On a Rankin–Selberg integral of the L-function for $\widetilde{\mathrm {SL}}_2\times {\mathrm {GL}}_2$

Abstract

We present a Rankin–Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{\mathrm {SL}}_2\times {\mathrm {GL}}_2$. As an application, we show that certain Fourier–Jacobi type periods on $G_2$ are non-vanishing.

1 Introduction

Let F be a global field with the ring of adeles ${\mathbb {A}}$. We assume that the characteristics of F is not 2. We present in this paper a Shimura type integral on the exceptional group $G_2$ which represents the L-function

$$\begin{aligned} L(s,\widetilde{\pi }\times (\chi \otimes \tau ))L(s,\widetilde{\pi }\otimes (\chi \otimes \omega _\tau )), \end{aligned}$$

where $\widetilde{\pi }$ is an irreducible genuine cuspidal representation of $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$, $\tau$ is an irreducible generic cuspidal representation of ${\mathrm {GL}}_2({\mathbb {A}})$ and $\chi$ is the quadratic character of $F^\times \backslash {\mathbb {A}}^\times$ defined by $\chi (a)=\prod _v (a_v,-1)_{F_v}$, where $a=(a_v)_v \in {\mathbb {A}}^\times$ and $(~,~)_{F_v}$ is the Hilbert symbol on $F_v$.

To give more details about the integral, we introduce some notations. The group $G_2$ has two simple roots and we label the short root by $\alpha$ and the long root by $\beta$. Let $P=MV$ (resp. $P'=M'V'$) be the maximal parabolic subgroup of $G_2$ such that the root space of $\beta$ is in the Levi M (resp. the root space of $\alpha$ is in the Levi $M'$). The Levi subgroups M and $M'$ are isomorphic to ${\mathrm {GL}}_2$. Let J be the subgroup of P which is isomorphic to ${\mathrm {SL}}_2 < imes V$. Let $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$ be the metaplectic double cover of ${\mathrm {SL}}_2({\mathbb {A}})$. There is a Weil representation $\omega _\psi$ of $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$ for a nontrivial additive character $\psi$ of $F\backslash {\mathbb {A}}$. Let $\widetilde{\theta }_\phi$ be a corresponding theta series associated with a function $\phi \in {\mathcal {S}}({\mathbb {A}})$. Let $\tau$ be an irreducible cuspidal automorphic representations of ${\mathrm {GL}}_2({\mathbb {A}})$. For $f_s\in {\mathrm {Ind}}_{P'({\mathbb {A}})}^{G_2({\mathbb {A}})}(\tau \otimes \delta _{P'}^s)$, we can form an Eisenstein series $E(g,f_s)$ on $G_2({\mathbb {A}})$. Let $\widetilde{\pi }$ be an irreducible genuine cuspidal automorphic forms of $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$. For a cusp form $\widetilde{\varphi }\in \widetilde{\pi }$, we consider the integral

$$\begin{aligned} I(\widetilde{\varphi }, \phi ,f_s)=\int _{{\mathrm {SL}}_2(F)\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\backslash V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)E(vg,f_s)dvdg. \end{aligned}$$

Our main result is the following

Theorem 1.1

The above integral is absolutely convergent for ${\mathrm {Re}}(s)\gg 0$ and can be meromorphically continued to all $s\in {\mathbb {C}}$. When ${\mathrm {Re}}(s)\gg 0$, the integral $I(\widetilde{\varphi },\phi ,f_s)$ is Eulerian. Moreover, at an unramified place v, the local integral represents the L-function

$$\begin{aligned} \frac{L(3s-1,\widetilde{\pi }_v\times (\chi _v\otimes \tau _v))L(6s-5/2,\widetilde{\pi }_v\otimes (\chi _v\otimes \omega _{\tau _v}))}{L(3s-1/2,\tau _v)L(6s-2,\omega _{\tau _v})L(9s-7/2,\tau _v\otimes \omega _{\tau _v})}. \end{aligned}$$

This is Theorem 3.1 and Proposition 4.6. We remark that Ginzburg–Rallis–Soudry gave integral representations for L-functions of generic cuspidal representations of $\widetilde{{\mathrm {Sp}}}_{2n}\times {\mathrm {GL}}_m$ in [8] using symplectic groups. It is still interesting to have different integral representations. As an application of Theorem 1.1, we show that if $Wd_\psi (\widetilde{\pi })=\chi \otimes \tau$, then a Shimura type period with respect to $\widetilde{\pi }$ and the residue of Eisenstein series on $G_2$ is non-vanishing, where $Wd_\psi$ is the Shimura–Waldspurger lift. It is an interesting theme in number theory to investigate the relations between poles of L-functions and non-vanishing of automorphic periods. There are many examples of this kind relations. See [5, 7, 9] for some examples. The non-vanishing results of automorphic periods have many interesting applications in automorphic forms. We expect the non-vanishing period in our case would be useful on problems related to the residue spectrum of $G_2$.

There are several known Rankin–Selberg integrals on $G_2$ which represents different L-functions and have many applications, see [4,5,6] for example. The integral $I(\widetilde{\varphi },\phi ,f_s)$ can be viewed as a dual integral of the standard $G_2$ L-function integral in [5] in the following sense. The integral $I(\widetilde{\varphi },\phi ,f_s)$ is an integral of a triple product of a cusp form on $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$, a theta series and an Eisenstein series on $G_2({\mathbb {A}})$, while the integral in [5] is an integral of a triple product of a cusp form on $G_2({\mathbb {A}})$, a theta series and an Eisenstein series on $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$. The integral in [6] is also in a similar pattern, which is an integral of a triple product of a cusp form on ${\mathrm {SL}}_2({\mathbb {A}})$, a theta series and an Eisenstein series on a cover of $G_2({\mathbb {A}})$. The results presented here were known for D. Ginzburg. But we still think that it might be useful to write up the details.

2 The group ${ {G}}_2$

2.1 Roots and Weyl group for ${ {G}}_2$

Let ${{G}}_2$ be the split algebraic reductive group of type ${{G}}_2$ (defined over ${\mathbb {Z}}$). The group ${{G}}_2$ has two simple roots, the short root $\alpha$ and the long root $\beta$. The set of the positive roots is $\Sigma ^+=\left\{ {\alpha , \beta , \alpha +\beta , 2\alpha +\beta , 3\alpha +\beta , 3\alpha +2\beta }\right\}$. Let ( ,  ) be the inner product in the root system and $\langle {~,~} \rangle$ be the pair defined by $\langle {\gamma _1,\gamma _2} \rangle =\frac{2(\gamma _1,\gamma _2)}{(\gamma _2,\gamma _2)}$. For the root space ${{G}}_2$, we have the relations:

$$\begin{aligned} \langle {\alpha ,\beta } \rangle =-1, \langle {\beta , \alpha } \rangle =-3. \end{aligned}$$

For a root $\gamma$, let $s_\gamma$ be the reflection defined by $\gamma$, i.e., $s_\gamma (\gamma ')=\gamma '-\langle {\gamma ',\gamma } \rangle \gamma$. We have the relation

$$\begin{aligned} s_\alpha (\beta )=3\alpha +\beta , s_\beta (\alpha )=\alpha +\beta . \end{aligned}$$

The Weyl group $\mathbf{{W}}=\mathbf{{W}}({{G}}_2)$ of ${{G}}_2$ has 12 elements, which is explicitly given by

$$\begin{aligned} \mathbf{{W}}= & {} \left\{ 1, s_\alpha , s_\beta , s_\alpha s_\beta , s_\beta s_\alpha , s_\alpha s_\beta s_\alpha , s_\beta s_\alpha s_\beta ,\right. \\&\quad \left. (s_\alpha s_\beta )^2, (s_\beta s_\alpha )^2, s_\beta (s_\alpha s_\beta )^2, s_\alpha (s_\beta s_\alpha )^2, (s_\alpha s_\beta )^3\right\} . \end{aligned}$$

For a root $\gamma$, let $U_\gamma \subset G$ be the root space of $\gamma$, and let $\mathbf{{x}}_\gamma : F\rightarrow U_\gamma$ be a fixed isomorphism which satisfies various Chevalley relations, see Chapter 3 of [14]. Among other things, $\mathbf{{x}}_{\gamma }$ satisfies the following commutator relations:

$$\begin{aligned} \begin{aligned} {[}\mathbf{{x}}_\alpha (x), \mathbf{{x}}_\beta (y)]&= \mathbf{{x}}_{\alpha +\beta }(-xy)\mathbf{{x}}_{2\alpha +\beta }(-x^2 y) \mathbf{{x}}_{3\alpha +\beta }( x^3 y)\mathbf{{x}}_{3\alpha +2\beta }(-2x^3 y^2)\\ {[}\mathbf{{x}}_\alpha (x), \mathbf{{x}}_{\alpha +\beta }(y)]&= \mathbf{{x}}_{2\alpha +\beta }( -2xy)\mathbf{{x}}_{3\alpha +\beta }(3x^2 y) \mathbf{{x}}_{3\alpha +2\beta }(3xy^2) \\ {[}\mathbf{{x}}_\alpha (x), \mathbf{{x}}_{2\alpha +\beta }(y)]&=\mathbf{{x}}_{3\alpha +\beta }( 3xy) \\ {[}\mathbf{{x}}_\beta (x), \mathbf{{x}}_{3\alpha +\beta }(y)]&=\mathbf{{x}}_{3\alpha +2\beta }(xy) \\ {[}\mathbf{{x}}_{\alpha +\beta }(x), \mathbf{{x}}_{2\alpha +\beta }(y)]&=\mathbf{{x}}_{3\alpha +2\beta }(3xy). \end{aligned} \end{aligned}$$

(2.1)

For all the other pairs of positive roots $\gamma _1, \gamma _2$, we have $[\mathbf{{x}}_{\gamma _1}(x), \mathbf{{x}}_{\gamma _2}(y)]=1$. Here $[g_1,g_2]=g_1^{-1}g_2^{-1}g_1g_2$ for $g_1,g_2\in G_2$. For these commutator relationships, see [12].

