = 1. Now, the group G is
GSpin∗ (2), associated with a quadratic extension K of k and it contains a copy ofGL(1) over k as its “center”. (Note that G is simply ResK/k GL(1) and abelian.)
The L-group L G is isomorphic to GSO(2, C) o Γk with the embedding of it intoGL(2, C) × Γk as in (3.3) again.

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

15

A cuspidal automorphic representation π of G(Ak ) is given as an idèle classcharacter χe of the quadratic extension K of k. Its restriction to the “center” is now
an idèle class character of k which we denote by ω. The transfer of π to GL(2, Ak )is now given by

O  W
ev ,
(4.3)
Π=
π IndWkKvv χv
Wev = χ
ev ⊕ χ
ev when v splits in K (i.e., Kv = kv ⊕ kv ), and it is thewhere IndWkKvv χ

Weil representation we described in Section 3.4.2 when v is inert in K. (See [GL,Appendix C].) By our earlier local description, it is again clear that Π ∼= Π̃ ⊗ ω andωΠ = ω · µ, where µ = µK/k is the quadratic idèle class character of k associatedwith the extension K/k.
We may assume below that n ≥ 2.
4.2. The Global Transfer. Let π be a irreducible, globally generic, unitary, cuspidal,
automorphic representation of G(Ak ). Let Π be a transfer of π to GL(2n, Ak ) as in [AS1,
Theorem 1.1] and Theorem 3.5. By the classification of automorphic representations ofgeneral linear groups [JS1, JS2] we know that Π is a constituent of some automorphic
representation
Σ = Ind (| det |r1 σ1 ⊗ · · · ⊗ | det |rt σt )(4.4)
with σi a unitary, cuspidal, automorphic representation of GL(ni , Ak ), ri ∈ R, andn1 + n2 + · · · + nt = 2n.

(4.5)

Let ω = ωπ denote the central character of π. Then ω is a unitary idèle class charactere ⊗ ω.
of k and we have shown that Π is nearly equivalent to Π
Our first goal in this section is to prove the fact that all the exponents ri = 0 in (4.4).
We start by introducing a necessary Eisenstein series and proving a lemma (Lemma 4.11
below) about twisted exterior and symmetric (partial) L-functions.
4.2.1. Eisenstein Series. Let m and n denote positive integers and assume n ≥ 2. Let π
be a globally generic, unitary, cuspidal, automorphic representation of G(Ak ) as before andlet τ be a unitary, cuspidal, automorphic representation of GL(m, Ak ). Denote their central
characters by ω = ωπ and ωτ , respectively. We construct a certain Eisenstein series for arepresentation induced from τ as follows.
Let H = H(m) be the split group GSpin(2m) if G = GSpin(2n + 1) and GSpin(2m + 1)
if G = GSpin(2n). In other words, H is of the opposite type to G. Consider the Siegel
parabolic Pm = Mm Nm in H(m) with Mm ∼= GL(m) × GL(1) and its representation0τs0 ;η = τ | det |s ⊗ η,

(4.6)

where η is an appropriately chosen idèle class character. (The choice of η and the isomorphism between Mm and GL(m) × GL(1) are made in [ACS1] so that the Zeta integral
we consider there is well-defined and the desired L-functions appear in the constant termof the Eisenstein series; see below.) Here, s0 = s − 1/2, with s ∈ C. (The shift in s is forconvenience here so that the parameter s appears in the Rankin-Selberg L-functions below.)
Extend τs0 ,η trivially across Nm (A) to obtain a representation of Pm (A). Consider thenormalized induced representation
H(A)ρs0 ;η = IndPm (A) (τs0 ;η ⊗ 1) .

(4.7)

16

MAHDI ASGARI AND FREYDOON SHAHIDI

For f = fτ,s0 ;η in this induced representation, construct the Eisenstein seriesX
E(h, fτ,s0 ;η ) =f (γh), h ∈ H(A).

