Holomorphy of adjoint L functions for quasisplit $A_2$

Abstract

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.

1 Introduction

Let F be a number field.¹ Let G be a quasisplit F-group, isomorphic to $SL_3$ over $\overline{F}.$ 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 $PGL_3(\mathbb {C})$ in the split case, or the semidirect product of $PGL_3(\mathbb {C})$ with ${\text {Gal}}(E/F)$ in the nonsplit case. In either case we have an action of $^LG$ on $PGL_3(\mathbb {C})$ by conjugation, which may be regarded as an action on $GL_3(\mathbb {C})$ which fixes the center. This then induces an action on the Lie algebra $\mathfrak {sl}_3(\mathbb {C})$ which we denote ${\text {Ad}}.$ Note that in the nonsplit case this does not coincide with the definition of ${\text {Ad}}$ in [12]. Rather, the nontrivial element ${\text {Fr}}$ of ${\text {Gal}}(E/F)$ will act by $X \mapsto -_tX,$ as this is the differential of its action on $PGL_3(\mathbb {C}).$ Let ${\text {Ad}}'$ denote the representation of $^LG$ considered in [12]. Thus ${\text {Ad}}' = {\text {Ad}}$ in the split case, but in the nonsplit case it is the representation of $^LG$ where $GL_3(\mathbb {C})$ acts by conjugation and ${\text {Fr}}$ acts by $X \mapsto \,{}_tX.$

Let $\pi$ be a globally generic irreducible cuspidal automorphic representation of G. Then we consider the adjoint L function $L(s, \pi , {\text {Ad}}).$ 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 $L(s, \pi , {\text {Ad}})$ 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 $P(F)\backslash P(\mathbb {A}),$ and $E_P$ be the corresponding Eisenstein series operator (these notions are reviewed in Sects. 2.2 and 2.6).

One may then define

$$\begin{aligned} I(s, \varphi , f) := \int \limits _{G(F)\backslash G(\mathbb {A})} \varphi (g) E_P.f(g,s) \, dg,\!\!\! \end{aligned}$$

(1.1)

where $s \in \mathbb {C},$ and $\varphi$ is a generic cusp form $G(F)\backslash G(\mathbb {A}) \rightarrow \mathbb {C}.$ 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 $I(s, \varphi , f)$ unfolds to a new integral $I(s, W_{\varphi }, f)$ where $W_{\varphi }$ is the Whittaker function attached to $\varphi .$ Assuming that $W_{\varphi }$ and f are factorizable, $I(s, W_{\varphi }, f)$ 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 $W_{\varphi }$ and f respectively. Moreover, if $W_v,f_v$ and $\chi _v$ are all unramified,² and $W_v$ and $f_v$ are normalized, then (see note³)

$$\begin{aligned} I(s, W_v, f_v) = \frac{L(3s-1, \pi _v, {\text {Ad}}'\otimes \chi _v)}{L(3s, \chi _v)L(6s-2, \chi _v^2) L(9s-3, \chi _v^3)}. \end{aligned}$$

Hence,

$$\begin{aligned} I(s, \varphi , f) = \frac{L^S(3s-1, \pi , {\text {Ad}}'\otimes \chi )}{L^S(3s, \chi )L^S(6s-2, \chi ^2) L^S(9s-3, \chi ^3)} \prod _{v \in S} I(s, W_v, f_v), \end{aligned}$$

(1.2)

where S is a finite set of places, away from which $W_v, f_v$ and $\chi _v$ are unramified. Notice that $L(s, \pi , {\text {Ad}}) = L(s, \pi , {\text {Ad}}')$ if G is split and $L(s, \pi , {\text {Ad}}' \otimes \chi _{E/F}),$ where $\chi _{E/F}$ 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 $L(s, \pi , {\text {Ad}}\otimes \chi )$ should be entire. Indeed, in the split case $L(s, \pi , {\text {Ad}}\times \chi ) = L(s, \pi \otimes \chi \times \widetilde{\pi })/L(s,\chi ),$ and it follows that the possible poles are precisely the zeros of $L(s, \chi ),$ unless $\chi$ is nontrivial and $\pi \otimes \chi \cong \widetilde{\pi },$ in which case there are additional simple poles at $s=0$ and $s=1.$ One expects that $L(s, \pi \otimes \chi \times \widetilde{\pi })$ is divisible by $L(s, \chi )$ and hence that the only actual poles are the simple poles at $s=0$ and $s=1$ which occur when $\chi$ is nontrivial and $\pi \otimes \chi \cong \widetilde{\pi }.$ In the special case when $\pi = \otimes '_v \pi _v$ and at least one component $\pi _v$ is supercuspidal, this was proved by Flicker [5].

In the nonsplit case, one must replace $L(s, \pi \times \widetilde{\pi })$ with the Asai L function of the stable base change lifting of $\pi$. One must also account for the image of certain theta liftings. Indeed, consider

$$\begin{aligned} \left\{ \begin{pmatrix}a&{} &{}b \\ &{}t&{} \\ c&{}&{}d \end{pmatrix}\right\} \cong GL_2(\mathbb {C}) \times GL_1(\mathbb {C}) \subset GL_3(\mathbb {C}) \end{aligned}$$

(1.3)

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 $U_{1,1} \times U_1$ (still attached to the same quadratic extension). See [20]. The above subgroup clearly stabilizes the one dimensional subspace of $\mathfrak {sl}_3(\mathbb {C})$ spanned by ${\text {diag}}(1, -2, 1),$ and so does the map $X \mapsto \, _t\!X:=J\ ^t\!X J,$ where $J = \left( {\begin{matrix}&{}&{}1\\ {} &{}1&{}\\ 1&{}&{}\end{matrix}}\right) .$ Thus we obtain a one dimensional space stable under the restriction of the representation ${\text {Ad}}'$ to $^L(U_{1,1} \times U_1),$ so that $L(s, \pi , {\text {Ad}}')$ should have a pole whenever $\pi$ is a lift from $U_{1,1} \times U_1.$ (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 $I(s, \varphi ,f)$ and does not depend on any property of a local component of $\pi$ was pioneered in [10]. If $I(s, \varphi , f)$ has a pole at $s=s_0$ then each of its negative Laurent coefficients is a global integral, similar to $I(s, \varphi , f),$ 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 $I(s, \varphi , f),$ when restricted to the subgroup $G \hookrightarrow G_2$ 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 $I(s, \varphi , f)$ for the case of trivial $\chi$ has only two poles in ${\text {Re}}(s) > \frac{1}{2},$ 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 $L(s, \pi ,{\text {Ad}}),$ this pole would occur at $s=1.$ Such a pole is impossible, because $L(s, \pi , {\text {Ad}}) = L(s, \pi \times \widetilde{\pi }) /\zeta (s),$ and $L(s, \pi \times \widetilde{\pi })$ and $\zeta (s)$ 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 $\chi .$ 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 $L(s, \pi , {\text {Ad}}\times \chi )$ at ${\text {Re}}(s) = \frac{1}{2}$ 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 $L(s, \pi , {\text {Ad}}\times \chi )$ in the half plane ${\text {Re}}(s) \ge \frac{1}{2}$ is simultaneously a zero of $L(s, \chi )$ 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 $SL(3, \mathbb {Z})$ 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.

2 Induced representations, interwining operators, and their poles

2.1 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 $\mathbf {X}_G$ for the complex manifold of characters of $G(\mathbb {A})$ trivial on G(F). These are groups and for the most part we write them additively. To reconcile with multiplicative notation for $G(\mathbb {A})$ and $\mathbb {C}^\times$, we use an exponential notation for the characters: the value of $\chi \in \mathbf {X}_G$ at $g \in G(\mathbb {A})$ is denoted $g^\chi .$ A similar notation is used for cocharacters. We identify $\chi \in X(G)$ with the character of $G(F) \backslash G(\mathbb {A})$ obtained by composing it with the absolute value on $\mathbb {A}^\times .$ This extends to a mapping of $X(G) \otimes _\mathbb {Z}\mathbb {C}$ into $\mathbf {X}_G.$ The image is the set of unramified characters, which we denote $\mathbf {X}_{G,{{\text {un}}}}.$ Similarly we denote the complex manifold of all characters of $G(F_v)$ by $\mathbf {X}_{G,v}$ and the image of $X(G) \otimes _\mathbb {Z}\mathbb {C}$ in it by $\mathbf {X}_{G,v,{{\text {un}}}}.$ We identify $\mathbf {X}_{GL_1,{{\text {un}}}}$ with $\mathbb {C}$ using the map $s \rightarrow |\ |^{s}.$ We denote the canonical pairing between characters and cocharacters by $\langle \ ,\ \rangle .$ If $\varphi ^\vee$ is a cocharacter, then $\langle \varphi ^\vee , \chi \rangle \in \mathbf {X}_{GL_1}.$

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 $\Phi (T,H)$ 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 $\{ x_{\alpha }: \mathbb {G}_a \rightarrow U_\alpha \} _{\alpha \in \Phi (G,T)},$ such that $x_{\alpha }(1)x_{-\alpha }(-1) x_{\alpha }(1) \in N_G(T)$ for each $\alpha .$ This product is then a representative for the simple reflection attached to $\alpha$ and one may select representatives for other elements of the Weyl group using them.