Following [14], we denote $w_\gamma (t)= \mathbf{{x}}_{\gamma }(t)\mathbf{{x}}_{-\gamma }(-t^{-1})\mathbf{{x}}_{\gamma }(t)$ and $w_\gamma =w_\gamma (1)$. Note that $w_\gamma$ is a representative of $s_\gamma$. Let $h_\gamma (t)=w_\gamma (t) w_\gamma ^{-1}$. Let T be the subgroup of G which consists of elements of the form $h_\alpha (t_1)h_\beta (t_2), t_1,t_2\in T$ and U be the subgroup of $G_2$ generated by $U_\gamma$ for all $\gamma \in \Sigma ^+$. Let $B=TU$, which is a Borel subgroup of $G_2$.

For $t_1,t_2\in \mathbb {G}_m,$ denote $h(t_1,t_2)=h_\alpha (t_1t_2)h_\beta (t_1^2t_2)$. From the Chevalley relation $h_{\gamma _1}(t)\mathbf{{x}}_{\gamma _2}(r) h_{\gamma _1}(t)^{-1}=\mathbf{{x}}_{\gamma _2}(t^{\langle {\gamma _2,\gamma _1} \rangle } r)$ (see [14, Lemma 20, (c)]), we can check the following relations

$$\begin{aligned} \begin{aligned} h^{-1}(t_1,t_2)\mathbf{{x}}_\alpha (r)h(t_1,t_2)&=\mathbf{{x}}_\alpha (t_2^{-1} r), \\ h^{-1}(t_1,t_2)\mathbf{{x}}_\beta (r)h(t_1,t_2)&=\mathbf{{x}}_\beta (t_1^{-1}t_2 r)\\ h^{-1}(t_1,t_2)\mathbf{{x}}_{\alpha +\beta }(r) h(t_1,t_2)&=\mathbf{{x}}_{\alpha +\beta }(t_1^{-1}r),\\ h^{-1}(t_1,t_2)\mathbf{{x}}_{2\alpha +\beta }(r) h(t_1,t_2)&= \mathbf{{x}}_{2\alpha +\beta }(t_1^{-1}t_2^{-1}r) \\ h^{-1}(t_1,t_2)\mathbf{{x}}_{3\alpha +\beta }(r)h(t_1,t_2)&=\mathbf{{x}}_{3\alpha +\beta }(t_1^{-1}t_2^{-2}r),\\ h^{-1}(t_1,t_2) \mathbf{{x}}_{3\alpha +2\beta }(r) h(t_1,t_2)&=\mathbf{{x}}_{3\alpha +2\beta }(t_1^{-2} t_2^{-1} r). \end{aligned} \end{aligned}$$

(2.2)

Thus the notation h(a, b) agrees with that of [5].

One can also check that

$$\begin{aligned} w_\alpha h(t_1,t_2) w_\alpha ^{-1}= h(t_1 t_2, t_2^{-1}), \quad w_\beta h(t_1,t_2) w_\beta ^{-1}= h(t_2,t_1). \end{aligned}$$

2.2 Subgroups

Let F be a field and denote $G=G_2(F)$. The group G has two proper parabolic subgroups. Let $P=M < imes V$ be the parabolic subgroup of G such that $U_\beta \subset M\cong {\mathrm {GL}}_2$. Thus the unipotent subgroup V is consisting of root spaces of $\alpha , \alpha +\beta , 2\alpha +\beta , 3\alpha +\beta ,3\alpha +2\beta$, and a typical element of V is of the form

$$\begin{aligned} \mathbf{{x}}_{\alpha }(r_1)\mathbf{{x}}_{\alpha +\beta }(r_2)\mathbf{{x}}_{2\alpha +\beta }(r_3)\mathbf{{x}}_{3\alpha +\beta }(r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5), r_i\in F. \end{aligned}$$

To ease the notation, we will write the above element as $[r_1,r_2,r_3,r_4,r_5]$. Denote by J the following subgroup of P

$$\begin{aligned} J={\mathrm {SL}}_2(F) < imes V. \end{aligned}$$

Let $V_1$ (resp. Z) be the subgroup of V which consists root spaces of $3\alpha +\beta$ and $3\alpha +2\beta$ (resp. $2\alpha +\beta , 3\alpha +\beta$ and $3\alpha +2\beta$). Note that P and hence J normalizes $V_1$ and Z. We will always view ${\mathrm {SL}}_2(F)$ as a subgroup of G via the inclusion ${\mathrm {SL}}_2(F)\subset M$. Denote by $A_{{\mathrm {SL}}_2}$, $N_{{\mathrm {SL}}_2}$ and $B_{{\mathrm {SL}}_2}$ the standard torus, the upper triangular unipotent subgroup and the upper triangular Borel subgroup of ${\mathrm {SL}}_2(F)$. Note that the torus element h(a, b) can be identified with

$$\begin{aligned} \begin{pmatrix}a&{}\\ &{}b \end{pmatrix}\in {\mathrm {GL}}_2(F)\cong M, \end{aligned}$$

and thus $A_{{\mathrm {SL}}_2}=\left\{ {h(a,a^{-1})|a\in F^\times }\right\}$ and $B_{{\mathrm {SL}}_2}=A_{{\mathrm {SL}}_2} < imes U_\beta$.

Let $P'=M'V'$ be the other maximal parabolic subgroup G with $U_\alpha$ in the Levi subgroup $M'$. The Levi $M'$ is isomorphic to ${\mathrm {GL}}_2(F)$, and from relations in (2.2), one can check that one isomorphism $M'\cong {\mathrm {GL}}_2(F)$ can be determined by

$$\begin{aligned} \begin{aligned}\mathbf{{x}}_\alpha (r)&\mapsto \begin{pmatrix}1&{} r\\ &{}1 \end{pmatrix},\\ h(a,b)&\mapsto \begin{pmatrix}ab&{}\\ &{}a \end{pmatrix}. \end{aligned} \end{aligned}$$

In particular, we see that $h(a,1)\in T\subset M'$ can be identified with ${\mathrm {diag}}(a,a).$ Let $\delta _{P'}$ be the modulus character of $P'$. One can check that $\delta _{P'}(m')=|\det (m')|^3$ for $m'\in M'$, where $\det (m')$ can be computed using the above isomorphism $M'\cong {\mathrm {GL}}_2(F)$.

2.3 Weil representation of $\widetilde{\mathrm {SL}}_2({\mathbb {A}}) < imes V({\mathbb {A}})$

In this subsection, we assume that F is a global field and ${\mathbb {A}}$ is its ring of adeles. In ${\mathrm {SL}}_2(F)$, we denote $t(a)={\mathrm {diag}}(a,a^{-1}),a\in F^\times$ and

$$\begin{aligned} n(b)=\begin{pmatrix}1&{} b\\ &{}1 \end{pmatrix}, b\in F. \end{aligned}$$

Denote $w^1=\begin{pmatrix}&{}1\\ -1&{} \end{pmatrix}$, which represents the unique nontrivial Weyl element of ${\mathrm {SL}}_2(F)$. Under the embedding ${\mathrm {SL}}_2(F)\subset M\subset G$, the element $w^1$ can be identified with $w_\beta$.

Let $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$ be the metaplectic double cover of ${\mathrm {SL}}_2({\mathbb {A}})$. Then we have an exact sequence

$$\begin{aligned} 0\rightarrow \mu _2\rightarrow \widetilde{\mathrm {SL}}_2({\mathbb {A}}) \rightarrow {\mathrm {SL}}_2({\mathbb {A}})\rightarrow 0, \end{aligned}$$

where $\mu _2=\left\{ {\pm 1}\right\}$.

We will identify ${\mathrm {SL}}_2({\mathbb {A}})$ with the symplectic group of ${\mathbb {A}}^2$ with symplectic structure defined by

$$\begin{aligned} \langle {(x_1,y_1),(x_2,y_2)} \rangle =-2x_1y_2+2x_2y_1. \end{aligned}$$

Let ${\mathscr {H}}({\mathbb {A}})$ be the Heisenberg group of the symplectic space $({\mathbb {A}}^2, \langle {~,~} \rangle )$, i.e., ${\mathscr {H}}({\mathbb {A}})={\mathbb {A}}^3$ with group law

$$\begin{aligned} (x_1,y_1,z_1)(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2, z_1+z_2-x_1y_2+y_1 x_2). \end{aligned}$$

Let ${\mathrm {SL}}_2({\mathbb {A}})$ act on ${\mathscr {H}}({\mathbb {A}})$ from the right side by

$$\begin{aligned} (x_1, y_1, z_1).g=((x_1,y_1)g, z_1), g\in {\mathrm {SL}}_2({\mathbb {A}}), \end{aligned}$$

where $(x_1,y_1)g$ is the usual matrix multiplication.

We then can form the semi-direct product ${\mathrm {SL}}_2({\mathbb {A}}) < imes {\mathscr {H}}({\mathbb {A}}),$ where the product is defined by

$$\begin{aligned} (g_1, h_1)(g_2,h_2)=(g_1g_2, (h_1.g_2 ) h_2), g_i\in {\mathrm {SL}}_2({\mathbb {A}}), h_i\in {\mathscr {H}}({\mathbb {A}}), i=1,2. \end{aligned}$$

Let $\psi$ be a nontrivial additive character of $F\backslash {\mathbb {A}}$. Then there is a Weil representation $\omega _\psi$ of $\widetilde{\mathrm {SL}}_2({\mathbb {A}}) < imes {\mathscr {H}}({\mathbb {A}})$. The space of $\omega _\psi$ is ${\mathcal {S}}({\mathbb {A}})$, the Bruhat–Schwartz functions on ${\mathbb {A}}$.

For $\phi \in {\mathcal {S}}({\mathbb {A}})$, we have the well-know formulas:

$$\begin{aligned} (\omega _\psi (n(b))\phi )(x)&=\psi (bx^2)\phi (x), b\in {\mathbb {A}}\\ (\omega _{\psi }((r_1,r_2,r_3))\phi )(x)&=\psi \left( r_3-2xr_2-r_1r_2\right) \phi (x+r_1), (r_1,r_2,r_3)\in {\mathscr {H}}({\mathbb {A}}), \end{aligned}$$

The above formulas could be found in [11].