(4.8)

γ∈Pm (k)\H(k)This Eisenstein series is absolutely convergent for <(s)  0, has a meromorphic continuation
to all of C with a finite number of poles, the continuation is of moderate growth, and it
converges uniformly on compact subsets away from the poles. Moreover, it has a functional
equation relating its values at s0 and 1 − s0 .
The Eisenstein series, along with a cusp form in the space of π, appears in a certain
Zeta integral we consider in [ACS1]. In particular, we establish the following statement in[ACS1], which we now borrow from that paper.
Proposition 4.9. With τ and π as before, if LS (s, π × τ ) has a pole at s0 with <(s0 ) ≥ 1,
then, for a choice of fτ,s0 ;η , the Eisenstein series E(h, fτ,s0 ;η ) has a pole at s = s0
The proposition is established by proving that the the Zeta integral in which it appears is
Eulerian and, up to a factor that can be made holomorphic and non-zero for an appropriate
choice of fτ,s0 ;η , is equal to the quotient of the Rankin-Selberg L-function LS (s, π × τ ) andeither LS (2s, τ, ∧2 ⊗ ω −1 ) (when G is odd) or LS (2s, τ, Sym2 ⊗ ω −1 ) (when G is even). We
carry out the details in [ACS1].
4.2.2. Twisted Symmetric and Exterior Square L-functions. We need a result on holomorphyof twisted L-function in the half plane <(s) > 1. This result, and much more, is the subjectof two works, one by Dustin Belt in his thesis at Purdue University, and the other by Suichiro
Takeda.
Proposition 4.10. ([Blt] and [Tk]) Assume that m is an arbitrary positive integer. Let χ
be an arbitrary idèle class character and let τ be a unitary, cuspidal, automorphic representation of GL(m, A). Let S be a finite set of places of k containing all the archimedean places
and all the non-archimedean places at which τ ramifies. Then the partial twisted L-functions
LS (s, τ, ∧2 ⊗ χ) and LS (s, τ, Sym2 ⊗ χ) are holomorphic in <(s) > 1.
We remark that Jacquet and Shalika proved that LS (s, τ, ∧2 ⊗ χ) has a meromorphic
continuation to a half plane <(s) > 1 − a with a > 0 depending on the representation [JS4,§8, Theorem 1]. Proposition 4.10 in the case of ∧2 ⊗ ω can also be dug out of their work.
However, D. Belt’s results show this for all s, with possible poles at s = 0, 1.
As far as we know, an analogue of Jacquet-Shalika’s result for twisted symmetric square
was not available. For m = 3 it follows from the results of W. Banks [Bnk] following theuntwisted (χ = 1) results of Bump and Ginzburg [BG]. S. Takeda’s results build on thisline of work. In the case of χ = 1, N. Grbac has recently established a full theory of theseL-functions in [Gr].
Lemma 4.11. Assume that m is an arbitrary positive integer. Let τ be an irreducible,
unitary, cuspidal, automorphic representation of GL(m, A). Let ω be an idèle class characterand let s ∈ C. Let S be a finite set of places of k including all the archimedean ones suchall data is unramified outside S.
(a) Both LS (s, τ, ∧2 ⊗ω −1 ) and LS (s, τ, Sym2 ⊗ω −1 ) are holomorphic and non-vanishingfor <(s) > 1.
(b) If either of the above L-functions has a pole at s = 1, then τ ∼= τe ⊗ ω.
Proof. We have
LS (s, τ ⊗ (τ ⊗ ω −1 )) = LS (s, τ, ∧2 ⊗ ω −1 )LS (s, τ, Sym2 ⊗ ω −1 ).

(4.12)

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

17

The left hand side is holomorphic and non-vanishing for <(s) > 1 by [JS2, Proposition(3.6)]. Moreover, by Proposition 4.10 both of the L-functions on the right hand side areholomorphic for <(s) > 1. Therefore, both are non-vanishing there, as well. This is part(a).
On the other hand, by [Sh5, Theorem 1.1] both L-functions on the right hand side are
non-vanishing on <(s) = 1. If one has a pole at s = 1, then the left hand side must havea pole at s = 1. Again by [JS2, Proposition (3.6)] the two representations τ and τ ⊗ ω −1
must be contragredient of each other, i.e., τ ∼= τe ⊗ ω. This is part (b).
Proposition 4.13. Assume that m is an arbitrary positive integer. Let τ be an irreducible,
unitary, cuspidal representation of GL(m, A). Fix s0 ∈ C with <(s0 ) ≥ 1 and assume that
the Eisenstein series E(h, fτ,s0 ;η ) introduced in (4.8) has a pole at s = s0 . Then, s0 = 1 andLS (s, τ, ∧2 ⊗ ω −1 ) has a simple pole at s = 1 in the odd case while LS (s, τ, Sym2 ⊗ ω −1 ) has
a simple pole at s = 1 in the even case.
Proof. We know from the general theory of Euler products of Langlands and the LanglandsShahidi method that the poles of E(h, fτ,s0 ;η ) come from its constant term along Pm .
For a decomposable section f = fτ,s0 ;η = ⊗v f (v) the constant term of E(h, fτ,s0 ;η ) along
Pm has the form
 LT (2s − 1, τ, ∧2 ⊗ ω −1 )Y
(4.14)
f (I) +M f (v)
LT (2s, τ, ∧2 ⊗ ω −1 )v∈Tin the odd case, andf (I) +