Let M be a standard Levi. Then we may identify $\mathbf {X}_M$ with $\{ \chi \in \mathbf {X}_T : \langle \chi , \alpha ^\vee \rangle = 0, \alpha \in \Phi (T,M)\}.$ Likewise, we may identify $\mathbf {X}_{M,{{\text {un}}}}$ with $\{ \chi \in \mathbf {X}_{T,{{\text {un}}}} : \langle \chi , \alpha ^\vee \rangle = 0, \alpha \in \Phi (T,M)\}.$ We would like to choose a complement $\mathbf {X}_{G,0}$ to $\mathbf {X}_{G,{{\text {un}}}}$ in $\mathbf {X}_G.$ 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 $\mathbb {Z}$-basis for X(G) and the subgroup $\mathbf {X}_{G,0}$ thus obtained is independent of the choice. If $G = G_{{\text {der}}} T$ where $G_{{\text {der}}}$ is the derived group and T is a torus, then restriction gives an embedding $\mathbf {X}_G \hookrightarrow \mathbf {X}_T,$ and we may apply the decomposition $\mathbf {X}_T = \mathbf {X}_{T,{{\text {un}}}} \oplus \mathbf {X}_{T,0}$ to obtain the corresponding decomposition of $\mathbf {X}_G.$

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 $\mathbf {X}_{G,v} = \mathbf {X}_{G,v,{{\text {un}}}} \oplus \mathbf {X}_{G,v,0}$ into unramified characters and normalized characters. For any character $\chi ,$ we define $\chi _{{{\text {un}}}}$ and $\chi _0$ to be the components relative to this decomposition. If $\chi = s + \chi _0 \in \mathbf {X}_{GL_1}$ (resp. $\mathbf {X}_{GL_1, v}$) we define $L(\chi )$ to be the usual global (resp. local) L function $L(s, \chi _0)$ (keeping in mind that $\mathbf {X}_{GL_1, {{\text {un}}}}$ has been identified with $\mathbb {C}$). Note that $\mathbf {X}_{GL_1,0}$ is identified with the group of Hecke characters $F^\times \backslash \mathbb {A}^\times \rightarrow \mathbb {C}^\times$ and that this group is normally written additively. We shall occasionally break with the practice of writing everything additively, and write $\mathbf {X}_{GL_1,0}$ multiplicatively, particularly when discussing L functions.

Take P a parabolic with Levi M and $\chi \in \mathbf {X}_M.$ Write $\rho _P = \frac{1}{2} \sum _{\alpha \in \Phi (P,T)} \alpha .$ We define $I_P^G(\chi )$ to be the normalized K-finite induced representation of $G(\mathbb {A})$ and ${\text {Ind}}_P^G(\chi )$ to be the non-normalized version. For $\chi \in \mathbf {X}_{M,v}$ we define $I_P^G(\chi )$ to be the normalized $K_v$-finite induced representation of $G(F_v)$ and ${\text {Ind}}_P^G(\chi )$ the non-normalized version. In either case, $I_P^G(\chi ) = {\text {Ind}}_P^G( \chi + \rho _P),$ and is a subset of $I_B^G(\chi + \rho _P - \rho _B).$ In the important special case when the Levi of P is rank one with unique root $\alpha ,$ this becomes $I_B^G( \chi - \frac{\alpha }{2}).$

2.2 Flat sections

Fix a reductive group G and standard Levi M,  and a normalized character $\chi _0 \in \mathbf {X}_M.$ We consider the family of induced representations $I_P^G(\chi )$ with $\chi$ in $\chi _0+ \mathbf {X}_{M,{{\text {un}}}}.$ Denote the family as a whole by $\mathcal I_P^G(\chi _0).$ By a section we mean a function

$$\begin{aligned} \chi \mapsto f_\chi , \quad (\chi \in \chi _0+ \mathbf {X}_{M,{{\text {un}}}}) \end{aligned}$$

such that

• (1) $f_\chi \in I_P^G(\chi )$ for each $\chi \in \chi _0+ \mathbf {X}_{M,{{\text {un}}}},$

• (2) f is smooth as a function $X_M \times G \rightarrow \mathbb {C}.$

We say that f is flat if $f( \chi _0+s, k)$ is independent of $s \in \mathbf {X}_{M, {{\text {un}}}}$ for $k \in K.$ (Here K is a fixed maximal compact subgroup.) The set of flat sections of $\mathcal I_P^G(\chi _0)$ is a complex vector space. Denote it ${\text {Flat}}(\chi _0).$

2.3 Coordinates on $\mathbf {X}_T$ in the case of $G_2$

Our main results deal with induced representations on the split exceptional group $G_2.$ I write $\alpha$ for the short root and $\beta$ for the long root. Unfortunately, this is the opposite of the notation used in [10]. I write $U_\gamma$ for the root subgroup attached to any root $\gamma .$ 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 $U_\alpha$ and $Q=LV$ for the one whose Levi contains $U_\beta .$ For $\chi _1, \chi _2 \in \mathbf {X}_{GL_1}$ let $[\chi _1, \chi _2]$ denote the element of $\mathbf {X}_T$ which satisfies

$$\begin{aligned} \langle [\chi _1, \chi _2],\ \alpha ^\vee \rangle = \chi _1,\quad \langle [\chi _1, \chi _2],\ \beta ^\vee \rangle = \chi _2. \end{aligned}$$

Thus $\varpi _1:=[1,0]$ and $\varpi _2:=[0,1]$ are the two fundamental weights. Note that $[\chi _1, \chi _2] \in \mathbf {X}_M \iff \chi _1 = 0$ and $[\chi _1, \chi _2] \in \mathbf {X}_L \iff \chi _2 = 0.$

2.4 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 $\chi = \prod _v \chi _v$ a character of $B_{GL_2}(\mathbb {A}).$ Let w be the unique nontrivial element of the Weyl group, and $\alpha$ the unique positive root.

Lemma 2.1

The normalized local intertwining operator

$$\begin{aligned} M^\star _v(w, \chi _v):= \frac{1}{L_v(\langle \chi _v , \alpha ^\vee \rangle )} M_v(w, \chi ) \end{aligned}$$

extends analytically to all of $\mathbf {X}_T.$ When $\langle \chi _v \circ \alpha ^\vee \rangle$ is unramified $I_{B_{GL_2}}^{GL_2}(\chi _v)$ has a normalized “spherical”⁵ vector which we denote $f^\circ _{\chi _v}.$ Then

$$\begin{aligned} M_v(w, \chi _v).f^\circ _{\chi _v} = \frac{L(\langle \chi _v, \alpha ^\vee \rangle )}{L(\langle \chi _v, \alpha ^\vee \rangle + 1 )}f^\circ _{w \chi _v}. \end{aligned}$$

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]. $\square$

Consequently if $f = \prod _{v \notin S} f_{\chi _v}^\circ \prod _{v\in S} f_v$ is an element of the global induced space $I_{B_{GL_2}}^{GL_2}(\chi )$, then

$$\begin{aligned} M(w, \chi ).f = \frac{L(\langle \chi , \alpha ^\vee \rangle ) }{L(\langle \chi , \alpha ^\vee \rangle +1) }\prod _{v \notin S} f_{\chi _v}^\circ \prod _{v\in S} L_v(\langle \chi _v, \alpha ^\vee \rangle +1) M_v^\star (w, \chi _v) f_v. \end{aligned}$$

(2.2)

We deduce that poles of $M(w, \chi )$ come in three classes

• (1) Poles of $\prod _{\alpha > 0, \ w\alpha < 0} L(\langle \chi , \alpha ^\vee \rangle ),$ which are not cancelled by poles of $\prod _{\alpha > 0, \ w\alpha < 0} L(\langle \chi , \alpha ^\vee \rangle +1),$ These occur at $\langle \chi , \alpha ^\vee \rangle =1,$ and in particular do not occur unless $\langle \chi , \alpha ^\vee \rangle _0=0.$

• (2) Zeros of $L(\langle \chi , \alpha ^\vee \rangle +1).$ These are all in the strip $-1< {\text {Re}}\langle \chi , \alpha ^\vee \rangle _{{{\text {un}}}} <0.$

• (3) Poles of $\prod _{v\in S}L_v(\langle \chi _v, \alpha ^\vee \rangle +1).$ These are all in the half plane ${\text {Re}}\langle \chi , \alpha ^\vee \rangle _{{{\text {un}}}} \le -1.$

2.5 Poles of intertwining operators on the principal series: general case

Now let G be a general split reductive group, B its Borel, $\chi = \prod _v \chi _v$ a character of $B(\mathbb {A}),$ and w any element of the Weyl group. For each root $\alpha$ we have a map $SL_2 \rightarrow G$ and can decompose the standard intertwining operator $M(w, \chi )$ as a composite of intertwining operators indexed by $\{ \alpha > 0 : w\alpha < 0 \}.$ Poles of the intertwining operator attached to a root $\alpha$ are of the same three types, along hyperplanes $\langle \alpha ^\vee , \chi \rangle = c$ in the space $\mathbf {X}_{B, {{\text {un}}}}$ defined using the corresponding coroot.