Recall that for $r_1,r_2,r_3,r_4,r_5\in {\mathbb {A}}$, the notation $[r_1,r_2,r_3,r_4,r_5]$ is an abbreviation of

$$\begin{aligned} \mathbf{{x}}_\alpha (r_1)\mathbf{{x}}_{\alpha +\beta }(r_2)\mathbf{{x}}_{2\alpha +\beta }(r_3)\mathbf{{x}}_{3\alpha +\beta }(r_4) \mathbf{{x}}_{3\alpha +2\beta }(r_5)\in V({\mathbb {A}}). \end{aligned}$$

Define a map ${\mathrm {pr}}: V({\mathbb {A}})\rightarrow {\mathscr {H}}({\mathbb {A}})$

$$\begin{aligned} {\mathrm {pr}}([r_1,r_2,r_3,r_4,r_5])=(r_1,r_2,r_3-r_1 r_2) . \end{aligned}$$

From the commutator relation (2.1), we can check that ${\mathrm {pr}}$ is a group homomorphism and defines an exact sequence

$$\begin{aligned} 0\rightarrow V_1({\mathbb {A}}) \rightarrow V({\mathbb {A}})\rightarrow {\mathscr {H}}({\mathbb {A}})\rightarrow 0. \end{aligned}$$

Recall that $V_1$ is the subgroup of V which is generated by the root space of $3\alpha +\beta , 3\alpha +2\beta$. Note that there is a typo in the formula of the projection map ${\mathrm {pr}}$ in [5, p.316].

For $g=\begin{pmatrix}a &{} b\\ c&{} d \end{pmatrix}\in {\mathrm {SL}}_2(F) \subset M$, we can check that

$$\begin{aligned} g^{-1}[r_1,r_2,r_3,0,0] g=[r_1', r_2', r_3', r_4',r_5'], \end{aligned}$$

where $r_1'=a r_1- cr_2, r_2'=-b r_1 + dr_2, r_3'-r_1'r_2'=r_3-r_1r_2$.

Consider the map $\overline{{\mathrm {pr}}}: J({\mathbb {A}})={\mathrm {SL}}_2({\mathbb {A}}) < imes V({\mathbb {A}})\rightarrow {\mathrm {SL}}_2({\mathbb {A}}) < imes {\mathscr {H}}({\mathbb {A}})$,

$$\begin{aligned} (g,v)\mapsto (g^*, {\mathrm {pr}}(v)), g\in {\mathrm {SL}}_2({\mathbb {A}}), v\in V({\mathbb {A}}). \end{aligned}$$

where $g^*=\begin{pmatrix} a&{} -b \\ -c &{} d \end{pmatrix}=d_1 g d_1^{-1}$, where $d_1={\mathrm {diag}}(1,-1)\in {\mathrm {GL}}_2(F)$. From the above discussion, the map $\overline{{\mathrm {pr}}}$ is a group homomorphism and its kernel is also $V_1({\mathbb {A}})$. We will also view $\overline{\text {pr}}$ as a homomorphism $\widetilde{{\mathrm {SL}}}_2({\mathbb {A}}) < imes V({\mathbb {A}})\rightarrow \widetilde{{\mathrm {SL}}}_2({\mathbb {A}}) < imes {\mathscr {H}}({\mathbb {A}})$.

In the following, we will also view $\omega _\psi$ as a representation of $\widetilde{{\mathrm {SL}}_2}({\mathbb {A}}) < imes V({\mathbb {A}})$ via the projection map $\overline{{\mathrm {pr}}}$. For $\phi \in {\mathcal {S}}({\mathbb {A}})$, we form the theta series

$$\begin{aligned} \widetilde{\theta }_\phi (vg)=\sum _{\xi \in F}\omega _\psi (vh)\phi (\xi ),v\in V({\mathbb {A}}),g\in \widetilde{\mathrm {SL}}_2({\mathbb {A}}). \end{aligned}$$

Note that given a genuine cusp form $\widetilde{\varphi }$ on $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$, the product

$$\begin{aligned} \widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg),v\in V({\mathbb {A}}),g\in \widetilde{\mathrm {SL}}_2({\mathbb {A}}) \end{aligned}$$

can be viewed as a function on $J({\mathbb {A}})={\mathrm {SL}}_2({\mathbb {A}}) < imes V({\mathbb {A}})$.

2.4 An Eisenstein series on ${ {G}}_2$

In this subsection and in the rest of the paper, every representation appeared is assumed to be irreducible. Let $\tau$ be a cuspidal automorphic representation on ${\mathrm {GL}}_2({\mathbb {A}})$. We will view $\tau$ as a representation of $M'({\mathbb {A}})$ via the identification $M'\cong {\mathrm {GL}}_2$. We then consider the induced representation $I(s,\tau )={\mathrm {Ind}}_{P'({\mathbb {A}})}^{{{G}}_2({\mathbb {A}})}(\tau \otimes \delta _{P'}^s)$. A section $f_s\in I(s,\tau )$ is a smooth function satisfying

$$\begin{aligned} f_s(v'm'g)=\delta _{P'}(m')^sf_s(g), \forall v'\in V'({\mathbb {A}}), m'\in M'({\mathbb {A}}),g\in { {G}}_2({\mathbb {A}}). \end{aligned}$$

For $f_s\in I(s,\tau )$, we consider the Eisenstein series

$$\begin{aligned} E(g,f_s)=\sum _{\delta \in P'(F){\setminus }{{G}}_2(F)}f_s(\delta g),g\in {{G}}_2({\mathbb {A}}). \end{aligned}$$

3 A global integral

Let $\widetilde{\pi }$ be a genuine cuspidal automorphic representation on $\widetilde{\mathrm {SL}}_2({\mathbb {A}}),$ and $\tau$ be a cuspidal automorphic representation of ${\mathrm {GL}}_2({\mathbb {A}})$. For $\widetilde{\varphi }\in V_\pi , \phi \in {\mathcal {S}}({\mathbb {A}})$ and $f_s\in I(s,\tau )$, we consider the integral

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\int _{{\mathrm {SL}}_2(F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\setminus V({\mathbb {A}})}\widetilde{\varphi }(g) \widetilde{\theta }_\phi (vg) E(vg,f_s)dvdg. \end{aligned}$$

Let $\gamma =w_\beta w_\alpha w_\beta w_\alpha \in {{G}}_2(F)$.

Theorem 3.1

The integral $I(\widetilde{\varphi },\phi ,f_s)$ is absolutely convergent when ${\mathrm {Re}}(s)\gg 0$ and can be meromorphically continued to all $s\in {\mathbb {C}}$. Moreover, when ${\mathrm {Re}}(s)\gg 0$, we have

$$\begin{aligned} I(\tilde{\varphi },\phi ,f_s)=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}}){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }({\mathbb {A}}){\setminus }V({\mathbb {A}})} W_{\widetilde{\varphi }}(g)\omega _\psi (vg)\phi (1)W_{f_s}(\gamma vg)dvdg, \end{aligned}$$

where

$$\begin{aligned} W_{\widetilde{\varphi }}(g)=\int _{F{\setminus }{\mathbb {A}}}\widetilde{\varphi }(\mathbf{{x}}_\beta (r)g)\psi (r)dr, \end{aligned}$$

and

$$\begin{aligned} W_{f_s}(\gamma vg)=\int _{F{\setminus }{\mathbb {A}}}f_s(\mathbf{{x}}_\alpha (r)\gamma vg)\psi (-2r)dr. \end{aligned}$$

Proof

The first assertion is standard. We only show that the above integral is Eulerian when ${\mathrm {Re}}(s)\gg 0$. Unfolding the Eisenstein series, we can get

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\sum _{\delta \in P'(F){\setminus }{{G}}_2(F)/P(F)}\int _{{\mathrm {SL}}_2^\delta (F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V^\delta (F){\setminus }V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)f_s(\delta vg)dvdg, \end{aligned}$$

where $X^\delta =\delta ^{-1} P'\delta \cap X$ for $X\subset G_2(F)$. We can check that a set of representatives of the double coset $P'(F){\setminus }{{G}}_2(F)/P(F)$ can be taken as $\left\{ {1,w_\beta w_\alpha , \gamma = w_\beta w_\alpha w_\beta w_\alpha }\right\} .$ For $\delta =1, w_\beta w_\alpha ,$ or $\gamma =w_\beta w_\alpha w_\beta w_\alpha$, denote

$$\begin{aligned} I_\delta =\int _{{\mathrm {SL}}_2^\delta (F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V^\delta (F)\setminus V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)f_s(\delta vg)dvdg. \end{aligned}$$

If $\delta =1$, the above integral $I_\delta$ has an inner integral

$$\begin{aligned} \int _{U_{2\alpha +\beta }(F){\setminus }U_{2\alpha +\beta }({\mathbb {A}})}\widetilde{\theta }_\phi (\mathbf{{x}}_{2\alpha +\beta }(r)vg)f_s(\mathbf{{x}}_{2\alpha +\beta }(r)vg)dr, \end{aligned}$$

which is zero because $f_s(\mathbf{{x}}_{2\alpha +\beta }(r)vg)=f_s(vg) , \widetilde{\theta }_\phi (\mathbf{{x}}_{2\alpha +\beta }(r)vg)=\psi (r)\tilde{\theta }_\phi (vg)$ and $\int _{F{\setminus }{\mathbb {A}}}\psi (r)dr=0.$ The last equation follows from the fact that $\psi$ is non-trivial.

