Holomorphy of adjoint L functions for quasisplit A 2

We study the poles of the twisted adjoint L function of a generic cuspidal automorphic representation of GL(3) or a quasisplit unitary group using a method pioneered by Ginzburg and Jiang and based on the theory of integral representations


Introduction
Let F be a number field. 1 Let G be a quasisplit F -group, isomorphic to SL 3 over F. Thus, F is either SL 3 , or a quasisplit unitary group attached to some quadratic extension E/F. The finite Galois form of G's L-group is then PGL 3 (C) in the split case, or the semidirect product of PGL 3 (C) with Gal(E/F ) in the nonsplit case. In either case we have an action of L G on PGL 3 (C) by conjugation, which may be regarded as an action on GL 3 (C) which fixes the center. This then induces an action on the Lie algebra sl 3 (C) which we denote Ad . Note that in the nonsplit case this does not coincide with the definition of Ad in [12]. Rather, the nontrivial element Fr of Gal(E/F ) will act by X → − t X, as this is the differential of its action on PGL 3 (C). Let Ad denote the representation of L G considered in [12]. Thus Ad = Ad in the split case, but in the nonsplit case it is the representation of L G where GL 3 (C) acts by conjugation and Fr acts by X → t X.
Let π be a globally generic irreducible cuspidal automorphic representation of G. Then we consider the adjoint L function L(s, π, Ad). One would like understand the poles of this L function. We discuss an attack based on the integral representation given in [9,12] and a strengthening the results of [10]. Our proof also applies to certain twisted L functions.
We briefly review the local zeta integral for L(s, π, Ad) presented in [9,12]. First, one fixes an embedding of G into the split exceptional group of type G 2 . Let P be the maximal standard parabolic subgroup of G 2 , whose Levi contains the root subgroup attached to the short root. Let f be a flat section of the family of induced representations attached to a family of characters of P(F )\P(A), and E P be the corresponding Eisenstein series operator (these notions are reviewed in Sects. 2.2 and 2.6).
One may then define I(s, ϕ, f ) := G(F )\G(A) ϕ(g)E P .f (g, s) dg, (1.1) where s ∈ C, and ϕ is a generic cusp form G(F )\G(A) → C. Both [9], and [12] present the argument under the assumption that the characters are unramified, but the extension to the general case is direct so we may regard the theorem as proved in the ramified case as well. The integral I(s, ϕ, f ) unfolds to a new integral I(s, W ϕ , f ) where W ϕ is the Whittaker function attached to ϕ. Assuming that W ϕ and f are factorizable, I(s, W ϕ , f ) then factors as a product of local zeta integrals I(s, W v , f v ).
Here W v and f v are the local components at a place v of W ϕ and f respectively. Moreover, if W v , f v and χ v are all unramified, 2 and W v and f v are normalized, then (see note 3 ) . Hence, where S is a finite set of places, away from which W v , f v and χ v are unramified. Notice that L(s, π, Ad) = L(s, π, Ad ) if G is split and L(s, π, Ad ⊗χ E/F ), where χ E/F is the quadratic character attached to the extension E/F by class field theory if G is not split. One expects that in general the L function L(s, π, Ad ⊗χ) should be entire. Indeed, in the split case L(s, π, Ad ×χ) = L(s, π ⊗ χ × π)/L(s, χ), and it follows that the possible poles are precisely the zeros of L(s, χ), unless χ is nontrivial and π ⊗ χ ∼ = π , in which case there are additional simple poles at s = 0 and s = 1. One expects that L(s, π ⊗ χ × π) is divisible by L(s, χ) and hence that the only actual poles are the simple poles at s = 0 and s = 1 which occur when χ is nontrivial and π ⊗ χ ∼ = π. In the special case when π = ⊗ v π v and at least one component π v is supercuspidal, this was proved by Flicker [5].
In the nonsplit case, one must replace L(s, π × π) with the Asai L function of the stable base change lifting of π. One must also account for the image of certain theta liftings. Indeed, consider This subgroup is stable under the outer automorphism which realizes the action of the nontrivial Galois element of the L group. Thus we obtain a subgroup of the L group. This subgroup may be realized as the image of an L-homomorphism from the L group of a product of smaller unitary groups U 1,1 ×U 1 (still attached to the same quadratic extension). See [20]. The above subgroup clearly stabilizes the one dimensional subspace of sl 3 (C) spanned by diag(1, −2, 1), and so does the map X → t X := J t XJ, where J = we obtain a one dimensional space stable under the restriction of the representation Ad to L (U 1,1 × U 1 ), so that L(s, π, Ad ) should have a pole whenever π is a lift from U 1,1 × U 1 .