 LT (2s − 1, τ, Sym2 ⊗ ω −1 )M f (v)
LT (2s, τ, Sym2 ⊗ ω −1 )v∈TY

(4.15)

in the even case, where T is a finite set of places of k containing S.
We should recall that in the construction the Eisenstein series we used s − 21 in (4.6)
instead of the usual s. This is responsible for the appearance of 2s − 1 and 2s instead of theusual 2s and 2s + 1 in the constant term.
The terms M f (v) , the local intertwining operators at I, are holomorphic for <(s) ≥ 1
for all v [Sh2, Sh3]. Therefore, if E(h, fτ,s0 ;η ) has a pole at s = s0 , thenLT (2s − 1, τ, ∧2 ⊗ ω −1 )LT (2s, τ, ∧2 ⊗ ω −1 )(4.16)

has a pole at s = s0 in the odd case, orLT (2s − 1, τ, Sym2 ⊗ ω −1 )LT (2s, τ, Sym2 ⊗ ω −1 )(4.17)

has a pole at s = s0 in the even case.
Assume that E(h, fτ,s0 ;η ) does have a pole at s = s0 with <(s0 ) ≥ 1. Then <(2s0 ) ≥ 2
and by [KS2, Prop. 7.3] the denominator in both (4.16) and (4.17) is non-vanishing for<(s) ≥ 1. Therefore, the numerator has a pole at s = s0 . Because <(2s0 − 1) ≥ 1, Lemma
4.11 implies that s0 = 1 and the proof is complete.

We use Proposition 4.9 in the proof of the following theorem.
Theorem 4.18. Assume that n ≥ 2 is an integer. Let π be an irreducible, unitary, globally
generic, cuspidal, automorphic representation of G(n, A). Let τ be an irreducible, unitary,cuspidal, automorphic representation of GL(m, A). Assume that S is a sufficiently large
finite set of places including all the archimedean places of k.

18

MAHDI ASGARI AND FREYDOON SHAHIDI(a) The L-function LS (s, π × τ ) is holomorphic for <(s) > 1.
(b) If LS (s, π × τ ) has a pole at s0 with <(s0 ) = 1, then s0 = 1 and LS (s, τ, ∧2 ⊗ ωπ−1 )has a pole at s = 1 in the odd case and LS (s, τ, Sym2 ⊗ ωπ−1 ) has a pole at s = 1 inthe even case. In particular, τ ∼= τe ⊗ ωπ . Such a pole would be simple.
Remark 4.19. The partial L-functions in the above statements may be replaced by thecompleted L-functions as Hundley and Sayag have pointed out in [HS2, §19.3]. The pointis that, using the description of generic unitary representations of the general linear groups
and analytic properties of local intertwining operators and L-functions, one can show thatthe local twisted symmetric or exterior square L-functions above are holomorphic and nonvanishing for <(s) ≥ 1. (They do it for the twisted symmetric square L-function and at
s = 1, but the same argument works for the twisted exterior square L-function and for<(s) ≥ 1.) We thank the referee for this remark.
Proof. Assume that LS (s, π × τ ) has a pole at s = s0 with <(s0 ) ≥ 1. By [KS2, Prop.
7.3] we know that both LS (s, τ, ∧2 ⊗ ωπ−1 ) and LS (s, τ, Sym2 ⊗ ωπ−1 ) are holomorphic andnon-vanishing at s = 2s0 . By Proposition 4.9, the Eisenstein series E(h, fτ,s0 ;η ) must have apole at s = s0 . Proposition 4.13 now implies that s0 = 1 and LS (s, τ, ∧2 ⊗ ω −1 ) in the odd
case, or LS (s, τ, Sym2 ⊗ ω −1 ) in the even case, has a simple pole at s = 1. This proves (a)and (b).