We next consider the term when $\delta =w_\beta w_\alpha$. We write

$$\begin{aligned} \widetilde{\theta }_\phi (vg)=\omega _\psi (vg)\phi (0)+\sum _{\xi \in F^\times }\omega _\psi (vg)\phi (\xi ). \end{aligned}$$

The contribution of the first term to the integral $I_{\delta }$ is

$$\begin{aligned} \int _{{\mathrm {SL}}_2^\delta (F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V^\delta (F){\setminus } V({\mathbb {A}})}\widetilde{\varphi }(g)\omega _\psi (vg)\phi (0)f_s(\delta vg)dvdg. \end{aligned}$$

Note that $\delta \mathbf{{x}}_\beta (r)\delta ^{-1}\subset U_{2\alpha +\beta }\subset V'$, we have $f_s(\delta v \mathbf{{x}}_\beta (r)g)=f_s(\delta \mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g)$. On the other hand, we have $\omega _\psi (\mathbf{{x}}_\beta (r)vg)\phi (0)=\omega _\psi (vg)\phi (0)$ . After a changing variable on v, we can see that the above integral contains an inner integral

$$\begin{aligned} \int _{F{\setminus }{\mathbb {A}}}\widetilde{\varphi }(\mathbf{{x}}_\beta (r)vg)dr, \end{aligned}$$

which is zero since $\widetilde{\varphi }$ is cuspidal. Thus the contribution of the term $\omega _\psi (vg)\phi (0)$ is zero when $\delta =w_\beta w_\alpha$. The contribution of $\sum _{\xi \in F^\times }\omega _\psi (vg)\phi (\xi )$ is

$$\begin{aligned} \int _{{\mathrm {SL}}_2^\delta (F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V^\delta (F){\setminus } V({\mathbb {A}})}\widetilde{\varphi }(g) \sum _{\xi \in F^\times }\omega _\psi (vg)\phi (\xi )f_s(\delta vg)dvdg. \end{aligned}$$

We consider the inner integral on $U_{\alpha +\beta }(F){\setminus }U_{\alpha +\beta }({\mathbb {A}})$. Note that $U_{\alpha +\beta }\subset V$ and $\delta U_{\alpha +\beta } \delta ^{-1}=U_{2\alpha +\beta }\subset V'$, we get $f_s(\delta \mathbf{{x}}_{\alpha +\beta }(r)vg)=f_s(\delta vg)$. On the other hand, we have $\omega _{\psi }(\mathbf{{x}}_{\alpha +\beta }(r)vg)\phi (\xi )=\psi (-2r\xi )\omega _\psi (vg)\phi (\xi )$. Thus the above integral has an inner integral

$$\begin{aligned} \int _{F{\setminus }{\mathbb {A}}}\sum _{\xi \in F^\times }\psi (-2r\xi )\omega _\psi (vg)\phi (\xi )dr=\sum _{\xi \in F^\times }\omega _\psi (vg)\phi (\xi )\int _{F{\setminus }{\mathbb {A}}}\psi (-2r\xi )dr=0. \end{aligned}$$

Thus when $\delta =w_\beta w_\alpha$, the corresponding term is zero. Thus we get

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\int _{{\mathrm {SL}}_2^\gamma (F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{V^\gamma (F){\setminus }V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)f_s(\gamma vg)dvdg. \end{aligned}$$

We have ${\mathrm {SL}}_2^\gamma =B_{{\mathrm {SL}}_2}$ and $V^\gamma =U_{\alpha +\beta }$. We decompose $\widetilde{\theta }_\phi$ as

$$\begin{aligned} \widetilde{\theta }_\phi (vg)=\omega _\psi (vg)\phi (0)+\sum _{\xi \in F^\times }\omega _\psi (vg)\phi (\xi )=\omega _\psi (vg)\phi (0)+\sum _{a\in F^\times }\omega _\psi (t(a)vg)\phi (1). \end{aligned}$$

Recall that $t(a)={\mathrm {diag}}(a,a^{-1})$. Since $\gamma U_\beta \gamma ^{-1}\subset U_{3\alpha +\beta }\subset V'$, we have

$$\begin{aligned} f_s(\gamma v\mathbf{{x}}_\beta (r)g)=f_s(\gamma \mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g). \end{aligned}$$

On the other hand we have $\omega _\psi (v\mathbf{{x}}_\beta (r)g)\phi (0)=\omega _\psi (\mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g)\phi (0)$. Thus after a changing variable on v, we can get that the contribution of $\omega _\psi (vg)\phi (0)$ to $I(\widetilde{\varphi },\phi ,f_s)$ has an inner integral

$$\begin{aligned} \int _{F{\setminus }{\mathbb {A}}}\widetilde{\varphi }(\mathbf{{x}}_\beta (r)g)dr, \end{aligned}$$

which is zero by the cuspidality of $\widetilde{\varphi }$. Thus we get

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\int _{B_{{\mathrm {SL}}_2}(F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }(F){\setminus }V({\mathbb {A}})}\widetilde{\varphi }(g)\sum _{a\in F^\times }\omega _\psi (t(a)vg)\phi (1)f_s(\gamma vg)dvdg. \end{aligned}$$

Collapsing the summation with the integration, we then get

$$\begin{aligned}&\quad I(\widetilde{\varphi },\phi ,f_s)\\&=\int _{N_{{\mathrm {SL}}_2}(F){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }(F){\setminus }V({\mathbb {A}})}\widetilde{\varphi }(g)\omega _\psi (vg)\phi (1)f_s(\gamma vg)dvdg\\&=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}}){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }(F){\setminus }V({\mathbb {A}})}\int _{F{\setminus }{\mathbb {A}}}\widetilde{\varphi }(\mathbf{{x}}_\beta (r)g)\omega _\psi (v\mathbf{{x}}_\beta (r)g)\phi (1)f_s(\gamma v\mathbf{{x}}_\beta (r)g)drdvdg. \end{aligned}$$

Note that we have $\omega _\psi (v\mathbf{{x}}_\beta (r)g)\phi (1)=\omega _\psi (\mathbf{{x}}_\beta (r)\mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g)\phi (1)=\psi (r) \omega _\psi (\mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g)\phi (1)$. On the other hand, we have $\gamma \mathbf{{x}}_\beta (r)\gamma ^{-1}\subset U_{3\alpha +\beta }\subset V'$. Thus $f_s(\gamma v\mathbf{{x}}_\beta (r)g)=f_s(\gamma \mathbf{{x}}_\beta (-r)v\mathbf{{x}}_\beta (r)g) .$ After a changing of variable on v, we get

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}}){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }(F){\setminus }V({\mathbb {A}})}W_{\widetilde{\varphi }}(g)\omega _\psi (vg)\phi (1)f_s(\gamma vg)dvdg, \end{aligned}$$

where

$$\begin{aligned} W_{\widetilde{\varphi }}(g)=\int _{F{\setminus }{\mathbb {A}}}\tilde{\varphi }(\mathbf{{x}}_\beta (r)g)\psi (r)dr. \end{aligned}$$

We can further decompose the above integral as

$$\begin{aligned}&I(\widetilde{\varphi },\phi ,f_s)=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}}){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }({\mathbb {A}}){\setminus }V({\mathbb {A}})}\\&\quad \int _{F{\setminus }{\mathbb {A}}}W_{\widetilde{\varphi }}(g)\omega _\psi (\mathbf{{x}}_{\alpha +\beta }(r)vg)\phi (1)f_s(\gamma \mathbf{{x}}_{\alpha +\beta }(r)vg)dr dvdg. \end{aligned}$$

Note that $\omega _\psi (\mathbf{{x}}_{\alpha +\beta }(r)vg)\phi (1)=\psi (-2r)\omega _\psi (vg)\phi (1)$ and $f_s(\gamma \mathbf{{x}}_{\alpha +\beta }(r)vg)=f_s(\mathbf{{x}}_\alpha (r)\gamma vg)$ since $\gamma \mathbf{{x}}_{\alpha +\beta }(r)\gamma ^{-1}=\mathbf{{x}}_\alpha (r)$. We then get

$$\begin{aligned} I(\widetilde{\varphi },\phi ,f_s)=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}}){\setminus }{\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }({\mathbb {A}}){\setminus }V({\mathbb {A}})} W_{\widetilde{\varphi }}(g)\omega _\psi (vg)\phi (1)W_{f_s}(\gamma vg)dvdg, \end{aligned}$$

where

$$\begin{aligned} W_{f_s}(\gamma vg)=\int _{F{\setminus }{\mathbb {A}}}f_s(\mathbf{{x}}_\alpha (r)\gamma vg)\psi (-2r)dr. \end{aligned}$$

This concludes the proof.$\square$

4 Unramified calculation

In this section, let F be a p-adic field with $p\ne 2$. Let ${\mathfrak {o}}$ be the ring of integers of F, and let p be a uniformizer of ${\mathfrak {o}}$ by abuse of notation. Let q be the cardinality of the residue field ${\mathfrak {o}}/(p)$.

4.1 Local Weil representations

Let $\psi$ be an additive character of F and let $\gamma (\psi )$ be the Weil index and let $\mu _\psi (a)=\frac{\gamma (\psi )}{\gamma (\psi _a)}$. Let $\omega _\psi$ be the Weil representation of $\widetilde{\mathrm {SL}}_2(F) < imes V$ on ${\mathcal {S}}(F)$ via the projection $\widetilde{\mathrm {SL}}_2(F) < imes V\rightarrow \widetilde{\mathrm {SL}}_2(F) < imes {\mathscr {H}}$. For $\phi \in {\mathcal {S}}(F)$, we have the well-know formulas:

$$\begin{aligned} (\omega _{\psi }(w^1)\phi )(x)&=\gamma (\psi )\hat{\phi }(x),\\ (\omega _\psi (n(b))\phi )(x)&=\psi (bx^2)\phi (x), b\in F\\ (\omega _\psi (t(a))\phi )(x)&=|a|^{1/2} \mu _\psi (a)\phi (ax), a\in F^\times \\ (\omega _{\psi }((r_1,r_2,r_3))\phi )(x)&=\psi \left( r_3-2xr_2-r_1r_2\right) \phi (x+r_1), (r_1,r_2,r_3)\in {\mathscr {H}}(F). \end{aligned}$$

where $\hat{\phi }(x)=\int _F \phi (y)\psi (2xy)dy$ is the Fourier transform of $\phi$ with respect to $\psi$. Note that under the embedding ${\mathrm {SL}}_2(F)\hookrightarrow {{G}}_2(F)$, we have $w^1=w_\beta ,n(b)=\mathbf{{x}}_\beta (b)$ and $t(a)=h(a,a^{-1})$.

4.2 Unramified calculation

In this subsection, we compute the local integral in last section. The strategy is similar to the unramified calculation in [6].