(In the split case, this issue does not arise: the group (1.3) is the L group of a Levi subgroup, the corresponding "functorial lifting" is realized by formation of Eisenstein series, and an element of its image can never be cuspidal). An approach to controlling poles of twisted adjoint L functions, which is based on the study of I(s, ϕ, f ) and does not depend on any property of a local component of π was pioneered in [10]. If I(s, ϕ, f ) has a pole at s = s 0 then each of its negative Laurent coefficients is a global integral, similar to I(s, ϕ, f ), but with the Eisenstein series replaced by its corresponding negative Laurent coefficient. Thus, it suffices to show that the negative Laurent coefficients of the Eisenstein series used in I(s, ϕ, f ), when restricted to the subgroup G → G 2 are "orthogonal" 4 to cusp forms. Following [10], this can be done by expressing such Laurent coefficients in terms of Eisenstein series induced from characters of the other maximal parabolic subgroup of G 2 , and then checking that the restrictions of these Eisenstein series are "orthogonal" to G-cuspforms. In [10], it is shown that the Eisenstein series appearing in the construction of I(s, ϕ, f ) for the case of trivial χ has only two poles in Re(s) > 1 2 , with one being simple and the other double. The residue of the simple pole is a constant function and thus obviously orthogonal to cusp forms. At the double pole, a "first term identity" is proved, which expresses the leading term of the Laurent expansion in terms of an Eisenstein series from the other parabolic. This rules out a double pole of the adjoint L function. In order to rule out a simple pole by this method, one would need a "second term identity." In Sect. 3.3 we prove an identity of this type. It is my understanding that such an identity was first obtained by Jiang in unpublished work.
That being said, if the second pole of the Eisenstein series gave rise to a pole of the global adjoint L function L(s, π, Ad), this pole would occur at s = 1. Such a pole is impossible, because L(s, π, Ad) = L(s, π × π)/ζ (s), and L(s, π × π ) and ζ (s) both have simple poles at s = 1.
In this paper we pursue the approach pioneered by Ginzburg and Jiang. First, we analyze the poles of the Eisenstein series in the case of nontrivial χ. This allows us to deduce information about the poles of the local zeta integral. We also prove a key vanishing result needed to deduce holomorphy of L(s, π, Ad ×χ) at Re(s) = 1 2 from (1.2). Then, we prove a weak result regarding local zeta integrals at ramified and Archimedean primes. While preparing this manuscript, I have learned that a stronger result-a local functional equation-has been obtained by Qing Zhang. The weaker result proved here suffices for our application, permitting us to deduce that each pole of the partial adjoint L function must be a pole of the global zeta integral for some choice of data.
Our main result is Theorem 6.1, which states that, in the split case, every pole of L(s, π, Ad ×χ) in the half plane Re(s) ≥ 1 2 is simultaneously a zero of L(s, χ) and a pole of the finite product over the ramified and Archimedean places. Using this result, together with knowledge of the form of the Gamma factor and the zeros of the Riemann zeta function, Buttcane and Zhou [3] were able to show holomorphy of the complete adjoint L function (and hence also all partial L functions) for an SL(3, Z) Maass form with trivial central character (such a form generates a representation unramified at all finite places).
Since then, Qing Zhang has been able to strengthen the main result of this paper by treating ramified nonarchimedean places. Thus, one can take the finite product only over Archimedean places. This immediately gives an extension of the result of Buttcane and Zhou to Maass forms with trivial central character attached to congruence subgroups.