2.6 Eisenstein series

Now suppose that G is a reductive group and P a parabolic subgroup. We fix a suitable maximal compact subgroup $K = \prod _v K_v$ of $G(\mathbb {A})$ and let $\mathcal A(G)$ denote the space of automorphic forms (relative to K) $G(\mathbb {A}) \rightarrow \mathbb {C},$ that is, the space of smooth functions $\phi : G(F)\backslash G(\mathbb {A}) \rightarrow \mathbb {C}$ of moderate growth which are finite under the action of K and the center, $\mathfrak z_G$ of the universal enveloping algebra of the Lie algebra of $G(F_\infty ).$

Fix $\chi _0 \in \mathbf {X}_{M,0}$ and let ${\text {Flat}}(\chi _0)$ denote the space of flat sections of $\mathcal I_P^G(\chi _0).$ For $f \in {\text {Flat}}(\chi _0),$ we define the Eisenstein series

$$\begin{aligned} E_P.f: G(\mathbb {A}) \times (\chi _0 + \mathbf {X}_{M,{{\text {un}}}}) \rightarrow \mathbb {C}\end{aligned}$$

by

$$\begin{aligned} E_P.f(g,\chi ) = \sum _{\gamma \in P(F) \backslash G(F) } f_\chi (\gamma g) \end{aligned}$$

for values of $\chi$ 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 $\chi$ away from the poles, $f \mapsto E_P.f(\cdot , \chi )$ is an intertwining operator $I_P^G(\chi ) \rightarrow \mathcal A(G).$ We denote it $E_P(\chi ).$

2.7 $G_2$ 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 $\mathbf {X}_{M,{{\text {un}}}}$ is one dimensional and can conveniently be identified with $\mathbb {C}$ using the mapping $s \mapsto \delta _P^s.$ Here $\delta _P$ is the modular quasicharacter. In the notation of Sect. 3.2, $\delta _P^s = [0, 3s].$ Equivalently, the half-sum of the roots of P is $\rho _P = [0,\frac{3}{2}].$ Thus ${\text {Ind}}_P^G(\delta _P^s) = I_P^G([0, 3s-\frac{3}{2}]) \subset I_B^G[-1, 3s-1]).$ In order to generalize the construction of [9, 12] to get $L(s, \pi , {\text {Ad}}' \times \chi )$ for general $\chi ,$ we would use $I_P^G([0, 3s- \frac{3}{2} + \chi ]),$

2.8 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 $c(u, \chi _0) = \frac{L(u, \chi _0)}{L(u+1, \chi _0)}.$ For $\chi \in \mathbf {X}_T$ and $w \in W$ let $c(w, \chi )= \prod _{\alpha > 0, \ w\alpha < 0 } c(\langle \alpha ^\vee , \chi \rangle ).$ Then decomposing $M(w, \chi )$ as a composite of operators attached simple reflections and letting $M^*(w,\chi )$ denote the corresponding composite of normalized operators yields the generalization of (2.2):

$$\begin{aligned} M(w, \chi ).f = \prod _{\alpha > 0, \ w\alpha < 0}\left( \frac{L(\langle \chi , \alpha ^\vee \rangle ) }{L(\langle \chi , \alpha ^\vee \rangle +1) } \prod _{v\in S} L_v(\langle \chi _v, \alpha ^\vee \rangle +1) \right) \prod _{v \notin S} f_{\chi _v}^\circ \prod _{v\in S} M_v^\star (w, \chi _v) f_v. \end{aligned}$$

The constant term of our Eisenstein series may be expressed as a sum over $w \in W$ such that $w\alpha > 0.$ Here $\alpha$ 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 $\chi = [-1,3s-1+\chi _0]$ and (see note⁶)

$$\begin{aligned} \mathcal C:= (c( 3s-1, \chi _0) ,\ c(9s-4, \chi _0^3) ,\ c(6s-3, \chi _0^2) ,\ c(9s-5, \chi _0^3) ,\ c(3s-2, \chi _0)), \end{aligned}$$

then $c(w_i, \chi ) = \prod _{j=1}^i \mathcal C_j.$ 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 $\chi _0$ is trivial; at 5 / 9 and 2 / 3 if $\chi _0^3$ is trivial, and at 2 / 3 if $\chi _0^2$ is trivial. If $\chi _0$ 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 ${\text {Re}}(s) < \frac{1}{2}.$ For example, $c(9s-5, \chi _0^3)$ could have poles as far right at ${\text {Re}}(s) = 5/9-\varepsilon ,$ but it never occurs without $c(9s-4, \chi _0^3)$ so the zeros of the L function in its denominator are always cancelled. We only get poles from the zeros of $L(9s-3, \chi _0^3),$ and these are to the left of $s =4/9.$

• (3) Poles which arise from local L functions are all in the half plane ${\text {Re}}(s) \le 4/9.$

It follows that the only possible poles of our Eisenstein series in the half plane ${\text {Re}}(s) \ge 1/2$ are at 5 / 9, 2 / 3 and 1,  and can occur only if $\chi _0$ is trivial, quadratic, or cubic. Notice that the pole at 2 / 3 would correspond to a pole of $L(s, \pi , {\text {Ad}}' \times \chi )$ at 1. More detailed information regarding the poles of the intertwining operators in ${\text {Re}}(s) \ge \frac{1}{2}$ is recorded in the following table.

$\chi$	Trivial	Nontrivial quadratic	Nontrivial cubic
$w_0$	Holomorphic	Holomorphic	Holomorphic
$w_1$	2/3	Holomorphic	Holomorphic
$w_2$	2/3, 5/9	Holomorphic	5/9
$w_3$	2/3 (double), 5/9	2/3	5/9
$w_4$	2/3 (triple), 5/9	2/3	2/3, 5/9
$w_5$	2/3 (triple), 5/9, 1	2/3	2/3, 5/9

2.9 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 $\chi _0$ 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 $M(w_5, \chi ) = M(s_\beta , w_4\chi ) M(w_4, \chi ).$ Here, $s_\beta$ is the simple reflection in the Weyl group attached to the long root $\beta$ ($s_\alpha$ is defined similarly). We check that when $\chi _0$ is trivial, $\langle \beta ^\vee , w_4 \chi \rangle$ vanishes at the point corresponding to $s=2/3$ It follows from [15, Proposition 6.3] that $M(s_\beta , w_4 \chi )$ is the scalar operator $-1$ at this point. Hence $M(w_4, \chi ) + M(w_5, \chi )$ is equal to the composition of $M(w_4, \chi )$ and an operator which vanishes at $s=2/3.$ Similarly, the four terms which give poles at 5 / 9 form two pairs such that the sum of each pair is holomorphic at $s=5/9.$

Theorem 2.4

Assume that $\chi _0$ is nontrivial quadratic or cubic. Then $E_P$ has a simple pole at $\frac{2}{3}$ and is otherwise holomorphic in ${\text {Re}}(s) \ge \frac{1}{2}.$

Proof

We have

$$\begin{aligned} M(w_5, \chi ) = M(s_\beta , w_4\chi )M(s_\alpha , w_3\chi ) M(s_\beta , w_2\chi ) M(s_\alpha , w_1\chi ) M(s_\beta , \chi ) \end{aligned}$$

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(..., $\chi$)	$\text { Maps to }I_B^{G_2}(\dots )$	(s = 5/9 , 3$\chi _0$ = 0)	(s=2/3, 3 $\chi _0$ = 0)	(s = 2/3 , 2$\chi _0$ =0)
$w_ 0$	[$-$1 ,$\chi _0$ + 3s$-$1 ]	[$-$1, $\chi _0$ + 2/3]	[$-$1, $\chi _0$ + 1 ]	[$-$1, $\chi _0$ + 1]
$w_1$	[ 3$\chi _0$ + 9s$-$4 ,$-$$\chi _0$ $-$ 3s+ 1	[1, $-$$\chi _0$ - 2/3]	[2, $-$$\chi _0$ $-$1]	[$\chi _0$ + 2, $\chi _0$ $-$1]]
$w_2$	[ $-$3$\chi _0$ $-$ 9s+4, 2$\chi _0$ + 6s$-$3 ]	[$-$1, $-$$\chi _0$ + 1/3]	[$-$2, $-$$\chi _0$ + 1]	[$\chi _0$ $-$2, 1]
$w_3$	[ 3$\chi _0$ + 9s$-$5, $-$2$\chi _0$ $-$ 6s+3]	[0, $\chi _0$ $-$1/3]	[ 1,$\chi _0$ $-$1]	[$\chi _0$ +1, $-$1]
$w_4$	[ $-$3$\chi _0$ $-$ 9s+5, $\chi _0$ + 3s$-$2]	[0, $\chi _0$ $-$ 1/3]	[$-$1, $\chi _0$]	[$\chi _0$ $-$1, $\chi _0$]
$w_5$	[ $-$1,$-$$\chi _0$ $-$ 3s+2 ]	[$-$1, $-$$\chi _0$ + 1/3]	[$-$1 , $-$$\chi _0$]	[$-$1, $\chi _0$]

And next the elements of $\mathbf {X}_{GL_1}$ which will determine the poles of the rank one operators.

Pairing	General	(s = 5/9 , 3$\chi _0$ = 0)	(s=2/3, 3 $\chi _0$ = 0)	(s = 2/3 , 2$\chi _0$ = 0)
$\langle$ $\beta ^\vee$, $\chi$ $\rangle$	$\chi _0$+3s$-$1	$\chi _0$+2/3	$\chi _0$+1	$\chi _0$+1
$\langle$ $\alpha ^\vee$, $w_1\chi$ $\rangle$	3$\chi _0$ + 9s$-$4	1	2	$\chi _0$+2
$\langle$ $\beta ^\vee$, $w_2\chi$ $\rangle$	2$\chi _0$ +6s$-$3	$-$$\chi _0$ +1/3	$-$$\chi _0$ + 1	1
$\langle$ $\alpha ^\vee$, $w_3\chi$ $\rangle$	3$\chi _0$ + 9s$-$5	0	1	$\chi _0$ + 1
$\langle$ $\beta ^\vee$, $w_4\chi$ $\rangle$	$\chi _0$ + 3s$-$2	$\chi _0$ $-$ 1/3	$\chi _0$	$\chi _0$

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. From this we can see that $M(w_i, [-1, \chi _0 + 3s-1])$ has a pole at $s=2/3$ if $2\chi _0 = 0$ and $i \ge 3$ or if $3\chi _0 = 0$ and $i \ge 4$.⁷ In either case, the operators $M(w_i, [-1, \chi _0+1])$ all land in different spaces. Hence there is no possibility of cancellation among them and the poles of the individual intertwining operators are inherited by their sum (i.e., the constant term) and then by the Eisenstein series itself.