Let $\widetilde{\pi }$ be an unramified genuine representation of $\widetilde{\mathrm {SL}}_2(F)$ with Satake parameter a, and let $\tau$ be an unramified irreducible representation of ${\mathrm {GL}}_2(F)$ with Satake parameters $b_1,b_2$. Let $\widetilde{W}\in {\mathcal {W}}(\tilde{\pi }, \psi )$ with $\widetilde{W}(1)=1$. Let $v_0\in V_\tau$ be an unramified vector and $\lambda \in {\mathrm {Hom}}_{N}(V_\tau ,\psi )$ such that $\lambda (v_0)=1$. Let $f_s:G_2\rightarrow V_\tau$ be the unramified section in $I(s,\tau )$ with $f_s(e)=v_0$. Let

$$\begin{aligned} W_{f_s}:G_2\times {\mathrm {GL}}_2(F)\rightarrow {\mathbb {C}}\end{aligned}$$

be the function $W_{f_s}(g,a)=\lambda (\tau (a)f_s(g)).$ We will write $W_{f_s}(g)$ for $W_{f_s}(g,1)$ in the following. By assumption and Shintani formula, we have

$$\begin{aligned} \begin{aligned} W_{f_s}(h(p^k,p^l))&=q^{-3s(2k+l)}\lambda (\tau ({\mathrm {diag}}(p^{k+l},p^k))v_0)\\&=q^{-3s(2k+l)}W_{v_0}({\mathrm {diag}}(p^{k+l},p^k))\\&=\left\{ \begin{array}{lll}q^{-3s(2k+l)}\frac{(b_1b_2)^kq^{-l/2}}{b_1-b_2}(b_1^{l+1}-b_2^{l+1}), &{} \text { if } l\ge 0,\\ 0, &{} \text { if } l<0. \end{array} \right. \end{aligned} \end{aligned}$$

(4.1)

Let $\phi \in {\mathcal {S}}(F)$ be the characteristic function of ${\mathfrak {o}}$. We need to compute the integral

$$\begin{aligned} I(\widetilde{W}, W_{f_s},\phi )=\int _{N_2{\setminus } {\mathrm {SL}}_2(F)}\int _{U_{\alpha +\beta }{\setminus }V}\widetilde{W}(g)\omega _\psi (vg)\phi (1)W_{f_s}(\gamma vg)dvdg. \end{aligned}$$

In the following, we fix the Haar measure such that ${\mathrm {vol}}(dr,{\mathfrak {o}})=1$. Thus ${\mathrm {vol}}(d^*r,{\mathfrak {o}}^\times )=1-q^{-1}$.

Using the Iwasawa decomposition ${\mathrm {SL}}_2(F)=N_2(F)A_2(F){\mathrm {SL}}_2({\mathfrak {o}})$, we have

$$\begin{aligned}&I(\widetilde{W}, W_{f_s},\phi )\\ =&\int _{F^\times }\int _{F^4}\widetilde{W}(t(a))\omega _\psi ([r_1,0,r_3]t(a))\phi (1)W_{f_s}\\&(\gamma (r_1,0,r_3,r_4,r_5)t(a))|a|^{-2}dr_1dr_3dr_4dr_5 d^\times a\\ =&\int _{F^\times }\int _{F^4}\widetilde{W}(t(a))\omega _\psi (t(a)[r_1,0,r_3])\phi (1)W_{f_s}\\&(\gamma t(a) (r_1,0,r_3,r_4,r_5))|a|^{-3}dr_1dr_3dr_4dr_5 d^\times a \end{aligned}$$

If $\widetilde{W}(t(a))\ne 0$, then $|a|\le 1$. On the other hand, we have

$$\begin{aligned} \omega _\psi (t(a)[r_1,0,r_3])\phi (1)=\mu _\psi (a)|a|^{1/2}\psi (r_3)\phi (a+r_1). \end{aligned}$$

If $\phi (a+r_1)\ne 0$ and $a\in {\mathfrak {o}}$, then $r_1\in {\mathfrak {o}}$. Thus the domain for a and $r_1$ in the above integral is $\left\{ {a\in F^\times \cap {\mathfrak {o}},r_1\in {\mathfrak {o}}}\right\}$. Note that $\gamma t(a)=h(1,a)\gamma =h(1,a)w_\beta w_\alpha w_\beta w_\alpha$. Thus, if we conjugate $w_\alpha \mathbf{{x}}_\alpha (r_1)$ to the right side, we can get

$$\begin{aligned} h(1,a)\gamma [r_1,0,r_3,r_4,r_5]=h(1,a)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(-r_3)\mathbf{{x}}_\beta (-r_4-3r_1r_3)\mathbf{{x}}_{3\alpha +2\beta }(r_5)w_\alpha \mathbf{{x}}_\alpha (r_1). \end{aligned}$$

Since $w_\alpha \mathbf{{x}}_\alpha (r_1)\in K$ for $r_1\in {\mathfrak {o}}$, by changing of variables, we get

$$\begin{aligned}&I(\widetilde{W}, W_{f_s},\phi )\\ =&\int _{|a|\le 1}\widetilde{W}(t(a))|a|^{-5/2}\mu _\psi (a) \\&\quad \cdot \int _{F^3} W_{f_s}(h(1,a)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(r_3)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-r_3)dr_3dr_4dr_5 d^*a\\ =&\sum _{n\ge 0} \widetilde{W}(t(p^n)) q^{5n/2}\mu _\psi (p^n) J(n), \end{aligned}$$

where

$$\begin{aligned} J(n)= \int _{F^3} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(r_3)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-r_3)dr_3dr_4dr_5. \end{aligned}$$

By dividing the domain of $r_3$ into two parts, we can write $J(n)=J_1(n)+J_2(n)$, where

$$\begin{aligned} J_1(n)&=\int _{|r_3|\le 1}\int _{F^2} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(r_3)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-r_3)dr_3dr_4dr_5\\&=\int _{F^2} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5, \end{aligned}$$

and

$$\begin{aligned} J_2(n)=\int _{|r_3|> 1}\int _{F^2} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(r_3)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-r_3)dr_3dr_4dr_5. \end{aligned}$$

Lemma 4.1

Set

$$\begin{aligned} I(n)=\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))dr. \end{aligned}$$

Then

$$\begin{aligned} I(n)&=\frac{q^{-(3s+1/2)n}}{b_1-b_2}\left[ (b_1^{n+1}-b_2^{n+1}) \right. \\&\quad +(1-q^{-1})\frac{b_1b_2X}{(1-b_1X) (1-b_2X)},\\&\quad \left. (b_1^n-b_2^n-b_1^{n+1}X+b_2^{n+1}X+b_1X(b_1b_2X)^n-b_2X(b_1b_2X)^n)\right] , \end{aligned}$$

where $X=q^{-(3s-3/2)}$.

Proof

We have

$$\begin{aligned} I(n)&=\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))dr\\&=\int _{|r|\le 1}W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))dr\\&\quad +\int _{|r|> 1}W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))dr\\&=W_{f_s}(h(1,p^n))+\int _{|r|> 1}W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))dr. \end{aligned}$$

To deal with the integral when $|r|>1$, we consider the following Iwasawa decomposition of $w_\beta \mathbf{{x}}_\beta (r)$:

$$\begin{aligned} w_\beta \mathbf{{x}}_\beta (r)=\mathbf{{x}}_\beta (-r^{-1})h(-r^{-1},-r)\mathbf{{x}}_{-\beta }(r^{-1}). \end{aligned}$$

Since $\mathbf{{x}}_{-\beta }(r^{-1})$ is in the maximal compact subgroup for $|r|>1$, we have

$$\begin{aligned} W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r))=W_{f_s}(h(1,p^n)\mathbf{{x}}_{\beta }(-r^{-1})h(-r^{-1},-r))=W_{f_s}(h(1,p^n)h(r^{-1},r)), \end{aligned}$$

where we used $U_\beta \subset V'$. For $|r|>1$, we can write $r= p^{-m}u$ for some $m\ge 1$ and $u\in {\mathfrak {o}}^\times$. We then have $dr=q^{m}du$. Note that ${\mathrm {vol}}({\mathfrak {o}}^\times )=1-q^{-1}$. Thus we have

$$\begin{aligned} I(n)=W_{f_s}(h(1,p^n))+\sum _{m\ge 1}(1-q^{-1})q^mW_{f_s}(h(p^m,p^{n-m})). \end{aligned}$$

Note that $h(p^m,1)\mapsto {\mathrm {diag}}(p^m,p^m)$ under the isomorphism $M'\cong {\mathrm {GL}}_2$. Thus we have

$$\begin{aligned} W_{f_s}(h(p^m,1)h(1,p^{n-m}))=q^{-6sm}\omega _\tau (p)^mW_{f_s}(h(1,p^{n-m})). \end{aligned}$$

Thus we get

$$\begin{aligned} I(n)=W_{f_s}(h(1,p^n))+\sum _{m\ge 1}(1-q^{-1})q^{(-6s+1)m}\omega _\tau (p)^mW_{f_s}(h(1,p^{n-m})). \end{aligned}$$

By (4.1), we have

$$\begin{aligned} W_{f_s}(h(1,p^{n-m}))&=\left\{ \begin{array}{lll} \frac{q^{-3s(n-m)-(n-m)/2}}{b_1-b_2}(b_1^{n-m+1}-b_2^{n-m+1}), &{} \text { if } n\ge m,\\ 0, &{} \text { if } n<m. \end{array} \right. \end{aligned}$$

Thus for $n\ge 1$, we have

$$\begin{aligned} I(n)&=\frac{q^{-(3s+1/2)n}}{b_1-b_2}\left( (b_1^{n+1}-b_2^{n+1})+\sum _{m= 1}^n(1-q^{-1})q^{-(3s-3/2)m}(b_1^{n+1}b_2^m-b_2^{n+1}b_1^m)\right) . \end{aligned}$$

Thus result can be computed using the geometric summation formula. One can check that the given formula also satisfies $I(0)=1$. $\square$

Lemma 4.2

We have

$$\begin{aligned} J_1(n)=\frac{1-q^{-6s+1}b_1b_2}{1-q^{-6s+2}b_1b_2}I(n). \end{aligned}$$