4.3. Description of the Image of Transfer. We can now describe the image in general.
Theorem 4.20. Let π be an irreducible, globally generic, unitary, cuspidal, automorphic
representation of G(n, A) with central character ω = ωπ and let Π be a transfer of π toGL(2n, A). Consider Π as a subquotient of Σ as in (4.4) with n1 + n2 + · · · + nt = 2n.
(a) We have r1 = r2 = · · · = rt = 0.
(b) The representations σi are pairwise inequivalent, and σi ∼ei ⊗ω for all i. Moreover,=σ
for S a sufficiently large finite set of places including all the archimedean ones, theL-function LS (s, σi , ∧2 ⊗ω −1 ) has a pole at s = 1 in the odd case, and the L-function
LS (s, σi , Sym2 ⊗ ω −1 ) has a pole at s = 1 in the even case.
Proof. By [AS1, Prop. 7.4] we know that Σ is induced from a representation of a Levi
subgroup of GL(2n) of type (a1 , . . . , aq , b1 · · · , b` , aq , · · · , a1 ) which can be written as
δ1 | det(·)|z1 ⊗ · · · ⊗ δq | det(·)|zq ⊗σ1 ⊗ σ2 ⊗ · · · ⊗ σ`−1e⊗(δq ⊗ ω )| det(·)|−zq ⊗ · · · (δe1 ⊗ ω −1 )| det(·)|−z1 ,(4.21)∼
where δi and σi are irreducible, unitary, cuspidal automorphic representations, σi = σei ⊗ ω,and2(a1 + · · · + aq ) + (b1 + · · · + b` ) = 2n.
(4.22)
Assume that q > 0. Rearranging if necessary, we may assume <(z1 ) ≤ · · · ≤ <(zq ) < 0.
Now, for S a sufficiently large finite set of places, we have
LS (s, π × δe1 )==

LS (s, Π × δe1 )q
YLS (s + zi , δi × δe1 )LS (s − zi , δei × δe1 ⊗ ω −1 )
i=1

·

`
Y
i=1

LS (s, σi × δe1 ).

(4.23)

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

19

The first term on the right hand side has a pole at s = 1 − z1 which can not be canceled by
the other terms because <(1 − z1 ± zi ) ≥ 1 and <(1 − z1 ) > 1. Therefore, the left hand side
has a pole at s = 1 − z1 .
We can apply Theorem 4.18(a) to conclude that <(z1 ) ≥ 0. This is a contradiction provingthat q = 0, i.e., there are no δi ’s. This proves part (a).
So far we have proved that Σ is induced from a representation of the form σ1 ⊗σ2 ⊗· · ·⊗σ`satisfying σi ∼ei ⊗ ω.
=σFix 1 ≤ j ≤ ` and consider
SS

L (s, π × σej ) = L (s, Π × σej ) =`Y

LS (s, σi × σ
ej ).

(4.24)

i=1

The right hand side has a pole and hence, so does the left hand side. Moreover, the σi ’s
are pairwise inequivalent because otherwise the left hand side of (4.24) would have a poleof higher order. Since σi ∼ei ⊗ ω we can apply Theorem 4.18(b).
=σWe conclude that the L-function LS (s, σi , ∧2 ⊗ ω −1 ) has a pole at s = 1 in the odd case,and the L-function LS (s, σi , Sym2 ⊗ ω −1 ) has a pole at s = 1 in the even case. This provespart (b).
Corollary 4.25. The representation Σ is irreducible and Π = Σ = σ1  · · ·  σt is ane ⊗ ω (not justisobaric sum of the σi . In particular, the transfer Π of π is unique and Π ∼=Πnearly equivalent as in [AS1, Theorem 1.1]).
Proof. The corollary immediately follows from the fact that r1 = · · · = rt = 0 and thatσi ∼ei ⊗ ω.
=σ
We continue to denote by π an irreducible, globally generic, unitary, cuspidal, automorphic representation of G(n, A). We proved that π has a unique transfer Π, an irreducible,
generic, automorphic representation of GL(2n, A). Moreover, we have shown that ωΠ = ω n µe ⊗ ω, where ω = ωπ denotes the central character of π, ωΠ denotes the centraland Π ∼
=Πcharacter of Π, and µ is a quadratic idèle class character.
Furthermore, Theorem 4.20 gives an “upper bound” for the image of the transfer from
G(n) groups to GL(2n). Combining this with the “lower bound” provided by Hundley andSayag (when each ni is even) in [HS1, HS2] and Ginzburg, Rallis, and Soudry [GRS] (whensome ni is odd), cf. [HS1, Remark 2.1.1 (3)], we obtain the full description of the image ofthis transfer. We summarize all these results as follows.
Theorem 4.26. Let k be a number field and let A = Ak be the ring of adèles of k. Assume
that n ≥ 1 is an integer. Denote by G(n) either the split GSpin(2n+1), the split GSpin(2n),or one of the non-split quasi-split groups GSpin∗ (2n) (associated with a quadratic extensionK/k.) Let π be an irreducible, globally generic, cuspidal, automorphic representation ofG(n, A) with central character ω = ωπ . Then π has a unique functorial transfer to an
automorphic representation Π of GL(2n, A) associated with the L-homomorphism ι describedin [AS1] (the split case) and Section 3 (the quasi-split non-split case). The transfer Πsatisfiese ⊗ ω.
Π∼=Π
Moreover,ωΠ = ωπn µ,where µ is a quadratic idèle class character which is trivial in the split case and non-trivial
in the quasi-split non-split case. In fact, µ defines the quadratic extension of k over which