Characters and degenerate induced representations
Let F be a number field as before. If G is an F -group, write X(G) for the group of rational characters of G and X G for the complex manifold of characters of G(A) trivial on G(F ). These are groups and for the most part we write them additively. To reconcile with multiplicative notation for G(A) and C × , we use an exponential notation for the characters: the value of χ ∈ X G at g ∈ G(A) is denoted g χ . A similar notation is used for cocharacters. We identify χ ∈ X(G) with the character of G(F )\G(A) obtained by composing it with the absolute value on A × . This extends to a mapping of X(G) ⊗ Z C into X G . The image is the set of unramified characters, which we denote X G,un . Similarly we denote the complex manifold of all characters of G(F v ) by X G,v and the image of X(G) ⊗ Z C in it by X G,v,un . We identify X GL 1 ,un with C using the map s → | | s . We denote the canonical pairing between characters and cocharacters by , . If ϕ ∨ is a cocharacter, then ϕ ∨ , χ ∈ X GL 1 .
We shall only require split connected reductive F -groups with simply connected derived groups. We always assume that each is equipped with a choice of split torus and Borel containing it. The torus is denoted T and the Borel B. Let G be such a group. For H a T -stable F -subgroup we write (T, H) for the roots of T in H. The Weyl group is denoted W. It is realized as a quotient of the normalizer, N G (T ), of T in G. We also assume G equipped with a realization, i.e. a family of isomorphisms {x α : G a → U α } α∈ (G,T ) , such that x α (1)x −α (−1)x α (1) ∈ N G (T ) for each α. This product is then a representative for the simple reflection attached to α and one may select representatives for other elements of the Weyl group using them.
Let M be a standard Levi. Then we may identify X M with {χ ∈ X T : χ, α ∨ = 0, α ∈ (T, M)}. Likewise, we may identify X M,un with {χ ∈ X T,un : χ, α ∨ = 0, α ∈ (T, M)}. We would like to choose a complement X G,0 to X G,un in X G . When G = GL 1 this is done by taking the normalized characters, i.e., those that are trivial on the multiplicative group of positive reals, embedded diagonally at all the infinite places. When G is a torus, it can be identified with several copies of GL 1 by choosing a Z-basis for X(G) and the subgroup X G,0 thus obtained is independent of the choice. If G = G der T where G der is the derived group and T is a torus, then restriction gives an embedding X G → X T , and we may apply the decomposition X T = X T,un ⊕ X T,0 to obtain the corresponding decomposition of X G .
For archimedean local fields, we again define a character to be normalized if it is trivial on the positive reals. For nonarchimedean fields, we first choose a uniformizer and then say that a character is normalized if it is trivial on the uniformizer. This leads to similar decompostions X G,v = X G,v,un ⊕ X G,v,0 into unramified characters and normalized characters. For any character χ, we define χ un and χ 0 to be the components relative to this decomposition. If χ = s + χ 0 ∈ X GL 1 (resp. X GL 1 ,v ) we define L(χ) to be the usual global (resp. local) L function L(s, χ 0 ) (keeping in mind that X GL 1 ,un has been identified with C). Note that X GL 1 ,0 is identified with the group of Hecke characters F × \A × → C × and that this group is normally written additively. We shall occasionally break with the practice of writing everything additively, and write X GL 1 ,0 multiplicatively, particularly when discussing L functions.
Take P a parabolic with Levi M and χ ∈ X M . Write ρ P = 1 2 α∈ (P,T ) α. We define I G P (χ) to be the normalized K -finite induced representation of G(A) and Ind G P (χ) to be the non-normalized version. For χ ∈ X M,v we define I G P (χ) to be the normalized K v -finite induced representation of G(F v ) and Ind G P (χ) the non-normalized version. In either case, In the important special case when the Levi of P is rank one with unique root α, this becomes I G B (χ − α 2 ).