Proof

To compute $J_1(n)$, we break up the domain of integration in $r_4$ and get

$$\begin{aligned} J_1(n)&=\int _{F}\int _{|r_4|\le 1} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5\\&\quad +\int _{F}\int _{|r_4|> 1} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5\\&:=J_{11}(n)+J_{12}(n), \end{aligned}$$

where

$$\begin{aligned} J_{11}(n)=&\int _{F}\int _{|r_4|\le 1}W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4 dr_5 \\ =&\int _{F}\int _{|r_4|\le 1}W_{f_s}(h(1,p^n) w_\beta w_\alpha w_\beta \mathbf{{x}}_{3\alpha +2\beta }(r_5)w_\beta ^{-1}w_\alpha ^{-1} w_\alpha w_\beta \mathbf{{x}}_\beta (r_4))dr_4dr_5\\ =&\int _{F}W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r_5))dr_5\\ =&I(n), \end{aligned}$$

and

$$\begin{aligned} J_{12}(n)&=\int _{F}\int _{|r_4|> 1}W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4 dr_5 \\&=\int _{F}\int _{|r_4|> 1}W_{f_s}(h(1,p^n) w_\beta w_\alpha w_\beta \mathbf{{x}}_{3\alpha +2\beta }(r_5)w_\beta ^{-1}w_\alpha ^{-1} w_\alpha w_\beta \mathbf{{x}}_\beta (r_4))dr_4dr_5\\&=\int _{F}\int _{|r_4|> 1}W_{f_s}(h(1,p^n) w_\beta \mathbf{{x}}_{\beta }(r_5) w_\alpha w_\beta \mathbf{{x}}_\beta (r_4))dr_4dr_5. \end{aligned}$$

We have the Iwasawa decomposition of $w_\beta \mathbf{{x}}_\beta (r_4)$:

$$\begin{aligned} w_\beta \mathbf{{x}}_\beta (r_4)=\mathbf{{x}}_\beta (-r_4^{-1})h(-r_4^{-1},-r_4)\mathbf{{x}}_{-\beta }(r_4^{-1}). \end{aligned}$$

Since $\mathbf{{x}}_{-\beta }(r_4^{-1})$ is in the maximal compact subgroup for $|r_4|>1$, we then get

$$\begin{aligned} J_{12}(n)&=\int _{F}\int _{|r_4|> 1}W_{f_s}(h(1,p^n) w_\beta \mathbf{{x}}_{\beta }(r_5) w_\alpha \mathbf{{x}}_\beta (-r_4^{-1})h(r_4^{-1},r_4))dr_4dr_5\\&=\int _F \int _{|r_4|>1}W_{f_s}(h(1,p^n) h(r_4^{-1},1)w_\beta \mathbf{{x}}_{\beta }(r_4^{-1}r_5))dr_4dr_5\\&=\int _F \int _{|r_4|>1}|r_4|W_{f_s}(h(1,p^n) h(r_4^{-1},1)w_\beta \mathbf{{x}}_{\beta }(r_5))dr_4dr_5\\&=\sum _{m\ge 1}(1-q^{-1})q^{2m} \int _F W_{f_s}(h(p^m,1)h(1,p^n)w_\beta \mathbf{{x}}_\beta (r_5))dr_5, \end{aligned}$$

where in the second equality, we conjugated $\mathbf{{x}}_\beta (-r_4^{-1})h(r_4^{-1},r_4)$ to the left, and in the third equality, we wrote $r_4=p^{-m}u$ for $m\ge 1, u\in {\mathfrak {o}}^\times$ and used $dr_4=q^mdu, {\mathrm {vol}}({\mathfrak {o}}^\times )=1-q^{-1}$. Note that $h(p^m,1)$ is in the center of $M'$, and thus

$$\begin{aligned} W_{f_s}(h(p^m,1)g)=q^{-6sm}\omega _\tau (p)^mW_{f_s}(g), \end{aligned}$$

we get

$$\begin{aligned} J_{12}(n)=(1-q^{-1})\sum _{m\ge 1}q^{-6sm+2m}\omega _\tau (p)^m\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r_5))dr_5. \end{aligned}$$

Thus we get

$$\begin{aligned} J_1(n)=I(n)+\sum _{m\ge 1}(1-q^{-1})q^{(-6s+2)m}(b_1b_2)^m I(n). \end{aligned}$$

A simple calculation gives the formula of $J_1(n)$. $\square$

We next consider the term

$$\begin{aligned} J_2(n)=\int _{|r_3|> 1}\int _{F^2} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(r_3)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-r_3)dr_3dr_4dr_5. \end{aligned}$$

For $|r_3|>1$, we can write $r_3\in p^{-m}u$ with $m\ge 1, u\in {\mathfrak {o}}^\times$. We then have,

$$\begin{aligned}&J_2(n)\\= & {} \int _{F^2}\sum _{m\ge 1}q^m W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-m}u)\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-p^{-m}u)dudr_4dr_5. \end{aligned}$$

Write $\mathbf{{x}}_{\alpha +\beta }(p^{-m}u)=h(u,u^{-1})\mathbf{{x}}_{\alpha +\beta }(p^{-m})h(u^{-1},u)$, and by conjugation and changing of variables, we get

$$\begin{aligned}&J_2(n)\\= & {} \int _{F^2}\sum _{m\ge 1}q^m W_{f_s}(h(u^{-1},p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-m})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-p^{-m}u)dudr_4dr_5, \end{aligned}$$

where we used $h(u,u^{-1})$ is in the maximal compact subgroup of ${ {G}}_2(F)$. Since $h(u^{-1},1)$ maps to the center of $M'$ and $|\omega _\tau (u)|=1$, we have

$$\begin{aligned}&W_{f_s}(h(u^{-1},p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-m})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\\= & {} W_{f_s}(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-m})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5)). \end{aligned}$$

Thus we get

$$\begin{aligned}&J_2(n)\\= & {} \int _{F^2}\sum _{m\ge 1}q^m W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-m})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))\psi (-p^{-m}u)dudr_4dr_5. \end{aligned}$$

Since

$$\begin{aligned} \int _{{\mathfrak {o}}^\times }\psi (p^ku)du=\left\{ \begin{array}{lll}1-q^{-1}, &{} \text { if } k\ge 0,\\ -q^{-1}, &{} \text { if } k=-1, \\ 0, &{} \text { if } k\le -2, \end{array}\right. \end{aligned}$$

we get $J_2(n)=-R(n)$, where

$$\begin{aligned} R(n)=\int _{F^2} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5. \end{aligned}$$

To evaluate R(n), we split the domain of $r_4$, and write $R(n)=R_1(n)+R_{2}(n)$, where

$$\begin{aligned} R_1(n)&=\int _{|r_4|\le 1} \int _{F} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5,\\&= \int _{F} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_5, \end{aligned}$$

and

$$\begin{aligned} R_2(n)=\int _{|r_4|>1} \int _{F} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5. \end{aligned}$$

We now compute $R_1(n)$. We conjugate $w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})$ to the right and then get

$$\begin{aligned} R_1(n)&=\int _{F} W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_{\beta }(r_5)w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1}))dr_5\\&=\int _{F} W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_{\beta }(r_5)w_\alpha \mathbf{{x}}_{\alpha }(-p^{-1}))dr_5 \end{aligned}$$

Next, we use the Iwasawa decomposition of $w_\alpha \mathbf{{x}}_\alpha (p^{-1})$:

$$\begin{aligned} w_\alpha \mathbf{{x}}_\alpha (-p^{-1})=\mathbf{{x}}_\alpha (p)h(p^{-1},p^2)\mathbf{{x}}_{-\alpha }(-p) \end{aligned}$$

to get

$$\begin{aligned} R_1(n)=\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r_5)\mathbf{{x}}_\alpha (p)h(p^{-1},p^2))dr_5. \end{aligned}$$

Next, we use the commutator relation

$$\begin{aligned} \mathbf{{x}}_\beta (r_5)\mathbf{{x}}_{\alpha }(p)=\mathbf{{x}}_{\alpha +\beta }(pr_5)u\mathbf{{x}}_\alpha (p)\mathbf{{x}}_\beta (r_5), \end{aligned}$$

where u is in the root space of $2\alpha +\beta ,3\alpha +\beta ,3\alpha +2\beta$. Then we get

$$\begin{aligned} R_1(n)=\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_{\alpha +\beta }(pr_5)u\mathbf{{x}}_\alpha (p)\mathbf{{x}}_\beta (r_5)h(p^{-1},p^2))dr_5. \end{aligned}$$

Note that $w_\beta u \mathbf{{x}}_\alpha (r) w_\beta (1)\in V'$, and $h(1,p^n)w_\beta \mathbf{{x}}_{\alpha +\beta }(pr_5)(h(1,p^n)w_\beta )^{-1}=\mathbf{{x}}_\alpha (-p^{n+1}r_5),$ and $W_{f_s}(\mathbf{{x}}_\alpha (r)g)=\psi (2r)W_{f_s}(g)$, we get

$$\begin{aligned} R_1(n)&=\int _F W_{f_s}(h(1,p^n)w_\beta \mathbf{{x}}_\beta (r_5)h(p^{-1},p^2))\psi (-2p^{n+1}r_5)dr_5\\&=\int _F W_{f_s}(h(p^2,1)h(1,p^{n-1})w_\beta \mathbf{{x}}_\beta (p^3r_5))\psi (-2p^{n+1}r_5)dr_5\\&=q^{-12s+3}\omega _\tau (p^2)\int _{F}W_{f_s}(h(1,p^{n-1})w_\beta \mathbf{{x}}_\beta (r_5))\psi (-2 p^{n-2}r_5)dr_5, \end{aligned}$$

where the last equality comes from a changing of variable on $r_5$ and the fact that $h(p^2,1)\mapsto {\mathrm {diag}}(p^2,p^2)$ under the isomorphism $M'\cong {\mathrm {GL}}_2$. We next break up the integral on $r_5$ and get

$$\begin{aligned} R_1(n)&=q^{-12s+3}\omega _\tau (p^2)W_{f_s}(h(1,p^{n-1}))\int _{|r_5|\le 1}\psi (-2p^{n-2}r_5)dr_5\\&\quad +q^{-12s+3}\omega _\tau (p^2)\int _{|r_5|>1}W_{f_s}(h(1,p^{n-1})w_\beta \mathbf{{x}}_\beta (r_5))\psi (-2p^{n-2}r_5)dr_5. \end{aligned}$$

Using the Iwasawa decomposition of $w_\beta \mathbf{{x}}_\beta (r_5)$, we have

$$\begin{aligned} R_1(n)&=q^{-12s+3}\omega _\tau (p^2)\\&\quad \left( W_{f_s}(h(1,p^{n-1}))\int _{|r_5|\le 1}\psi (-2p^{n-2}r_5)dr_5 \right. \\&\quad \left. +\sum _{m=1}^\infty W_{f_s}(h(p^{m},p^{n-m-1}))q^m\int _{{\mathfrak {o}}^\times }\psi (-2p^{n-m-2}u)du \right) . \end{aligned}$$

Lemma 4.3

We have $R_1(n)=0$ if $n\le 1$, and

$$\begin{aligned} R_1(n)=q^{-12s+3}\omega _\tau (p)^2I(n-1)-q^{-6s(n+1)+n+2}\omega _\tau (p)^{n+1}, \end{aligned}$$

for $n\ge 2$.