20

MAHDI ASGARI AND FREYDOON SHAHIDI(the non-split) GSpin∗ (2n) splits. (The triviality or non-triviality of ωΠ ω −n can tell apartthe split and quasi-split non-split cases.)
The automorphic representation Π is an isobaric sum of the form
Π = Ind (Π1 ⊗ · · · ⊗ Πt ) = Π1  · · ·  Πt ,where each Πi is a unitary, cuspidal, automorphic representation of GL(ni , A) such that forT a sufficiently large finite set of places of k containing the archimedean places, the partial
L-function LT (s, Πi , ∧2 ⊗ω −1 ) has a pole at s = 1 in the odd case and LT (s, Πi , Sym2 ⊗ω −1 )has a pole at s = 1 in the even case (both split and quasi-split non-split cases). We haveΠi ∼6 Πj if i 6= j and n1 + · · · + nt = 2n.
=Conversely, any automorphic representation Π of GL(2n, A) satisfying the above conditions is a functorial transfer of some irreducible, globally generic, cuspidal, automorphic
representation π of G(n, A).
Again we should point out that the partial L-functions in the statement above may bereplaced by the completed L-functions (cf. Remark 4.19).
5. Applications
5.1. Local Representations at the Ramified Places. The local components of theautomorphic representation Π = ⊗v Πv are well understood for the archimedean v as well as
those non-archimedean v outside of the finite set S through our construction of the candidate
transfer. However, the converse theorem tells us nothing about Πv for v ∈ S. Having proved
Theorem 4.26 we can now get some information for these places as well. This shows thatwhile we did not have control over places v ∈ S, the automorphic representation Π does
indeed turn out to have the right local components in S.
Proposition 5.1. Let πv be an irreducible, admissible, generic representation of G(n, kv ),where v is a non-archimedean place of k. Assume that πv appears as the local component of a
globally generic cuspidal representation of G(n, Ak ). Then, there exists a unique irreducible,
admissible, generic representation Πv of GL(2n, kv ) which is the local transfer of πv in thefollowing sense:
If πv is the local component at v of any globally generic cuspidal automorphic representation π, then, as the name suggests, Πv is the local component at v of the unique transfer
Π of π to GL(2n, Ak ) whose existence and uniqueness we established in Theorem 4.26.
To be more explicit, let πv be of the formπv ∼= Ind π1,v | det |b1,v ⊗ · · · ⊗ πr,v | det |br,v ⊗ π0,v ,where each πi,v is a tempered representation of GL(ni , kv ), b1,v > · · · > br,v and π0,v is atempered, generic representation of some smaller G(m, kv ) with n1 + · · · + nr + m = n.
Denote the central character of πv by ωv . Then Πv is of the formΠv∼=

Ind (π1,v | det |b1,v ⊗ · · · ⊗ πr,v | det |br,v ⊗ Π0,v ⊗(eπr,v ⊗ ωv )| det |−br,v ⊗ · · · ⊗ (eπ1,v ⊗ ωv )| det |−b1,v .