Flat sections
Fix a reductive group G and standard Levi M, and a normalized character χ 0 ∈ X M . We consider the family of induced representations I G P (χ) with χ in χ 0 + X M,un . Denote the family as a whole by I G P (χ 0 ). By a section we mean a function

Coordinates on X T in the case of G 2
Our main results deal with induced representations on the split exceptional group G 2 . I write α for the short root and β for the long root. Unfortunately, this is the opposite of the notation used in [10]. I write U γ for the root subgroup attached to any root γ . I assume G 2 to be equipped with a choice of Borel and of maximal torus. These are B and T. I write P = MU for the standard parabolic subgroup whose Levi contains U α and Q = LV for the one whose Levi contains U β . For χ 1 , Thus 1 := [1, 0] and 2 := [0, 1] are the two fundamental weights. Note that [χ 1 ,

Normalization and poles of intertwining operators: GL 2 case
We study poles of intertwining operators. The theory is fairly uniform for split groups, and reduces to the special case of GL 2 . First we consider the case of GL 2 . Write B GL 2 for the standard Borel of GL 2 consisting of upper triangular matrices. Take χ = v χ v a character of B GL 2 (A). Let w be the unique nontrivial element of the Weyl group, and α the unique positive root.

Lemma 2.1 The normalized local intertwining operator
extends analytically to all of X T . When Proof In the nonarchimedean case, both assertions can be verified by fairly direct computations. Alternatively, the first assertion is a special case of a result of Winarsky, [23], and the second assertion is a special case of the result in Sect. 4 of Langlands, Euler products [17]. Over the reals, both assertions can be deduced from Proposition 2.6.3 of [2]. The second assertion is also the simplest case of the formula of Gindikin and Karpalevic [8], first proved for GL n by Bhanu Murti [1]. Over the complex numbers, both assertions follow from Lemma 7.23 of [22]. See also [6] and additional references therein. The first assertion over either archimedean field also follows from the generalization found on p. 110 of [21].

Poles of intertwining operators on the principal series: general case
Now let G be a general split reductive group, B its Borel, χ = v χ v a character of B(A), and w any element of the Weyl group. For each root α we have a map SL 2 → G and can decompose the standard intertwining operator M(w, χ) as a composite of intertwining operators indexed by {α > 0 : wα < 0}. Poles of the intertwining operator attached to a root α are of the same three types, along hyperplanes α ∨ , χ = c in the space X B,un defined using the corresponding coroot.

Eisenstein series
Now suppose that G is a reductive group and P a parabolic subgroup. We fix a suitable maximal compact subgroup K = v K v of G(A) and let A(G) denote the space of automorphic forms (relative to K ) G(A) → C, that is, the space of smooth functions φ : G(F )\G(A) → C of moderate growth which are finite under the action of K and the center, z G of the universal enveloping algebra of the Lie algebra of G(F ∞ ). Fix χ 0 ∈ X M,0 and let Flat(χ 0 ) denote the space of flat sections of I G P (χ 0 ). For f ∈ Flat(χ 0 ), we define the Eisenstein series for values of χ such that this sum is convergent and by meromorphic continuation elsewhere. Outside the domain of convergence, one encounters poles of finite order along a locally finite set of root hyperplanes. For each χ away from the poles, f → E P .f (·, χ) is an intertwining operator I G P (χ) → A(G). We denote it E P (χ).

G 2 Eisenstein series
We briefly recall the Eisenstein series which appear in [9,12] and their normalization. The Eisenstein series in question are attached to the parabolic P = MU as in Sect. 3.2. In [9,12], unramified Eisenstein series are considered. The space X M,un is one dimensional and can conveniently be identified with C using the mapping s → δ s P . Here δ P is the modular quasicharacter. In the notation of Sect. 3.2, δ s P = [0, 3s]. Equivalently, the half-sum of the roots of P is ρ P = [0, 3 2 . In order to generalize the construction of [9,12] to get L(s, π, Ad ×χ) for general χ, we would use