Proof

Note that $\int _{|r|\le 1}\psi (p^kr)dr=0$ if $k<0$ and $\int _{|r|\le 1}\psi (p^kr)dr=1$ if $k\ge 0$. Moreover, we have

$$\begin{aligned} \int _{{\mathfrak {o}}^\times }\psi (p^ku)du=\left\{ \begin{array}{lll}1-q^{-1}, &{} \text { if } k\ge 0,\\ -q^{-1}, &{} \text { if } k=-1, \\ 0, &{} \text { if } k\le -2. \end{array}\right. \end{aligned}$$

Thus we get $R_1(n)=0$ for $n\le 1$. For $n\ge 2$, we have

$$\begin{aligned} R_1(n)&= q^{-12s+3}\omega _\tau (p^2)\\&\quad \cdot \left( W_{f_s}(h(1,p^{n-1}))+\sum _{m=1}^{n-2}(1-q^{-1})q^mW_{f_s}( h(p^{m},p^{n-m-1}))\right. \\&\quad \left. -q^{-1}q^{n-1}W_{f_s}(h(p^{(n-1)},1))\right) =q^{-12s+3}\omega _\tau (p^2)\\&\quad \cdot \left( W_{f_s}(h(1,p^{n-1}))+\sum _{m=1}^{n-1}(1-q^{-1})q^mW_{f_s}( h(p^{m},p^{n-m-1}))\right. \\&\quad \left. -q^{n-1}W_{f_s}(h(p^{(n-1)},1)) \right) =q^{-12s+3}\\&\quad \omega _\tau (p)^2I(n-1)-q^{-12s+3+n-1}\omega _\tau (p)^2W_{f_s}(h(p^{n-1},1)), \end{aligned}$$

where in the last equation, we used the formula in the computation of I(n). Since $h(p^{n-1},1)$ is in the center of $M'$, we have $W_{f_s}(h(p^{n-1},1))=q^{-6s(n-1)}\omega _\tau (p)^{n-1}$. The result follows. $\square$

We next consider

$$\begin{aligned} R_2(n)&=\int _{|r_4|>1} \int _{F} W_{f_s}(h(1,p^n)w_\beta w_\alpha w_\beta \mathbf{{x}}_{\alpha +\beta }(p^{-1})\mathbf{{x}}_\beta (r_4)\mathbf{{x}}_{3\alpha +2\beta }(r_5))dr_4dr_5. \end{aligned}$$

Conjugating $w_\beta$ to the right side and using the Iwasawa decomposition of $w_\beta \mathbf{{x}}_\beta (r_4)$, we can get

$$\begin{aligned} R_2(n)=\int _F \int _{|r_4|>1}W_{f_s}(h(1,p^n)w_\beta w_\alpha \mathbf{{x}}_\alpha (p^{-1})\mathbf{{x}}_{3\alpha +\beta }(r_5)\mathbf{{x}}_\beta (r_4^{-1})h(r_4^{-1},r_4))dr_4dr_5. \end{aligned}$$

From the commutator relation, we have

$$\begin{aligned} \mathbf{{x}}_\alpha (p^{-1})\mathbf{{x}}_\beta (r_4^{-1})=\mathbf{{x}}_\beta (r_4^{-1})\mathbf{{x}}_\alpha (p^{-1})\mathbf{{x}}_{2\alpha +\beta }(p^{-2}r_4^{-1})u, \end{aligned}$$

for some u in the group generated by roots subgroups of $\alpha +\beta ,3\alpha +\beta ,3\alpha +2\beta$. Like in the computation of $R_1(n)$, we have

$$\begin{aligned} R_2(n)&=\int _F\int _{|r_4|>1}W_{f_s}(h(1,p^n)w_\beta w_\alpha \mathbf{{x}}_\alpha (p^{-1})\mathbf{{x}}_{3\alpha +\beta }(r_5)h(r_4^{-1},r_4))\psi \\&\quad (-2p^{n-2}r_4^{-1})dr_4dr_5\\&=\int _F \int _{|r_4|>1}W_{f_s}(h(1,p^n)h(r_4^{-1},1)w_\beta \mathbf{{x}}_\beta (r_5r_4^{-1})w_\alpha \mathbf{{x}}_\alpha (p^{-1}r_4^{-1}))\psi (-2p^{n-2}r_4^{-1})\\&\quad dr_4dr_5\\&=\int _F \int _{|r_4|>1}|r_4|W_{f_s}(h(1,p^n)h(r_4^{-1},1)w_\beta \mathbf{{x}}_\beta (r))\psi (-2p^{n-2}r_4^{-1})dr_4dr\\&=I(n)\int _{|r_4|>1}|r_4|^{-6s+1}\omega _\tau (r_4^{-1})\psi (-2p^{n-2}r_4^{-1})dr_4\\&=I(n)\sum _{m= 1}^\infty q^{(-6s+2)m}\omega _\tau (p)^{m}\int _{{\mathfrak {o}}^\times }\psi (-2p^{m+n-2}u)du. \end{aligned}$$

Lemma 4.4

We have

$$\begin{aligned} R_2(n)=\left\{ \begin{array}{lll}I(0)q^{-6s+2}\omega _\tau (p)\left( -q^{-1}+(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)} \right) ,&{} n=0, \\ I(n)(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)},&{} n\ge 1 \end{array}\right. \end{aligned}$$

Proof

If $n\ge 1$, then $\int _{{\mathfrak {o}}^\times }\psi (p^{m+n-2}u)du=(1-q^{-1})$ for $m\ge 1$. Thus, we have

$$\begin{aligned} R_2(n)&=I(n)\sum _{m= 1}^\infty q^{(-6s+2)m}\omega _\tau (p)^{m}(1-q^{-1})\\&=I(n)(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)}. \end{aligned}$$

If $n=0$, then $\int _{{\mathfrak {o}}^\times }\psi (p^{m+n-2}u)du=(1-q^{-1})$ for $m\ge 2$, and $\int _{{\mathfrak {o}}^\times }\psi (p^{m+n-2}u)du=-q^{-1}$ for $m=1$. Thus, we have

$$\begin{aligned} R_2(0)&=I(0)(-q^{-1}q^{-6s+2}\omega _\tau (p)+(1-q^{-1})\sum _{m=2}^\infty q^{(-6s+2)m}\omega _\tau (p)^{m})\\&=I(0)q^{-6s+2}\omega _\tau (p)\left( -q^{-1}+(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)} \right) . \end{aligned}$$

The completes the proof of the lemma. $\square$

Combining the above results, we get the following

Lemma 4.5

We have

$$\begin{aligned} R(n)=\left\{ \begin{array}{lll}-I(0)q^{-6s+1}\omega _\tau (p)\frac{1-q^{-6s+3}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)}, &{} n=0,\\ I(1)(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)}, &{} n=1,\\ q^{-12s+3}\omega _\tau (p)^2I(n-1)-q^{-6s(n+1)+n+2}\omega _\tau (p)^{n+1} &{}\\ \qquad \qquad \qquad +I(n)(1-q^{-1})\frac{q^{-6s+2}\omega _\tau (p)}{1-q^{-6s+2}\omega _\tau (p)}, &{} n\ge 2, \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} J(n)&=J_1(n)-R(n)\\&=\left\{ \begin{array}{lll}1+Y, &{} n=0\\ I(1), &{} n=1,\\ I(n)-q^{-1}Y^2 I(n-1)+q^{-n}Y^{n+1}, &{} n\ge 2. \end{array} \right. \end{aligned}$$

where $Y=q^{-6s+2}\omega _\tau (p)$

By the main result of [1], we have

$$\begin{aligned} \widetilde{W}(t(p^n))=\frac{\mu _\psi (p^{n})q^{-n}}{a-a^{-1}}\left( (1-\chi (p)q^{-1/2}a^{-1})a^{n+1}-(1-\chi (p)q^{-1/2}a)a^{-(n+1)}\right) , \end{aligned}$$

where $\chi (p)=(p,p)_F=(p,-1)_F.$ Note that the notation $\gamma (a)$ in [1] is our $\mu _\psi (a)^{-1}$. Note that $\mu _\psi (p^n)\mu _\psi (p^n)=(p^n,p^n)_F=\chi (p)^n$. Thus

$$\begin{aligned}&I(\widetilde{W},W_{f_s},\phi )=\sum _{n\ge 0}\frac{q^{3n/2}\chi (p)^n}{a-a^{-1}}\\&\quad \left( (1-\chi (p)q^{-1/2}a^{-1})a^{n+1}-(1-\chi (p)q^{-1/2}a)a^{-(n+1)}\right) J(n). \end{aligned}$$

Plugging the formula J(n) into the above equation, we can get that

$$\begin{aligned}&I(\widetilde{W},W_f,\phi )\\&\quad =\frac{(1-b_1q^{-1}X)(1-b_2q^{-1}X)(1-b_1b_2q^{-1}X^2)(1-b_1^2b_2q^{-1}X^3)(1-b_1b_2^2q^{-1}X^3)}{(1-\chi (p)a^{-1}b_1b_2q^{-1/2}X^2)(1-\chi (p)ab_1b_2q^{-1/2}X^2)}\\&\quad \cdot \frac{1}{\prod _{i=1}^2(1-\chi (p)a^{-1}b_iq^{-1/2}X)\prod _{i=1}^2(1-\chi (p)ab_iq^{-1/2}X)}\\&\quad =\frac{L(3s-1,\widetilde{\pi }\times (\chi \otimes \tau ))L(6s-5/2,\widetilde{\pi }\otimes (\chi \otimes \omega _\tau ))}{L(3s-1/2,\tau )L(6s-2,\omega _\tau )L(9s-7/2,\tau \otimes \omega _\tau )}. \end{aligned}$$