(5.2)

Here, Π0,v appears only if m > 0. It is a tempered representation of GL(2m, kv ) which is
the local transfer of Π0,v .
In particular, if S is the non-empty finite set of non-archimedean places in [AS1, Thm.
1.1.] or Theorem 3.5, then the local components Πv , v ∈ S, are uniquely determined by thecorresponding local components πv .

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

21

Proof. The argument proceeds the same way as in the proof of [AS2, Prop. 2.5.], which
proved the analogous result for the case of GSp(4) = GSpin(5). We briefly mention thesteps for completeness.
Given πv , let π be as in the statement of the proposition and let Π be its unique transfer
to GL(2n, Ak ). The fact that Π is an isobaric sum of unitary, cuspidal representations ofgeneral linear groups, Theorem 4.26, implies that every local component of Π is generic andfull induced from tempered representations in the sense of the Langlands Classification. In
particular, so is Πv .
The first step is to show that
γ(s, πv × ρv , ψv ) = γ(s, Πv × ρv , ψv )for every supercuspidal representation ρv of GL(a, kv ). To do this we “embed” the local
representation ρv in a unitary, cuspidal representation ρ of GL(a, A) whose other local
components are unramified [Sh3, Prop. 5.1] and apply the converse theorem with S 0 =S − {v} just as in [CKPSS2, Prop. 7.2]. Moreover, by multiplicativity of the γ-factors we
obtain the equality for ρv in the discrete series, as well.
Next, assume that πv is tempered. We claim that Πv is also tempered. Here again
the main tool is multiplicativity of the γ-factors and the proof is exactly as in [CKPSS2,
Lemma 7.1]. In addition to several applications of multiplicativity, it involves information
about generic unitary representations of the general linear groups, with the only different
ingredient in the general spin case being the so-called tempered L-function conjecture, thatthe local L-functions for tempered representations are holomorphic for <(s) > 0. For thegeneral spin groups, this is available in [A], and even more generally in [HO]. This proves
the Proposition for r = 0.
Now consider the case of r > 0. Apply the case of r = 0 to πv = π0,v and take the resulting
tempered representation of GL(2m, kv ) to be Π0,v . To show that this representation satisfies
the requirements of the proposition we use the converse theorem again. Let T = {w} consistof a single non-archimedean place w 6= v at which π is unramified and consider the global
representation Π0 of GL(2n, A) whose local components are the same as those of Π outside of
T and Π0w is the irreducible, induced representations on the right hand side of (5.2). We canapply the converse theorem, Theorem 3.11, to Π0 and T because L(s, Π0 × τ ) = L(s, π × τ )and similarly the contragredients as well as the -factors for all τ. The conclusion is that Π0
is a transfer of π (outside of T ) and by the uniqueness of the transfer, Theorem 4.26, wehave Π0v ∼= Πv for v ∈ S. This completes the proof.
5.2. Ramanujan Estimates. Following [CKPSS2], we introduce the following notation.
Let Π = ⊗v Πv be a unitary, cuspidal, automorphic representation of GL(m, Ak ). For
each place v the representation Πv is unitary generic and can be written as a full induced
representation
Πv ∼
(5.3)
= Ind (Π1,v | det |a1,v ⊗ · · · ⊗ Πr,v | det |ar,v )with a1,v > · · · > ar,v and each Πi,v tempered.
Definition 5.4. We say Π satisfies H(θm ) with θm ≥ 0 if for all places v we have−θm ≤ ai,v ≤ θm .
The classification of the generic unitary dual of GL(m), [Td, V], trivially gives H(1/2).
The best result currently known for a general number field is θm = 1/2 − 1/(m2 + 1) provedin [LRS] with a few better results known for small values of m and over Q. The Ramanujan
conjecture for GL(m) demands H(0).