Relevant intertwining operators for G 2 and their poles
We apply the material from Sects. 2.4 and 2.5 to the intertwining operators that appear in the constant term of our G 2 Eisenstein series. Write c(u, Then decomposing M(w, χ) as a composite of operators attached simple reflections and letting M * (w, χ) denote the corresponding composite of normalized operators yields the generalization of (2.2): The constant term of our Eisenstein series may be expressed as a sum over w ∈ W such that wα > 0. Here α is the short simple root. There are six such w, one of each length from 0 to 5. We write w i for the element of length i. If χ = [−1, 3s − 1 + χ 0 ] and (see note 6 ) Recall that poles of the intertwining operators come in three classes. We analyze each class.
(1) Poles which arise from the pole of the zeta function at 1 will occur at 2/3 and 1 if χ 0 is trivial; at 5/9 and 2/3 if χ 3 0 is trivial, and at 2/3 if χ 2 0 is trivial. If χ 0 is trivial then the pole at 2/3 can be a triple pole. Otherwise it is simple. The other poles are always simple.
(2) Poles which arise from zeros of a global L function are in the half plane Re(s) < 1 2 . For example, c(9s − 5, χ 3 0 ) could have poles as far right at Re(s) = 5/9 − ε, but it never occurs without c(9s − 4, χ 3 0 ) so the zeros of the L function in its denominator are always cancelled. We only get poles from the zeros of L(9s − 3, χ 3 0 ), and these are to the left of s = 4/9.

Application to the constant term of our G 2 Eisenstein series
Now, the poles of the Eisenstein series (and their orders) are the same as the poles of the constant term (and their orders), which is a sum of intertwining operators. Having determined the poles of the summands, and their orders, the next step is to account for the possibility of cancellation in the sum. In the unramified case this is done in [10].

Theorem 2.3 [Ginzburg-Jiang]
Assume that χ 0 is trivial. Then E P has a simple pole at s = 1, and a double pole at s = 2/3. At s = 5/9 it is holomorphic.
We sketch a proof which is slightly different than the one given in [10]. The technique is similar to [13] and will be worked out in detail for the ramified case below. We consider the two terms which have triple poles at s = 2/3. They correspond to the Weyl elements w 4 and w 5 . We may write M(w 5  The corresponding expression for w i with i < 5 is obtained by taking only the rightmost operators in this composite. We tabulate key data. First, the spaces that the six operators map into M(…, χ) Maps to I G 2 B (. . . ) (s = 5/9 , 3χ 0 = 0) (s=2/3, 3 χ 0 = 0) (s = 2/3 , 2χ 0 =0) And next the elements of X GL 1 which will determine the poles of the rank one operators. Pairing The key facts are the following: when the pairing is 1 the corresponding rank-one operator has a pole. When it is zero, the corresponding rank-one operator is the scalar operator −1. When it is −1 the corresponding rank-one operator has a kernel. Otherwise, the rank-one operator is an isomorphism. We also see that M(w i , [−1, χ 0 + 3s − 1]) has a simple pole at s = 5/9 if i ≥ 2 and 3χ 0 = 0. In this case we have two pairs of operators which land in the same space. To study them, set u = 3s − 5/3 = 3(s − 5 9 ) so that 3s − 1 = u + 2 3 . Thus u is a convenient local coordinate in a neighborhood of s = 5 9 . The first key point is that  7 We have reverted to writing X GL1,un additively: 2χ 0 instead of χ 2 0 and 3χ 0 instead of Once again we have an operator with a simple pole composed with an operator that has a zero. This completes the proof.

Remark 2.5
The existence of the pole of E P at 2/3 in the cubic case can also be deduced from the existence of a pole of the adjoint L function: cuspidal representations of GL 3 (A) satisfying π ∼ = π ⊗ χ exist by Theorem 2.4(iv) of [4]. Thanks to David Loeffler for explaining this to me. For such a representation L(s, π, Ad ⊗χ) = L(s, π × π × χ)/L(s, χ) will have a pole at s = 1. As local L functions are nonvanishing this pole will be inherited by the partial L function, and then, by Theorem 5.1 below by the global zeta integral. In the case when χ is nontrivial quadratic the existence of a pole at s = 2/3 can also be deduced from Theorem 3.4 below.