We also see that $M(w_i, [-1, \chi _0+3s-1])$ has a simple pole at $s=5/9$ if $i \ge 2$ and $3\chi _0 = 0.$ In this case we have two pairs of operators which land in the same space. To study them, set $u=3s-5/3=3(s-\frac{5}{9})$ so that $3s-1=u+\frac{2}{3}.$ Thus u is a convenient local coordinate in a neighborhood of $s=\frac{5}{9}.$ The first key point is that

$$\begin{aligned} \langle \alpha ^\vee , w_3[-1, \chi _0+2/3] \rangle =0 \implies M(s_\alpha , w_3 [-1, \chi _0+u+2/3]) = -1 + O(u). \end{aligned}$$

Hence

$$\begin{aligned}&M(w_3, [-1, \chi _0+u+2/3]) + M(w_4, [-1, \chi _0+u+2/3])\\&\quad = (1+M(s_\alpha , w_3 [-1, \chi _0+u+2/3]) )\circ M(w_3, [-1, \chi _0+u+2/3]) \end{aligned}$$

is the composite of an operator with a simple pole at $u=0$ and an operator which vanishes at $u=0.$ Thus, it is holomorphic at $u=0.$

The second key point is that $M(s_\beta , -s_\beta \chi ) = M(s_\beta , \chi )^{-1}.$ (This is an identity of meromorphic functions.) Hence

$$\begin{aligned} H(u):= & {} M(s_\beta , [-3u, \chi _0+u-1/3])M(s_\alpha , [3u, \chi _0 -2u -1/3]) \\&\times \,M(s_\beta , [-3u-1, 2u+1/3-\chi _0] )\\= & {} (M(s_\beta , [0, \chi _0 - 1/3])+O(u))(-1+O(u))\times (M(s_\beta , [-1, 1/3-\chi _0])+O(u))\\= & {} -1+O(u). \end{aligned}$$

But

$$\begin{aligned}&M(w_2, [-1, \chi _0 + u + 2/3]) + M(w_5, [-1, \chi _0 + u + 2/3])s\\&\quad =(1+H(u))\circ M(w_2, [-1, \chi _0 + u + 2/3]). \end{aligned}$$

Once again we have an operator with a simple pole composed with an operator that has a zero. This completes the proof. $\square$

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 $GL_3(\mathbb {A})$ satisfying $\pi \cong \pi \otimes \chi$ exist by Theorem 2.4(iv) of [4]. Thanks to David Loeffler for explaining this to me. For such a representation $L(s, \pi , {\text {Ad}}\otimes \chi )= L(s, \pi \times \widetilde{\pi }\times \chi )/L(s, \chi )$ 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 $\chi$ is nontrivial quadratic the existence of a pole at $s=2/3$ can also be deduced from Theorem 3.4 below.

2.10 Key vanishing property of the Eisenstein series

Proposition 2.6

Assume that $2 \chi _0=0.$ Then $E_P([-1, 3s-1+\chi _0])$ vanishes at $s=\frac{1}{2}.$

Proof

The proof is similar to that of Theorem 2.4. Note that $w_5[-1, \frac{1}{2} + \chi _0] = [-1, \frac{1}{2}+ \chi _0],$ $w_4[-1, \frac{1}{2} + \chi _0] = w_1[-1, \frac{1}{2}+ \chi _0]$ and $w_3[-1, \frac{1}{2} + \chi _0] = w_2[-1, \frac{1}{2}+ \chi _0].$ Write $w_4 = w_{4,1}w_1$ and $w_3= w_{3,2}w_2.$ As in the proof of Theorem 2.4 one readily checks that $M(w_5, [-1, 3s-1 + \chi _0]), M(w_{4,1}, w_1 [-1, 3s-1 + \chi _0])$ and $M(w_{3,2}, w_2 [-1, 3s-1 + \chi _0])$ are all $-1+ O(s-\frac{1}{2})$ at $s=\frac{1}{2}.$ This time none of the intertwining operators has a pole, so the sum vanishes at $s= \frac{1}{2}.$

Alternatively, having established that $M(w_5, [-1, 3s-1 + \chi _0]) = -1+O(s-\frac{1}{2}),$ We can deduce vanishing of the Eisenstein series from the fact that it is holomorphic and satisfies the functional equation

$$\begin{aligned} E_P([-1, 3s-1+\chi _0]) = E_P( [-1, 2-3s+\chi _0])\circ M(w_5, [-1, 3s-1+\chi _0]). \end{aligned}$$

$\square$

3 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.

3.1 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 $\chi \in \mathbf {X}_{M,v} \subset \mathbf {X}_{T,v}.$ Then the standard intertwining operator $M(w_\alpha , \chi + \frac{\alpha }{2})$ maps $I_B^G( \chi + \frac{\alpha }{2})$ to $I_P^G (\chi ) \subset I_B^G (\chi - \frac{\alpha }{2}).$ We sketch a proof that the map is surjective. Take any $f \in I_P^G (\chi )$ and let $f_s^\circ$ denote the spherical vector in ${\text {Ind}}_B^G (s+\frac{1}{2})\alpha .$ Then let $\widetilde{f} := f\cdot f_s^\circ ,$ which lies in the space $I_B^G (\chi + s\alpha ).$ It follows immediately from the integral formula for the standard intertwining operator that

$$\begin{aligned} M(w_\alpha ,\chi + s\alpha )\widetilde{f} = f M(w_\alpha ,s\alpha )\cdot f_s^\circ =f \frac{\zeta (2s)}{\zeta (2s+1)} f_{-s}^\circ . \end{aligned}$$

But $f_{-1/2}^\circ$ is just the constant function 1, so when $s=1/2$ we obtain a nonzero scalar multiple of f.

In the global setting, this fails because $\zeta (2s)$ has a pole at $s=1/2.$ Let

$$\begin{aligned} M^*( w_\alpha , \chi ) = (\langle \alpha ^\vee , \chi \rangle _{{{\text {un}}}}-1) M( w_\alpha , \chi ). \end{aligned}$$

(Recall that $\langle \alpha ^\vee , \chi \rangle _{{{\text {un}}}}\in \mathbf {X}_{GL_1,{{\text {un}}}}$ which has been identified with $\mathbb {C}.$) Then $M^*(w_\alpha , \chi )$ has no poles in ${\text {Re}}(s) \ge 0,$ and for $\langle \chi , \alpha ^\vee \rangle =0$ maps $I_B^G(\chi + \frac{\alpha }{2})$ (by which we now mean the global version) surjectively onto $I_P^G (\chi ) \subset I_B^G ( \chi - \alpha /2).$

Proposition 3.1

If $\Phi (T,M) = \{ \pm \alpha \}$ and f is a flat section of $\mathcal I_P^G(\chi _0),$ then there exists a flat section $\widetilde{f}$ of $\mathcal I_B^G(\chi _0)$ such that $M^*(w_\alpha , \chi +\frac{\alpha }{2}).\widetilde{f}_{\chi + \frac{\alpha }{2}} = f_\chi$ for all $\chi \in \chi _0 + \mathbf {X}_{M,{{\text {un}}}}.$

Proof

As before, we construct $\widetilde{f}$ by taking the product of f and the normalized spherical vector in ${\text {Ind}}_B^G (s+\frac{1}{2})\alpha .$ (Note that $\mathbf {X}_{T,{{\text {un}}}} = \mathbf {X}_{M,{{\text {un}}}} + \mathbb {C}\alpha$.) Then for $\chi \in \mathbf {X}_M$ and $s \in \mathbb {C}$

$$\begin{aligned} M^*(w_\alpha , \chi +s\alpha ).\widetilde{f}_{\chi + s\alpha } = f_\chi \cdot (2s-1) \cdot \frac{\zeta (2s)}{\zeta (2s+1)}\cdot f_{-s}. \end{aligned}$$

Again, $f_{-1/2}$ is the constant function 1,  so we get a nonzero scalar multiple of $f_{\chi }.$

Remark 3.2

For each fixed $\chi ,$ the operator $M^*(w_\alpha , \chi +\frac{\alpha }{2})$ maps $I_B^G(\chi + \frac{\alpha }{2})$ onto $I_P^G(\chi ) \subset I_B^G(\chi - \frac{\alpha }{2}).$ The key point is that this extends to a map from flat sections of $\mathcal I_B^G(\chi _0)$ to flat sections of $\mathcal I_P^G(\chi _0).$

Proposition 3.3

Take $P = MU$ with $\Phi (T,M) = \{ \pm \alpha \}$ and f a flat section of $\mathcal I_P^G(\chi _0).$ Assume $\langle \alpha ^\vee , \chi _0\rangle = 0.$ Choose $\widetilde{f}$ a flat section of $\mathcal I_B^G(\chi _0)$ such that $M^*(w_\alpha , \chi +\frac{\alpha }{2}).\widetilde{f}|_{\chi _0 + X_{M,un}} = f.$ Define $E_B^{\alpha *}\widetilde{f}(g,\chi ) =( \langle \chi , \alpha ^\vee \rangle _{{\text {un}}}-1) E_B.\widetilde{f}(g,\chi + \frac{\alpha }{2})$ Then

$$\begin{aligned} E_B^{\alpha *}.\widetilde{f}(g,\chi )= E_P.f(g, \chi ), \quad ( \forall \chi \in \chi _0+ X_{M,un}, \ g \in G(\mathbb {A})). \end{aligned}$$

Proof

The two sides have the same constant term, namely

$$\begin{aligned} \sum _{w \alpha> 0 } M\left( w, \chi -\frac{\alpha }{2}\right) \cdot M^*\left( w_\alpha , \chi +\frac{\alpha }{2}\right) \cdot \widetilde{f} = \sum _{w \alpha > 0 } M\left( w,\chi - \frac{\alpha }{2}\right) \cdot f. \end{aligned}$$

Hence their difference is both a cusp form and a linear combination of Eisenstein series. As such, it is zero.