22

MAHDI ASGARI AND FREYDOON SHAHIDI

Similarly, if π = ⊗v πv is a unitary, generic, cuspidal, automorphic representation ofG(n, Ak ) each πv can be written as a full induced representation
∼ Ind π1,v | det |b1,v ⊗ · · · ⊗ πr,v | det |br,v ⊗ τv ,
πv =
(5.5)
where each πi,v is a tempered representation of some GL(ni , kv ) and τv is a tempered,generic representation of some G(m, kv ) with n1 + · · · + nt + m = n.
Definition 5.6. We say π satisfies H(θn ) with θn ≥ 0 if for all places v we have−θm ≤ bi,v ≤ θm .
Again, we would have the bound H(1) trivially as a consequence of the classification of
the generic unitary dual and the Ramanujan conjecture demands H(0).
Proposition 5.7. Let k be a number field and assume that all the unitary, cuspidal, automorphic representations of GL(m, Ak ) satisfy H(θm ) for 2 ≤ m ≤ 2n. Then any globally
generic, unitary, cuspidal, automorphic representation π of G(n, Ak ) satisfies H(θ), whereθ = max θi . In fact, if π transfers to a non-cuspidal representation Π = Π1  · · ·  Πt ,2≤i≤2nthen π satisfies the possibly better bound of H(θ) where θ = max{θn1 , θn2 , . . . , θnt }. Here,Πi is a unitary, cuspidal representation of GL(ni , Ak ).
Proof. The argument is exactly the same as the proof of [AS2, Theorem 3.3] and we do
not repeat it here. Note that our Proposition 5.1 is used for the ramified non-archimedean
places.

Corollary 5.8. Every globally generic, unitary, cuspidal, automorphic representation π ofG(n, Ak ) satisfy
4n2 − 1.
H
2(4n2 + 1)
If π transfers to a non-cuspidal, automorphic representationΠ = Π1  · · ·  Πt ,then we can replace n with the size of the largest GL block appearing, resulting in a better
estimate.
Proof. This is immediate if we combine Proposition 5.7 with the GL(m) estimate of 1/2 −1/(m2 + 1).
We should remark that for small values of n it is possible to obtain better estimates
because much better estimates are available for small general linear groups (and also for
k = Q). For an example, see [AS2, §3.1]
Corollary 5.9. The Ramanujan conjecture for the unitary, cuspidal representations ofGL(m, Ak ) for m ≤ 2n implies the Ramanujan conjecture for the generic spectrum ofG(n, Ak ).
Proof. This is an immediate corollary of Proposition 5.7 where all the θ’s are zero.



5.3. Image of Kim’s exterior square. H. Kim proved the exterior square transfer of
automorphic representations from GL(4, Ak ) to GL(6, Ak ) [K2, H2]. A. Raghuram and thefirst author gave a complete cuspidality criterion for this transfer, determining when theimage of this transfer is not cuspidal [AR]. A natural question about the image of thistransfer is which automorphic representations of GL(6, Ak ) are indeed in the image of thistransfer. We can now answer this question as an application of our Theorem 4.26.

IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS

23

Proposition 5.10. Let Π be a cuspidal, automorphic representation of GL(6, Ak ). Then
there is a cuspidal automorphic representation π of GL(4, Ak ) such that Π = ∧2 π if and only
if there is an idèle class character ω such that the partial L-function LS (s, Π, Sym2 ⊗ ω −1 )has a pole at s = 1 for S a sufficiently large finite set of places of k including all thearchimedean ones.
Proof. The proposition follows immediately from our Theorem 4.26 if we recall that Kim’s
exterior square transfer from GL(4) to GL(6) is a special case of the transfer in the split
even case of our theorem when m = 3, i.e., the transfer from GSpin(6) to GL(6) [AS1, Prop.
7.6].
If we assume that Π is the transfer of π, then we have proved that we can take ω =ωπ , the central character of π. The opposite direction requires the descent method in
our cases and would follow from J. Hundley and E. Sayag’s “lower bound” result for ourtransfer [HS1, HS2] because Kim’s ∧2 is a special case of transfer from GSpin(6) to GL(6)as mentioned above.

Another natural question regarding the image of Kim’s exterior square transfer is todetermine “the fiber” for each cuspidal Π which is indeed in the image. In other words,
determine all representations π such that Π = ∧2 π.
A further interesting question would be to explore possible overlaps between varioustransfers to cuspidal representations of GL(6). As pointed out in [CPSS2, §6] (for the
untwisted ω = 1 case) and as it is apparent from our Theorem 4.26 there can be no overlap
between the images of transfers from GSpin(7) and quasi-split forms of GSpin(6) (which
includes Kim’s transfer) to GL(6). However, there may be potential overlaps with thetransfer from unitary groups or the Kim-Shahidi transfer [KS1] from GL(2) × GL(3) toGL(6).
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[KS1]

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078–1058, USA
E-mail address: asgari@math.okstate.edu
Mathematics Department, Purdue University, West Lafayette, IN 47907, USA
E-mail address: shahidi@math.purdue.edu