Siegel-Weil type identities
In this section we prove identities relating degenerate Eisenstein series induced from the two different parabolic subgroups of G 2 . Such identities are sometimes called Siegel-Weil nth term identities. A conceptual explanation for their existence comes from embeddings of degenerate induced representations into principal series representations induced from the Borel, together with the symmetry of the principal series (see [14]). We first prove a technical result which extends this philosophy to flat sections and Eisenstein series.

Surjectivity property of intertwining operators
In the next few sections we consider an alternate normalization of the intertwining operator, which is different than the one considered in Sect. 2.
Take χ ∈ X M,v ⊂ X T,v . Then the standard intertwining operator M(w α , χ + α 2 ) maps I G B (χ + α 2 ) to I G P (χ) ⊂ I G B (χ − α 2 ). We sketch a proof that the map is surjective. Take any f ∈ I G P (χ) and let f • s denote the spherical vector in Ind G B (s + 1 2 )α. Then let f := f · f • s , which lies in the space I G B (χ + sα). It follows immediately from the integral formula for the standard intertwining operator that But f • −1/2 is just the constant function 1, so when s = 1/2 we obtain a nonzero scalar multiple of f.

Proposition 3.1 If (T, M) = {±α} and f is a flat section of I G P (χ 0 ), then there exists a flat section f of I
Proof As before, we construct f by taking the product of f and the normalized spherical vector in Ind G B (s + 1 2 )α. (Note that X T,un = X M,un + Cα.) Then for χ ∈ X M and s ∈ C Again, f −1/2 is the constant function 1, so we get a nonzero scalar multiple of f χ .
The key point is that this extends to a map from flat sections of I G B (χ 0 ) to flat sections of I G P (χ 0 ).

Proposition 3.3 Take P = MU with (T, M) = {±α} and f a flat section of
Proof The two sides have the same constant term, namely Hence their difference is both a cusp form and a linear combination of Eisenstein series. As such, it is zero.

An identity of ramified Eisenstein series on G
It follows that Now, It follows that the residue of E P · f at s = 1 2 and that of E Q · h at u = 1 2 are two different expressions for the value of the meromorphic continuation of (s

An identity of unramified Eisenstein series on G 2
In this section we prove an intriguing identity between unramified G 2 Eisenstein series.
Also, c(s) has a simple zero at s = −1 a simple pole at s = 1 and is holomorphic and nonvanishing at each other integer. Write c i,j for the ith Laurent coefficient at j, so that Also c(s)c(−s) = 1 and c(0) = −1, which implies that u+1] vanishes identically at u = 0. Proof It suffices to prove that the constant term vanishes. We compute the constant terms of E Q · f • [3u+2, −1] and

Theorem 3.5 The meromorphic function
The terms are grouped according to which T -eigenspace they reside in when u = 0. The proof is by direct computation. For each T -eigenspace we consider the terms in the Laurent expansion up to O(u). For example take the character [−1, 1] of T. −1] , which nontrivial, but vanishes at u = 0 because c(3u − 1) vanishes at u = 0 and none of the other terms has a pole at u = 0. This is a fairly simple example.
We consider one additional example, which is a little more complex. Recall that where N is the unipotent radical of B. Hence Write c, d for the mapping (ntk) → log |t 1 |c + log |t 2 |d, (c, d ∈ Z, n ∈ N (A), t ∈ T (A), k ∈ K ).
We simplify the expression in brackets in ([1,-1],Q) Multiplying by c(s and now substituting s = 3u yields (3.7). The other eigenspaces are treated similarly.