3.2 An identity of ramified Eisenstein series on $G_2$

Theorem 3.4

Take $\eta \in \mathbf {X}_{GL_1},$ nontrivial quadratic, and $\widetilde{f}$ a flat section of $\mathcal I_B^G[0, \eta ].$ Let f be the flat section of $\mathcal I_P^G[0,\eta ]$ determined by $\widetilde{f}$ as in Proposition 3.1. The standard intertwining operator $M(w_\alpha w_\beta , [1, \eta -1])$ is an isomorphism. Let $\widetilde{h}$ be the flat section of $\mathcal I_B^G[\eta , 0]$ satisfying $\widetilde{h}_{[\eta , 1]} = M(w_\alpha w_\beta ,[1, \eta -1]).\widetilde{f}_{[1, \eta -1]}.$ Let h be the flat section of $\mathcal I_Q^G[\eta , 0]$ determined by $\widetilde{h}$ as in Proposition 3.1. Then $E_P.f$ has a pole at $[0, \eta +\frac{1}{2}],$ $E_Q.h$ has a pole at $[\eta +\frac{1}{2}, 0],$ and the two residues are the same.

Proof

Note that $\alpha = [2, -3]$ and $\beta = [-1,2].$ Note also that

$$\begin{aligned} w_\alpha w_\beta [0,1] = [-1,2] = \beta \equiv \varpi _1 \pmod 2, \quad w_\beta w_\alpha \alpha =[0,1]. \end{aligned}$$

It follows that

$$\begin{aligned} w_\alpha w_\beta [0, s+\eta ] + u \frac{\alpha }{2} = [u+\eta , 0] + s \frac{\beta }{2}. \end{aligned}$$

Now,

$$\begin{aligned}&E_P[0, s+\eta ]\cdot f = \left( u -\frac{1}{2}\right) E_B([0,s+\eta ]+u \frac{\alpha }{2})\widetilde{f}|_{u=\frac{1}{2}}, \\&\quad E_Q[u+\eta , 0]\cdot h = \left( s-\frac{1}{2}\right) E_B([u+\eta ,0]+ s\frac{\alpha }{2})\widetilde{h}|_{s=\frac{1}{2}}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} E_B\left( [0,s+\eta ]+u \frac{\alpha }{2}\right) \widetilde{f}&= E_B\left( [u+\eta ,0 ]+s \frac{\alpha }{2}\right) M\left( w_\alpha w_\beta , [0,s+\eta ]+u \frac{\alpha }{2}\right) \widetilde{f}\\&= E_B\left( [u+\eta ,0]+ s\frac{\alpha }{2}\right) \left( \widetilde{h}+ \text { higher order terms.}\right) \end{aligned} \end{aligned}$$

It follows that the residue of $E_P\cdot f$ at $s = \frac{1}{2}$ and that of $E_Q\cdot h$ at $u=\frac{1}{2}$ are two different expressions for the value of the meromorphic continuation of $(s-\frac{1}{2})(u-\frac{1}{2})E_B([0, s+\eta ]+ u\frac{\alpha }{2})\cdot \widetilde{f}$ to $s=u=\frac{1}{2}.$ $\square$

3.3 An identity of unramified Eisenstein series on $G_2$

In this section we prove an intriguing identity between unramified $G_2$ Eisenstein series. This result was previously obtained by D. Jiang in unpublished work.

Define [a, b] as before and write $f^\circ _{[a,b]}$ for the spherical vector in the corresponding induced representation. Let $c(s) = \frac{\zeta (s)}{\zeta (s+1)}.$ Identify $\mathbf {X}_{GL_1,{{\text {un}}}}$ with $\mathbb {C}$ as usual. Recall that

$$\begin{aligned} M(w,[a,b])\cdot f^\circ _{[a,b]} = \prod _{\alpha > 0, w \alpha < 0 } c( [a,b]\circ \alpha ^\vee )\cdot f_{w\cdot [a, b]}^\circ \end{aligned}$$

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

$$\begin{aligned} c(j+s) = \sum _{i = {\text {ord}}_j(c)}^\infty c_{i,j} s^i, \quad {\text {ord}}_j(c) = {\left\{ \begin{array}{ll} 1, &{} j = -1, \\ -1, &{} j = 1, \\ 0, &{} j \in \mathbb {Z}\smallsetminus \{ \pm 1\}. \end{array}\right. } \end{aligned}$$

Also $c(s) c(-s) = 1$ and $c(0) = -1,$ which implies that

$$\begin{aligned} c_{1,-1}= & {} -c_{-1,1}^{-1}\quad c_{2,-1} = -\frac{c_{0,1}}{c_{-1,1}^2}\quad c_{3,-1} =\frac{c_{1,1}}{c_{-1,1}^2}-\frac{c_{0,1}^2}{c_{-1,1}^3}\\ c_{0,0}= & {} -1,\quad c_{2,0} = -\frac{1}{2} c_{1,0}^2 . \end{aligned}$$

Theorem 3.5

The meromorphic function

$$\begin{aligned} E_Q f^\circ _{[3u+2,-1]} - \frac{1}{3} c(u) c(3u-1) E_P f^\circ _{[-1, u+1]} \end{aligned}$$

vanishes identically at $u=0.$

Remark 3.6

It would be interesting to make sense of this identity in terms of the functional equations of the Eisenstein series $E_B.f^\circ ,$ which are parametrized by the Weyl group of $G_2.$ However, this is not so trivial, as $[3u+2, -1]$ is not in the same Weyl orbit as $[-1, u+1]$ for $u \ne 0.$

Proof

It suffices to prove that the constant term vanishes. We compute the constant terms of $E_Q\cdot f^\circ _{[3u+2,-1]}$ and $E_P\cdot f^\circ _{[-1, u+1]}.$ We have

$$\begin{aligned} \begin{aligned} (E_Q\cdot f^\circ _{[3u+2,-1]})_B&= f^\circ _{[3u+2,-1]} + c(3u+2) f^\circ _{[-2-3u, 3u+1]} \\&\quad +\,c(3u+1)c(3u+2)\left( f^\circ _{[6u+1,-1-3u]}+c(3u)c(3u-1)c(6u+1)f^\circ _{[1-3u,-1]}\right) \\&\quad +\, c(3u+1)c(3u+2)c(6u+1) \left( f^\circ _{[-6u-1,3u]} + c(3u) f^\circ _{[3u-1,-3u]}\right) \\ (E_P.f^\circ _{[-1,u+1]})_B&=f^\circ _{[-1,u+1]} + c(u+1) f^\circ _{[3u+2,-1-u]} \\&\quad +\,c(u+1)c(3u+2) f^\circ _{[-3u-2,2u+1]} \\&\quad +\, c(u+1)c(3u+2)c(2u+1)f^\circ _{[3u+1,-2u-1]} \\&\quad +\, c(u+1)c(3u+2)c(2u+1)c(3u+1) \left( f^\circ _{[-3u-1,u]}+c(u)f^\circ _{[-1,-u]} \right) \end{aligned} \end{aligned}$$

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. In $(E_Q\cdot f^\circ _{[3u+2,-1]})_B$ it does not appear. In $- \frac{1}{3} c(u) c(3u-1)(E_P\cdot f^\circ _{[-1,u+1]})_B$ the contribution is $- \frac{1}{3} c(u) c(3u-1) f^\circ _{[3u+2,-1]},$ 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

$$\begin{aligned} f_{[a,b]}^\circ (ntk) = |t^{\varpi _1}|^{a+1} |t^{\varpi _2}|^{b+1}, \quad (n \in N(\mathbb {A}), \ t \in T(\mathbb {A}), \ k \in K), \end{aligned}$$

where N is the unipotent radical of B. Hence

$$\begin{aligned} f_{[a,b]+u[c,d]}^\circ (ntk) = f_{[a,b]}(ntk) \cdot \left( \sum _{m=0}^\infty \frac{1}{m!} [(\log |t^{\varpi _1}|c + \log |t^{\varpi _2}|d)\cdot u]^m \right) . \end{aligned}$$

Write $\langle c, d\rangle$ for the mapping

$$\begin{aligned} (ntk) \mapsto \log |t^{\varpi _1}|c + \log |t^{\varpi _2}|d, \quad (c,d \in \mathbb {Z}, n \in N(\mathbb {A}), \ t \in T(\mathbb {A}), k \in K). \end{aligned}$$

Then we have

$$\begin{aligned} f_{[a,b]+u[c,d]}^\circ = f_{[a,b]}^\circ \cdot \sum _{m=0}^\infty \frac{\langle c, d \rangle ^m}{m!} u^m. \end{aligned}$$

Note that $\langle c_1, d_1 \rangle + \langle c_2, d_2 \rangle = \langle c_1+c_2, d_1+d_2\rangle .$ We compare the contributions to the constant terms of $E_Q \cdot f^\circ _{[3u+2,-1]}$ and $E_P \cdot f^\circ _{[-1, u+1]}$ which lie in the $[1,-1]$ eigenspace when $u=0.$ The relevant portion of $\frac{1}{3} c(u)c(3u-1)(E_P\cdot f^\circ _{[-1, u+1]})_B$ is

while the relevant portion of $E_Q \cdot f^\circ _{[s+2,-1]}$ is

It makes sense to simplify ([1,-1],Q) before substituting $s=3u.$ However, notice that once this substitution is made, the factor of $c(3u+2)$ appears in both ([1,-1],P) and ([1,-1],Q). So, we may omit it from both. We expand the remainder of ([1,-1],P).

$$\begin{aligned}&\frac{1}{3} c(u) c(3u-1)c(u+1)c(2u+1)f^\circ _{[3u+1,-2u-1]} \nonumber \\= & {} \frac{1}{3}(-1+c_{1,0} u +O(u^2))\left( -\frac{3}{c_{-1,1}} u -9 \frac{c_{0,1}}{c_{-1,1}^2} u^2 +O(u^3)\right) \left( \frac{c_{-1,1}}{u} + c_{0,1} +O(u)\right) \nonumber \\&\quad \times \, \left( \frac{c_{-1,1}}{2u}+c_{0,1} + O(u)\right) f_{[1,-1]}^\circ (1+ \langle 3,-2\rangle u + O(u^2)) \nonumber \\= & {} f_{[1,-1]}^\circ \left( \frac{c_{-1,1}}{2u} -\frac{c_{-1,1}c_{1,0}}{2} + 3c_{0,1} +\frac{c_{-1,1}}{2} \langle 3 , -2 \rangle + O(u) \right) \end{aligned}$$

(3.7)

We simplify the expression in brackets in ([1,-1],Q)

$$\begin{aligned} \begin{aligned}&f^\circ _{[2s+1,-1-s]}+c(s)c(s-1)c(2s+1)f^\circ _{[1-s,-1]}\\&\quad = f_{[1,-1]}^\circ \left( 1+ s\langle 2, -1\rangle + (-1 + c_{1,0} s)\left( -\frac{s}{c_{-1,1}}- \frac{c_{0,1}s^2}{c_{-1,1}^2}\right) \right. \\&\left. \qquad \times \,\left( \frac{c_{-1,1}}{2s}+c_{0,1}\right) (1+\langle -1, 0\rangle s)+O(s^2) \right) \\&\quad = f_{[1,-1]}^\circ \left( 1+ s\langle 2, -1\rangle + \frac{1}{2} \left( 1+ s [ -c_{1,0} + \frac{3c_{0,1}}{c_{-1,1}} + \langle -1,0\rangle ]\right) \right) \\&\quad = f_{[1,-1]}^\circ \left( \frac{3}{2}+ \frac{s}{2}\left( \langle 3, -2\rangle -c_{1,0} + \frac{3c_{0,1}}{c_{-1,1}}\right) \right) \end{aligned} \end{aligned}$$

Multiplying by $c(s+1) = (c_{-1,1}/s+ c_{0,1} + O(s))$ yields

$$\begin{aligned} \begin{aligned}&f_{[1,-1]}^\circ \left( \frac{3c_{-1,1}}{2s}+ \frac{c_{-1,1}}{2}\left( \langle 3, -2\rangle -c_{1,0} + \frac{3c_{0,1}}{c_{-1,1}} \right) +\frac{3}{2} c_{0,1} +O(s) \right) \\&= f_{[1,-1]}^\circ \left( \frac{3c_{-1,1}}{2s}+ \frac{c_{-1,1}}{2}\langle 3, -2\rangle - \frac{c_{1,0}c_{-1,1}}{2} + {3c_{0,1}}+O(s) \right) , \end{aligned} \end{aligned}$$

and now substituting $s=3u$ yields (3.7). The other eigenspaces are treated similarly. $\square$

4 Global zeta integral involving a Q-Eisenstein series

In this section we consider the degenerate Eisenstein series $E_Q.h$ on $G_2(\mathbb {A}),$ where $Q= LV$ is the parabolic subgroup whose Levi contains the root subgroup attached to the long simple root. We let $H_\rho$ be a quasisplit subgroup of $G_2$ of type $A_2$ defined as in [12]. Thus $\rho \in F^\times$ and $H_\rho$ is isomorphic to $SL_3$ if $\rho$ is a square, and a quasisplit unitary group attached to the corresponding quadratic extension if F is $\rho$ is a nonsquare. For any normalized character $\chi _0$ of $L(F) \backslash L(\mathbb {A})$ and any flat section h of $I=\mathcal I_Q^{G_2}(\chi _0)$ the restriction of $E_P.h$ to $H_\rho (\mathbb {A})$ is a smooth function of moderate growth on $H_\rho (F) \backslash H_\rho (\mathbb {A}),$ so it may be integrated against a cuspform on $H_\rho .$ In the split case, this integral is identically zero for all $\chi _0, h,$ as shown in Proposition 4.6 of [10]. We extend the same idea to the general case.

First we need to analyze $Q(F) \backslash G_2(F)/H_\rho (F).$ Following [12], we regard $G_2$ as a subgroup of (split) $SO_8$ preserving the quadratic form ${\underline{\mathrm{x}}} \mapsto {}_t {\underline{\mathrm{x}}}\cdot {\underline{\mathrm{x}}}.$ Here $_t$ is the “other transpose” (as in [12]). Then $G_2$ is contained in the group $SO_7$ preserving the subspace $V_0:= \{ {\underline{\mathrm{x}}} ={} ^t \begin{bmatrix}x_1&\dots&x_8\end{bmatrix}: x_4 = x_5 \}.$ The group $H_\rho$ is the stablizer of $v_\rho : = {}^t\begin{bmatrix}0&0&1&0&0&\rho&0&0\end{bmatrix}$ in $G_2.$ Thus $G_2(F)/H_\rho (F)$ may be identified with the orbit $O_\rho := G_2(F) \cdot v_\rho ,$ and $Q(F) \backslash G_2(F)/H_\rho (F)$ can be identified with the set of Q(F)-orbits in $O_\rho .$

Lemma 4.1

The orbit $O_\rho$ is equal to $\{ {\underline{\mathrm{x}}} \in V_0: _t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho \}.$ If $\rho$ is not square it is a union of two Q(F) orbits namely

$$\begin{aligned} \left\{ {\underline{\mathrm{x}}} = \begin{bmatrix}x_1&\dots&x_8 \end{bmatrix}\in O_\rho : x_8 \ne 0\right\} ,\quad \left\{ {\underline{\mathrm{x}}} = \begin{bmatrix}x_1&\dots&x_8 \end{bmatrix}\in O_\rho : x_8 = 0\right\} . \end{aligned}$$

If $\rho = a^2$ then $\{ {\underline{\mathrm{x}}} = \begin{bmatrix}x_1&\dots&x_8 \end{bmatrix}\in O_\rho : x_8 \ne 0\}$ is still a single Q(F)-orbit, while

$$\begin{aligned} \left\{ {\underline{\mathrm{x}}} = \begin{bmatrix}x_1&\dots&x_8 \end{bmatrix}\in O_\rho : x_8 = 0\right\} \end{aligned}$$

is a union of three orbits, viz.

$$\begin{aligned}&\left\{ {\underline{\mathrm{x}}} = \begin{bmatrix}x_1&\dots&x_8 \end{bmatrix}\in O_\rho : x_8 = 0, \{ x_6, x_7\} \ne \{ 0\}\right\} ,\\&\left\{ \begin{bmatrix}x_1&x_2&x_3&a&a&0&0&0\end{bmatrix}\right\} , \quad \text { and } \left\{ \begin{bmatrix}x_1&x_2&x_3&-a&-a&0&0&0\end{bmatrix}\right\} . \end{aligned}$$

Proof

It’s clear that each element ${\underline{\mathrm{x}}}$ of $G_2(F) \cdot v_\rho$ satisfies $_t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho .$ Let $v_\rho ' = {}^t \begin{array}{lllll}[\rho&0&\dots&0&1 ].\end{array}$ Then a suitable representative for $s_\alpha s_\beta s_\alpha$ maps $v_\rho$ to $v_\rho '.$ One checks that the second, third, fourth, sixth, and seventh elements of the last column of an element of the standard maximal unipotent subgroup of $G_2(F)$ can be chosen arbitrarily and it follows that the Q(F) orbit of $v_\rho '$ (and also of $v_\rho$) contains every element of $\{ {\underline{\mathrm{x}}} = \begin{array}{lll}[ x_1&\dots&x_8 ]\end{array} \in O_\rho : x_8 \ne 0\}.$ On the other hand, this subset is clearly Q(F)-stable, so it is the Q(F) orbit of $v_\rho '.$ Likewise, a suitable representative of $s_\beta s_\alpha$ maps $v_\rho$ to $v_\rho ''=^t \begin{array}{lllllll}[ 0&\rho&0&\dots&0&1&0] \end{array}$ and the orbit of $v_\rho ''$ under the Borel of $G_2(F)$ is readily seen to be all ${\underline{\mathrm{x}}}$ with $_t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho , \ x_8 = 0, \ x_7 \ne 0.$ Clearly, each vector with $x_6, x_7$ not both zero is Q(F)-equivalent to one with $x_7 \ne 0.$ It follows that $\{ {\underline{\mathrm{x}}} : _t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho , \ x_8 = 0, \{ x_6, x_7\} \ne 0 \}$ is the Q(F) orbit of $v_\rho ''.$ Recall that ${\underline{\mathrm{x}}} \in V_0$ forces $x_4 = x_5.$ If $x_6=x_7=x_8= 0$ we get $_t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2x_4^2,$ so that $_t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho \implies \rho = x_4^2.$ If $\rho$ is not a square this is impossible. It follows that the two Q(F) orbits already described exhaust all vectors with $_t {\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho .$ If $\rho = a^2$ we have the two additional subsets described. Each is readily seen to be a Q(F)-orbit. Let $X_{-\alpha }$ 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 $-\alpha .$ Let $x_{-\alpha }(r) = \exp ( r X_{-\alpha }).$ (This is a polynomial formula, because $X_{-\alpha }$ is nilpotent.) Then

$$\begin{aligned} x_{-\alpha }(\pm a) \cdot v_{a^2} = \ ^t \begin{array}{llllllll}[0&0&1&\pm a&\pm a&0&0&0]\end{array}, \end{aligned}$$

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 $\{ {\underline{\mathrm{x}}} \in V_0: _t{\underline{\mathrm{x}}} \cdot {\underline{\mathrm{x}}} = 2\rho \}.$ This completes the proof. $\square$

Corollary 4.2

If $\rho$ is not a square, then $\{e, s_\alpha s_\beta \}$ is a set of representatives for $Q(F) \backslash G_2(F) / H_\rho (F).$ If $\rho = a^2,$ then $\{e, s_\alpha s_\beta , x_{-10}(a), x_{-10}(-a)\}$ is a set of representatives.