Global zeta integral involving a Q-Eisenstein series
In this section we consider the degenerate Eisenstein series E Q .h on G 2 (A), where Q = LV is the parabolic subgroup whose Levi contains the root subgroup attached to the long simple root. We let H ρ be a quasisplit subgroup of G 2 of type A 2 defined as in [12]. Thus ρ ∈ F × and H ρ is isomorphic to SL 3 if ρ is a square, and a quasisplit unitary group attached to the corresponding quadratic extension if F is ρ is a nonsquare. For any normalized character χ 0 of L(F )\L(A) and any flat section h of I = I G 2 Q (χ 0 ) the restriction of E P .h to H ρ (A) is a smooth function of moderate growth on H ρ (F )\H ρ (A), so it may be integrated against a cuspform on H ρ . In the split case, this integral is identically zero for all χ 0 , h, as shown in Proposition 4.6 of [10]. We extend the same idea to the general case.
First we need to analyze Q(F )\G 2 (F )/H ρ (F ). Following [12], we regard G 2 as a subgroup of (split) SO 8 preserving the quadratic form x → t x · x. Here t is the "other transpose" (as in [12]). Then G 2 is contained in the group SO 7 preserving the subspace V 0 := {x = t x 1 . . .
If ρ is not a square this is impossible. It follows that the two Q(F ) orbits already described exhaust all vectors with t x ·x = 2ρ. If ρ = a 2 we have the two additional subsets described. Each is readily seen to be a Q(F )-orbit. Let X −α be the matrix with a one at positions (2, 1), (4, 3) and (5, 3), a −1 at positions (8,7), (6,5) and (6,4), and zeros everywhere else. It spans the root subspace for the root −α. Let x −α (r) = exp(rX −α ). (This is a polynomial formula, because X −α is nilpotent.) Then which proves that our two additional Q(F )-orbits are still in the G 2 (F )-orbit of v a 2 . Clearly, the four taken together exhaust {x ∈ V 0 : t x · x = 2ρ}. This completes the proof.

A certain period
Let ϕ 32 : SL 2 → G 2 be the mapping If H ρ is split, then H ρ is a nonstandard Levi subgroup and is conjugate to a standard Levi subgroup. It suffices to show that the period along the standard Levi subgroup vanishes. As noted in [10] this follows easily from cuspidality.
Remark 4. 6 We may regard H ρ as SU 1,1 ⊂ U 1,1 ⊂ U 2,1 and one could enlarge our period P, to an integral over U 1,1 (F )\U 1,1 (A) against a character. Such periods are considered in [7], where they are used to characterize the image of the theta lifting. Recall that the Ad L function of a representation in the image of this lifting should have a pole at s = 1. So, the emergence of the H ρ period as an obstruction to proving holomorphy in general makes perfect sense. Proof In the case when χ is unramified this is a slight refinement of Proposition 4.6 of [10]. We know from Theorems 2.3 and 2.4 that the only possible poles in Re(s) ≥ 1 2 occur at s = 2 3 when χ 0 is trivial, quadratic, or cubic, and at s = 1 when χ 0 is trivial. When H ρ is split and χ 0 is trivial, the possibility of a pole at s = 1 or a double pole at s = 2/3 are ruled out by [10]. When χ 0 is quadratic, the possibility of a pole at 2 3 is ruled out by the same argument, using Theorem 3.4 and Proposition 4.4. The vanishing at 1 2 when χ 0 is quadratic follows from Proposition 2.6.