Proof

Let each element act on $v_\rho$ and consider the explicit description of the Q(F)-orbits in $O_\rho .$

4.1 A certain period

Let $\varphi _{32}: SL_2 \rightarrow G_2$ be the mapping

$$\begin{aligned} \begin{pmatrix}a&{} b \\ c &{} d \end{pmatrix}\mapsto \begin{pmatrix}a&{}&{}&{}&{}&{}&{}b&{}\\ &{}a&{}&{}&{}&{}&{}&{}b\\ &{}&{}1&{}&{}&{}&{}&{}\\ &{}&{}&{}1&{}&{}&{}&{}\\ &{}&{}&{}&{}1&{}&{}&{}\\ &{}&{}&{}&{}&{}1&{}&{}\\ c&{}&{}&{}&{}&{}&{}d&{}\\ &{}c&{}&{}&{}&{}&{}&{}d\\ \end{pmatrix}. \end{aligned}$$

The image is contained in $H_\rho$ for every $\rho .$ (Cf. (2) on p. 195 of [12].) If we embed $H_\rho (F)$ into $GL_3(E)$ using the identification $E^3$ with the six dimensional subspace of $F^8$ stabilized by $H_\rho$ as in [12] remarks 2, then $\varphi _{32}$ corresponds to the mapping

$$\begin{aligned} \begin{pmatrix}a&{}b\\ c&{}d \end{pmatrix}\mapsto \begin{pmatrix}a&{}&{} \frac{b}{2\tau }\\ &{}1&{}\\ 2\tau c &{}&{} d \end{pmatrix}, \end{aligned}$$

where $\tau \in E$ satisfies $\tau ^2 = \rho .$ From this point of view, the image of $\varphi _{32}$ is a smaller special unitary group, and we denote in $H'_\rho .$ We consider the period

$$\begin{aligned} \mathcal P(\varphi ) = \int \limits _{H'_\rho (F)\backslash H'_\rho (\mathbb {A})} \varphi (h) \, dh. \end{aligned}$$

(4.3)

We say that an automorphic representation $\pi$ is $H'_\rho$-distinguished if $\mathcal P$ does not vanish identically on it.

Proposition 4.4

Let h be a flat section of $\mathcal I_Q^{G_2}(\chi _0)$ and $\varphi$ a cuspform defined on $H_\rho (\mathbb {A}).$ Assume that $s \in \mathbf {X}_{Q, {{\text {un}}}}$ is not a pole of $E_Q.$ Then for a suitable measure on the space $H'_\rho (\mathbb {A})\backslash H_\rho (\mathbb {A})$ we have

$$\begin{aligned} \int \limits _{H_\rho (F)\backslash H_\rho (\mathbb {A})} \varphi (g) E_Q \cdot h(g, s) \, dg = \int \limits _{H'_\rho (\mathbb {A})\backslash H_\rho (\mathbb {A})} h(s_\alpha s_\beta g, s) \mathcal P( R(g)\cdot \varphi ) \, dg. \end{aligned}$$

(4.5)

(Here, R denotes right translation.) In particular, the integral vanishes identically on any cuspidal automorphic representation which is not $H'_\rho$-distinguished. If $H_\rho$ is split, then no cuspidal automorphic representation is $H'_\rho$-distinguished, and the integral (4.5) is always zero.

Proof

For s in the domain of convergence for $E_Q,$ our integral is equal to

$$\begin{aligned} \sum _{ \gamma \in Q(F) \backslash G_2(F) / H_\rho (F) }\int _{H_\rho (F) \cap \gamma ^{-1} Q(F) \gamma \backslash H_\rho (\mathbb {A})} h(\gamma g,s) \varphi (g) \, dg. \end{aligned}$$

If $\gamma$ is the identity, or if $\rho = a^2$ and $\gamma = x_{-\alpha }(\pm a)$ then $H_\rho \cap \gamma ^{-1} Q \gamma$ contains the unipotent radical of the standard Borel subgroup of $H_\rho .$ Hence the function $g \mapsto h(\gamma g,s)$ is invariant by this subgroup on the left, and the integral over $H_\rho (F) \cap \gamma ^{-1} Q(F) \gamma \backslash H_\rho (\mathbb {A})$ factors through the mapping that sends $\varphi$ to its constant term– which is zero. If $\gamma = s_\alpha s_\beta ,$ then $H_\rho \cap \gamma ^{-1} Q \gamma =H'_\rho ,$ and $\gamma H_\rho ' \gamma ^{-1}$ is contained in the derived group of the Levi L of Q. Hence the function $g \mapsto f(s_\alpha s_\beta g, s)$ is invariant by $H_\rho '(\mathbb {A})$ on the right. We are reduced to showing that

$$\begin{aligned} \int \limits _{H'_\rho (F) \backslash H_\rho (\mathbb {A})} h(s_\alpha s_\beta g, s) \varphi (g) \, dg = \int \limits _{H'_\rho (\mathbb {A}) \backslash H_\rho (\mathbb {A})} h(s_\alpha s_\beta g_1, s) \int \limits _{H'_\rho (F)\backslash H'_\rho (\mathbb {A})}\varphi (h'_\rho g_1)\, dh'_\rho \, dg_1 \end{aligned}$$

for a suitable measure $dg_1.$ Here, dg and $dh'_\rho$ are the Haar measures on the respective groups. The existence of a suitable invariant measure on each of the corresponding local homogeneous spaces $H'_\rho (F_v)\backslash H_\rho (F_v)$ follows from Proposition 4.3.5 of [2]. The existence of $dg_1$ then follows easily from expressing dg and $dh'_\rho$ in terms of the Haar measures on the local groups.

If $H_\rho$ is split, then $H'_\rho$ 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. $\square$

Remark 4.6

We may regard $H'_\rho$ as $SU_{1,1} \subset U_{1,1} \subset U_{2,1}$ and one could enlarge our period $\mathcal P,$ to an integral over $U_{1,1}(F)\backslash U_{1,1}(\mathbb {A})$ against a character. Such periods are considered in [7], where they are used to characterize the image of the theta lifting. Recall that the ${\text {Ad}}'$ L function of a representation in the image of this lifting should have a pole at $s=1.$ So, the emergence of the $H'_\rho$ period as an obstruction to proving holomorphy in general makes perfect sense.

4.2 Conclusions regarding poles of global zeta integrals

Theorem 4.7

Suppose that $\varphi$ generates a cuspidal automorphic representation which is not $H'_\rho$ distinguished. Then the global zeta integral $I(s, \varphi , f)$ defined in (1.1) has no poles in the half plane ${\text {Re}}(s) \ge \frac{1}{2},$ except for a simple pole at $s=\frac{2}{3}$ which can occur only when $\chi _0$ is cubic. Moreover, if $\chi _0$ is quadratic, then $I(\frac{1}{2}, \varphi , f )=0$ for all $\varphi$ and f.

Remark 4.8

In the split case, no cuspidal representation is $H'_\rho$ distinguished. For these purposes the trivial character is both cubic and quadratic.

Proof

In the case when $\chi$ 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 ${\text {Re}}(s)\ge \frac{1}{2}$ occur at $s=\frac{2}{3}$ when $\chi _0$ is trivial, quadratic, or cubic, and at $s= 1$ when $\chi _0$ is trivial. When $H_\rho$ is split and $\chi _0$ is trivial, the possibility of a pole at $s=1$ or a double pole at $s=2/3$ are ruled out by [10]. When $\chi _0$ is quadratic, the possibility of a pole at $\frac{2}{3}$ is ruled out by the same argument, using Theorem 3.4 and Proposition 4.4. The vanishing at $\frac{1}{2}$ when $\chi _0$ is quadratic follows from Proposition 2.6. $\square$

5 Nonvanishing of local zeta integrals

In this section we prove that for any fixed $s_0 \in \mathbb {C}$ 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” $L^S(3s, \chi )L^S(6s-2, \chi ^2) L^S(9s-3, \chi ).$ We take up some notations from [12]: for fixed $\rho \in F,$ $H_\rho$ 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. ), and $N_2$ is a two dimensional unipotent subgroup, with the property that $H_\rho \cap w_2^{-1} P w_2 = N_2 T_{{\text {sp}}},$ where $T_{{\text {sp}}}$ denotes the one dimensional maximal F-split torus contained in the standard Borel of $H_\rho .$ Finally, $\psi _N$ is a certain generic character of N. Details are found in [12]. It is convenient to identify $\mathbf {X}_{M,{{\text {un}}}}$ with $\mathbb {C}$ via the map $s \mapsto \delta _P^s.$

For $\rho \in F,$ $\pi$ a irreducible admissible $\psi _N$-generic representation of $H_\rho ,$ W in the $\psi _N$-Whittaker model $\mathcal W_{\psi _N}(\pi )$ of $\pi ,$ and $f \in {\text {Flat}}(\chi _0)$ we define

$$\begin{aligned} I(W, f; s)= \int _{N_2\backslash H_\rho } W(g) f_s( w_2 g) \, dg. \end{aligned}$$

Theorem 5.1

For any $\rho \in F^\times ,$ any irreducible admissible generic representation $\pi$ of $H_\rho ,$ any $\chi _0 \in \mathbf {X}_{M,0},$ and any fixed $s_0 \in \mathbb {C},$ there exist $W \in \mathcal W_{\psi _N}(\pi )$ and $f \in {\text {Flat}}(\chi _0)$ such that $I(W, f; s_0) \ne 0.$

Proof

Expressing the Haar measure on $H_\rho$ as a suitable product measure on the open Bruhat cell yields

$$\begin{aligned} I(W, f; s) = \int _{N_2 \backslash N} \int _{T} \int _{N^-} W( ntn^-) f_s( wntn^{-}) \delta _B^{-1}(t) \, dn^-\, dt\, dn. \end{aligned}$$

We may identify $N_2\backslash N$ with the complementary subgroup

$$\begin{aligned} N_2' := \left\{ \begin{pmatrix} 1&{}r&{}-\frac{r^2}{2}\\ &{}1&{}-r\\ &{}&{}1\end{pmatrix}: r \in F \right\} , \end{aligned}$$

which is $T_{{\text {sp}}}$-stable, and fix a fundamental domain $[T_{{\text {sp}}}\backslash T]$ for $T_{{\text {sp}}}$ in T,  to express I(W, f; s) as

where $\gamma$ is the common restriction of the two simple roots of $H_{\rho }$ to ${T_{{\text {sp}}}}$ (so that $\delta _B^{-1}(t_{{\text {sp}}})= t_{{\text {sp}}}^{-4\gamma }$) and

Note that the function $(n, t', n^- ) \mapsto P w_2 n t' n^-$ is a continuous injection of $N_2' \times [T_{{\text {sp}}}\backslash T] \times N^-$ into $P \backslash G_2.$ Now fix $s_0,$ and let $\phi _f(n, t', n^- ):= \delta _B(t')^{-1}f_{s_0}( w_2nt'n^{-}),$ which we view as a “test function” on $N_2' \times [T_{{\text {sp}}}\backslash T] \times N^-.$ First assume that F is nonarchimedean. Then for any smooth function $\phi _1$ of compact support defined on $N_2' \times [T_{{\text {sp}}}\backslash T] \times N^-$ we can choose f so that $\phi _f = \phi _1.$ But then

$$\begin{aligned} I(W, f; s_0) = I'(\phi _f*W; s_0), \end{aligned}$$

where $*$ denotes the action by convolution and $I'(W, s)$ is defined for $W\in \mathcal W_{\psi _N}(\pi )$ and $s\in \mathbb {C}$ by

$$\begin{aligned} I'(W, s):= \int _{T_{{\text {sp}}}} W(t_{{\text {sp}}}) t_{{\text {sp}}}^{w_2 \chi _0 + w_2 s -3 \gamma } \, dt_{{\text {sp}}}. \end{aligned}$$

Hence $I(W, f; s_0)$ vanishes for all $f \in {\text {Flat}}(\chi _0), W \in W_{\psi _N}(\pi )$ if and only if $I'(W; s_0)$ vanishes for all $W \in W_{\psi _N}(\pi ).$

But now let $\phi _2$ be a Schwartz function on F and $x: F \rightarrow H_\rho$ an embedding into N chosen so that $t_{{\text {sp}}}x(a) t_{{\text {sp}}}^{-1} = x(t_{{\text {sp}}}^\gamma a)$ and $\psi _N(x(a)) = \psi (a).$ Then

$$\begin{aligned} \begin{aligned} I'((\phi _2\circ x)*W, s)&= \int _{T_{{\text {sp}}}} \int _F W(t_{{\text {sp}}}x(a)) t_{{\text {sp}}}^{w_2 \chi _0 + w_2 s -3 \gamma }\phi _2(a) \, da \, dt_{{\text {sp}}}\\&= \int _{T_{{\text {sp}}}} \int _F W(t_{{\text {sp}}}) t_{{\text {sp}}}^{w_2 \chi _0 + w_2 s -3 \gamma }\psi (t_{{\text {sp}}}^\gamma a)\phi _2(a) \, da \, dt_{{\text {sp}}}\\&= \int _{T_{{\text {sp}}}} W(t_{{\text {sp}}}) t_{{\text {sp}}}^{w_2 \chi _0 + w_2 s -3 \gamma }\widehat{\phi }_2(t_{{\text {sp}}}^\gamma ) \, dt_{{\text {sp}}}. \end{aligned} \end{aligned}$$

Notice that $\widehat{\phi }_2$ is a Schwartz function which can be chosen arbitrarily. Clearly, we can now choose $W \in W_{\psi _N}(\pi )$ which does not vanish identically on $T_{{\text {sp}}}$ and then choose $\phi _2$ so that $I'((\phi _2\circ x)*W, s_0) \ne 0.$

This completes the proof in the nonarchimedean case. In the archimedean case, the same argument shows that the mapping $(W, f) \mapsto I(W, f; s_0)$ does not vanish identically on $\mathcal W_{\psi _N}(\pi ) \times {\text {SI}}_P^G (\chi _0 + s_0),$ where ${\text {SI}}$ 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 $(W, f) \mapsto I(W, f; s_0)$ can not vanish identically on $\mathcal W_{\psi _N}(\pi ) \times {\text {Ind}}_P^G (\chi _0 + s_0)$ either, completing the proof in this case. $\square$

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 $\pi _v$ or $\chi _v$ is ramified. Then the partial twisted adjoint L function $L^S(s, \pi , {\text {Ad}}\otimes \chi )$ has no poles in the half-plane ${\text {Re}}(s) \ge \frac{1}{2},$ except possibly for a simple pole at ${\text {Re}}(s) =1$ when $\chi$ is nontrivial and $\pi \cong \pi \otimes \chi$ (which forces $\chi$ to be cubic). If this pole is present, then it is inherited by the complete L function $L(s, \pi , {\text {Ad}}\times \chi ).$ Every other pole of $L(s, \pi , {\text {Ad}}\times \chi )$ in ${\text {Re}}(s)\ge \frac{1}{2}$ is a zero of the Hecke L function $L(s, \chi )$, and a pole of $\prod _{v \in S} L_v(s, \pi _v, {\text {Ad}}\times \chi _v).$

Proof

According to the results which we have proved so far, the global zeta integral $I(s, \varphi , f)$ has no poles in ${\text {Re}}(s)\ge \frac{1}{2}$ except possibly for a simple pole at ${\text {Re}}(s) = \frac{2}{3}$ which can occur only when $\chi$ is nontrivial cubic. By Theorem 5.1, these properties are inherited by the ratio of partial L functions⁸

$$\begin{aligned} \frac{L^S(3s-1, \pi , {\text {Ad}}' \otimes \chi )}{L^S(3s, \chi ) L^S(6s-2, \chi ^2) L^S(9s-3, \chi ^3)}, \end{aligned}$$

and then, since local L functions are nonvanishing meromorphic functions, by

$$\begin{aligned} \frac{L^S(3s-1, \pi , {\text {Ad}}' \otimes \chi )}{L(3s, \chi ) L(6s-2, \chi ^2) L(9s-3, \chi ^3)}. \end{aligned}$$

The product $L(3s, \chi ) L(6s-2, \chi ^2) L(9s-3, \chi ^3)$ has no poles in ${\text {Re}}(s)\ge \frac{1}{2}$ except for the simple pole of $L(6s-2, \chi ^2)$ at $s = \frac{1}{2}$ which occurs only if $\chi ^2$ is trivial. But we have seen that $I(\frac{1}{2} , \varphi , f)=0$ when $\chi$ 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 $L(s, \pi , {\text {Ad}}\times \chi ) = L(s, (\pi \otimes \chi ) \times \widetilde{\pi })/L(s, \chi ).$ 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 and occur if and only if $\pi \cong \pi \otimes \chi$. Thus, any other poles which appear must be zeros of the denominator. $\square$

Remark 6.3

The expression $L(s, \pi , {\text {Ad}}) = L(s, \pi \times \widetilde{\pi })/\zeta (s)$ also gives us a shorter proof that $L(s, \pi , {\text {Ad}})$ is holomorphic and nonvanishing at $s=1,$ without appealing to Theorem 3.5 and Proposition 4.4. The functional equations of the global Rankin–Selberg L-function and the global Dedekind zeta function give a functional equation of $L(s, \pi , {\text {Ad}}).$ By the functional equation, the set of poles of $L(s, \pi , {\text {Ad}})$ is symmetric as $s \mapsto 1-s.$

In the nonsplit case, the same argument gives the following result.

Theorem 6.4

Assume that G is nonsplit, and let S be a finite set of places, including all archimedean places such that $\pi _v$ and $\chi _v$ are unramified for $v \notin S.$ Then the partial twisted L function $L^S(s, \pi , {\text {Ad}}' \otimes \chi )$ has no poles in the half-plane ${\text {Re}}(s) \ge \frac{1}{2},$ except possibly at $s =1.$ The pole at $s=1$ can occur only when $\chi$ is trivial, quadratic, or cubic. If a pole occurs when $\chi$ is trivial or quadratic, then $\pi$ is $H'_\rho$-distinguished. The pole at $s=1$ is at most a double pole when $\chi$ is trivial, and at most a simple pole when $\chi$ is nontrivial quadratic or cubic.

Remark 6.5

In the nonsplit case we have $L(s, \pi , {\text {Ad}}' \times \chi ) = L(s, {\text {sbc}}(\pi ), {\text {Asai}} \times \chi )/L(s, \chi ),$ where ${\text {sbc}}$ denotes the stable base change lifting from $U_{2,1}$ to ${\text {Res}}_{E/F}GL_3,$ constructed in [16], and ${\text {Asai}}$ is the Asai representation. In the important special case $\chi = \chi _{E/F}$ this becomes $L(s, \pi , {\text {Ad}}) = L(s, {\text {sbc}}(\pi ), {\text {Asai}}\times \chi _{E/F})/L(s, \chi _{E/F}).$ It is proved in [16], that $L(s, {\text {sbc}}(\pi ), {\text {Asai}})$ will have a simple pole at $s=1$ if ${\text {sbc}}(\pi )$ is cuspidal, but also that ${\text {sbc}}(\pi )$ need not be cuspidal. Arguing as on p. 22 of [11], we may deduce that $L(s, {\text {sbc}}(\pi ), {\text {Asai}}\times \chi _{E/F})$ is holomorphic and nonvanishing at $s=1,$ and deduce that $L(s, \pi , {\text {Ad}})$ has the same property, still provided that ${\text {sbc}}(\pi )$ is cuspidal. A more thorough analysis of the nonsplit case is left to future work.