Nonvanishing of local zeta integrals
In this section we prove that for any fixed s 0 ∈ C there is a choice of data such that the local zeta integral for the adjoint L function is nonzero at s 0 . It then follows that any pole of the partial adjoint L function would give rise to a pole of the global zeta integral, except possibly at poles of the "normalizing factor" L S (3s, χ)L S (6s − 2, χ 2 )L S (9s − 3, χ). We take up some notations from [12]: for fixed ρ ∈ F, H ρ is defined as in the previous section. We equip it with a choice of Borel subgroup B = TN where T is a maximal torus and N is a maximal unipotent. (This departs from our previous usage of B, T and N for corresponding subgroups of G 2 .) Also w 2 is the second simple reflection in G 2 (attached to the long simple root; this departs from the usage of w 2 in Sect. 2.8), and N 2 is a two dimensional unipotent subgroup, with the property that H ρ ∩ w −1 2 Pw 2 = N 2 T sp , where T sp denotes the one dimensional maximal F -split torus contained in the standard Borel of H ρ . Finally, ψ N is a certain generic character of N. Details are found in [12]. It is convenient to identify X M,un with C via the map s → δ s P . For ρ ∈ F, π a irreducible admissible ψ N -generic representation of H ρ , W in the ψ N -Whittaker model W ψ N (π) of π, and f ∈ Flat(χ 0 ) we define Theorem 5.1 For any ρ ∈ F × , any irreducible admissible generic representation π of H ρ , any χ 0 ∈ X M,0 , and any fixed s 0 ∈ C, there exist W ∈ W ψ N (π) and f ∈ Flat(χ 0 ) such that I(W, f ; s 0 ) = 0.
Proof Expressing the Haar measure on H ρ as a suitable product measure on the open Bruhat cell yields We may identify N 2 \N with the complementary subgroup which is T sp -stable, and fix a fundamental domain [T sp \T ] for T sp in T, to express I(W, f ; s) as where γ is the common restriction of the two simple roots of H ρ to T sp (so that δ −1 Notice that φ 2 is a Schwartz function which can be chosen arbitrarily. Clearly, we can now choose W ∈ W ψ N (π) which does not vanish identically on T sp and then choose φ 2 so that I ((φ 2 • x) * W, s 0 ) = 0. This completes the proof in the nonarchimedean case. In the archimedean case, the same argument shows that the mapping (W, f ) → I(W, f ; s 0 ) does not vanish identically on W ψ N (π) × SI G P (χ 0 + s 0 ), where SI denotes smooth induction, as opposed to K -finite induction. But since the space of K -finite vectors is dense in the smooth induced representation, it then follows that (W, f ) → I(W, f ; s 0 ) can not vanish identically on W ψ N (π) × Ind G P (χ 0 + s 0 ) either, completing the proof in this case.
6 Application to poles of the adjoint L function Theorem 6.1 Assume that G is split, and let S be a finite set of places, including all archimedean places and all places where either π v or χ v is ramified. Then the partial twisted adjoint L function L S (s, π, Ad ⊗χ) has no poles in the half-plane Re(s) ≥ 1 2 , except possibly for a simple pole at Re(s) = 1 when χ is nontrivial and π ∼ = π ⊗ χ (which forces χ to be cubic). If this pole is present, then it is inherited by the complete L function L(s, π, Ad ×χ). Every other pole of L(s, π, Ad ×χ) in Re(s) ≥ 1 2 is a zero of the Hecke L function L(s, χ), and a pole of v∈S L v (s, π v , Ad ×χ v ).
The product L(3s, χ)L(6s − 2, χ 2 )L(9s − 3, χ 3 ) has no poles in Re(s) ≥ 1 2 except for the simple pole of L(6s − 2, χ 2 ) at s = 1 2 which occurs only if χ 2 is trivial. But we have seen that I( 1 2 , ϕ, f ) = 0 when χ is quadratic. This completes the proof of our assertions regarding the partial L function. Since local L functions are meromorphic but nonvanishing, passing from the partial to the completed L function may introduce additional poles, but will not cancel the pole at 1 in the case when it occurs. On the other hand, it follows immediately from the definitions that L(s, π, Ad ×χ) = L(s, (π ⊗ χ) × π)/L(s, χ). By a result of Moeglin and Waldspurger [19,Corollaire,p. 667], the numerator has at most two simple poles, which occur at 0 and 1 8 The particularly careful reader may have noticed that the equality I(s, W v , f v ) = L v (3s − 1, π, Ad ⊗χ) L v (3s, χ)L v (6s − 2, χ 2 )L v (9s − 3, χ 3 )

(6.2)
is only attained by taking normalized spherical vectors if ψ N,v is unramified, in addition to π v and χ v . We remark briefly on the places where π v and χ v are unramified and ψ N,v is not. The form of the global character ψ N emerges from the unfolding in [9,12] ensures that when ψ N,v is ramified, its orbit under the maximal F v -split torus contains an element which is unramified. Then W v (g) = W v (tg), where W v is in the Whittaker model attached to this unramified additive character, and t is a suitable element of the torus. Making a change of variables in the integral we find that (6.2) holds up to an exponential factor.