Here

$$\begin{aligned} \begin{aligned} L(s,\widetilde{\pi }\otimes (\chi \otimes \omega _{\tau } ))=\frac{1}{(1-a\chi (p)b_1b_2q^{-s})((1-a^{-1}\chi (p)b_1b_2q^{-s}))} \end{aligned} \end{aligned}$$

is the L function of $\widetilde{\pi }$ twisted by the character $\chi \otimes \omega _\tau$, and

$$\begin{aligned} L(s,\widetilde{\pi }\times (\chi \otimes \tau ))=\frac{1}{\prod _{i=1}^2(1-\chi (p)a^{-1}b_iq^{-s})\prod _{i=1}^2(1-\chi (p)ab_iq^{-s})} \end{aligned}$$

is the Rankin–Selberg L-function of $\widetilde{\pi }$ twisted by $\chi \otimes \tau$. We record the above calculation in the following

Proposition 4.6

Let $\widetilde{W}\in {\mathcal {W}}(\widetilde{\pi },\psi )$ be the normalized unramified Whittaker function, $f_s$ be the normalized unramified section in $I(s,\tau )$ and $\phi \in {\mathcal {S}}(F)$ is the characteristic function of ${\mathfrak {o}}$, we have

$$\begin{aligned} I(\widetilde{W},W_{f_s},\phi )=\frac{L(3s-1,\widetilde{\pi }\times (\chi \otimes \tau ))L(6s-5/2,\widetilde{\pi }\otimes (\chi \otimes \omega _\tau ))}{L(3s-1/2,\tau )L(6s-2,\omega _\tau )L(9s-7/2,\tau \otimes \omega _\tau )}. \end{aligned}$$

5 Some local theory

In this section, let F be a local field, which can be archimedean or non-archimedean. If F is non-archimedean, let ${\mathfrak {o}}$ be the ring of integers of F, p be a uniformizer of ${\mathfrak {o}}$ and $q={\mathfrak {o}}/(p)$. Let $\widetilde{\pi }$ be an irreducible genuine generic representation of $\widetilde{\mathrm {SL}}_2(F)$, $\tau$ be an irreducible generic representation of ${\mathrm {GL}}_2(F)$. Let $\psi$ be a nontrivial additive character of F.

Lemma 5.1

Let $\widetilde{W}\in {\mathcal {W}}(\widetilde{\pi },\psi ), f_s\in I(s,\tau ),\phi \in {\mathcal {S}}(F)$, then the integral $I(\widetilde{W},W_{f_s},\phi )$ converges absolutely for ${\mathrm {Re}}(s)$ large and has a meromorphic continuation to the whole s-plane. Moreover, if F is a p-adic field, then $I(\widetilde{W},W_{f_s},\phi )$ is a rational function in $q^{-s}$.

The proof is similar to [5, Lemma 4.2–4.7] and [6, Lemma 3.10, Lemma 3.3]. We omit the details.

Lemma 5.2

Let $s_0\in {\mathbb {C}}$. Then there exists $\widetilde{W}\in {\mathcal {W}}(\widetilde{\pi },\psi ),f_{s_0}\in I(s_0,\tau ),\phi \in {\mathcal {S}}(F)$ such that $I(\widetilde{W},W_{f_{s_0}},\phi )\ne 0$.

Proof

The proof is similar to the proof of [5, Lemma 4.4,4.7], [6, Proposition 3.4]. We omit the details. $\square$

6 Nonvanishing of certain periods on $G_2$

6.1 Poles of Eisenstein series on ${ {G}}_2$

Let $\tau$ be a cuspidal unitary representation of ${\mathrm {GL}}_2({\mathbb {A}})\cong M'({\mathbb {A}})$. Let K be a maximal compact subgroup of ${ {G}}_2({\mathbb {A}})$. Given a $K\cap {\mathrm {GL}}_2({\mathbb {A}})$-finite cusp form f in $\tau$, we can extend f to a function $\widetilde{f}:{ {G}}_2({\mathbb {A}})\rightarrow {\mathbb {C}}$ as in [13, §2]. We then define

$$\begin{aligned} \Phi _{\widetilde{f},s}(g)=\widetilde{f}(g) \delta _{P'}(m')^{s/3+1/2}, \end{aligned}$$

for $g=v'm'k$ with $v'\in V'({\mathbb {A}}),m'\in M'({\mathbb {A}}),k\in K$. Then $\Phi _{\widetilde{f},s}$ is well-defined and $\Phi _{\widetilde{f},s}\in I(\frac{s}{3}+\frac{1}{2},\tau )$. Then we can consider the Eisenstein series

$$\begin{aligned} E(s,\widetilde{f},g)=\sum _{P'(F)\backslash { {G}}_2(F)}\Phi _{\widetilde{f},s}(\gamma g). \end{aligned}$$

Proposition 6.1

The Eisenstein series $E(s,\widetilde{f}, g)$ has a pole on the half plane ${\mathrm {Re}}(s)>0$ if and only if $s=\frac{1}{2}, \omega _\tau =1$ and $L(\frac{1}{2},\tau )\ne 0$.

For a proof of the above proposition, see [16, §1] or [10, §5]. If $\omega _\tau =1$ and $L(\frac{1}{2},\tau )\ne 0$, denote by ${\mathcal {R}}(\frac{1}{2},\tau )$ the space generated by the residues of Eisenstein series $E(s,\widetilde{f},g)$ defined as above. Note that an element $R\in {\mathcal {R}}(\frac{1}{2},\tau )$ is an automorphic form on $G_2({\mathbb {A}})$.

6.2 On the Shimura–Waldspurger lift

Let $\widetilde{\pi }$ be a genuine cuspidal automorphic representation of $\widetilde{{\mathrm {SL}}}_2({\mathbb {A}})$. Let $Wd_\psi (\widetilde{\pi })$ be the Shimura–Waldspurger lift of $\widetilde{\pi }$. Then $Wd_\psi (\widetilde{\pi })$ is a cuspidal representation of ${\mathrm {P}}{\mathrm {GL}}_2({\mathbb {A}})$. A cuspidal automorphic representation $\tau$ is in the image of $Wd_\psi$ if and only if $L(\frac{1}{2},\tau )\ne 0$. Moreover, the correspondence $\widetilde{\pi }\mapsto Wd_\psi (\widetilde{\pi })$ respects the Rankin-Selberg L-functions. For these assertions, see [15] or [2].

6.3 A period on $G_2$

Theorem 6.2

Let $\widetilde{\pi }$ be a genuine cuspidal automorphic representation of $\widetilde{\mathrm {SL}}_2({\mathbb {A}})$ and $\tau$ be a unitary cuspidal automorphic representation of ${\mathrm {GL}}_2({\mathbb {A}})$. Assume that $\omega _\tau =1$ and $L(\frac{1}{2},\tau )\ne 0$. In particular, $\tau$ can be viewed as a cuspidal automorphic representation of ${\mathrm {P}}{\mathrm {GL}}_2({\mathbb {A}})$. If $Wd_\psi (\widetilde{\pi })=\chi \otimes \tau ,$ then there exists $\widetilde{\varphi }\in V_{\widetilde{\pi }}, \phi \in {\mathcal {S}}({\mathbb {A}}), R\in {\mathcal {S}}(\frac{1}{2},\tau )$ such that the period

$$\begin{aligned} \begin{aligned} {\mathcal {P}}(\widetilde{\varphi }, \widetilde{\theta }_\phi ,R)=\int _{{\mathrm {SL}}_2(F)\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\backslash V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)R(vg)dvdg \end{aligned} \end{aligned}$$

is non-vanishing.

Proof

For $\widetilde{\varphi }\in V_\pi , \phi \in {\mathcal {S}}({\mathbb {A}})$ and a good section $\Phi _{\widetilde{f}, s}$ as in Sect. 6.1, by Theorem 3.1 and Proposition 4.6, we have

$$\begin{aligned} I(\widetilde{\varphi },\phi ,\widetilde{f},s)&=\int _{{\mathrm {SL}}_2(F)\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\backslash V({\mathbb {A}})} \widetilde{\varphi }(g)\widetilde{\theta }_{\phi }(vg)E(vg,\Phi _{\widetilde{f},s})dvdg\\&=\int _{N_{{\mathrm {SL}}_2}({\mathbb {A}})\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{U_{\alpha +\beta }({\mathbb {A}})\backslash V({\mathbb {A}})}W_{\widetilde{\varphi }}(g)\omega _\psi (vg)\phi (1)W_{\Phi _{\widetilde{f},s}}(\gamma vg)dvdg \nonumber \\&=I_S \cdot \frac{L^S(s+\frac{1}{2},\widetilde{\pi }\times (\chi \otimes \tau ))L^S(2s+\frac{1}{2},\widetilde{\pi }\otimes (\chi \otimes \omega _\tau ))}{L^S(s+1,\tau )L^S(2s+1,\omega _\tau )L^S(3s+1,\tau \otimes \omega _\tau )}. \nonumber \end{aligned}$$

Here S is a finite set of places of F such that for $v\notin S$, $\pi _v,\tau _v$ are unramified, and $I_S$ is the product of the local zeta integrals over all places $v\in S$ and $L^S$ denotes the partial L-function which is the product of all local L-function as the place v runs over $v\notin S$. Note that $\tau \cong \tau ^\vee$ since $\omega _\tau =1$. Suppose that $Wd_\psi (\widetilde{\pi })=\chi \otimes \tau =\chi \otimes \tau ^\vee$, then $L^S(s+1/2, \widetilde{\pi }\times (\chi \otimes \tau ))$ has a pole at $s=1/2$. Note that at $s=\frac{1}{2}$, $L^{S}(2s+1/2,\widetilde{\pi }\otimes (\chi \otimes \omega _\tau ))$ is holomorphic and nonzero, while $L^S(s+1,\tau )L^S(2s+1,\omega _\tau )L^S(3s+1,\tau \otimes \omega _\tau )$ is holomorphic. Moreover, $I_S$ can be chosen to be nonzero. Thus we get that $I(\widetilde{\varphi },\phi , \widetilde{f},s)$ has a pole at $s=1/2$, which means that there exists a residue R(g) of $E(s,\widetilde{f},g)$ such that

$$\begin{aligned} \begin{aligned} {\mathcal {P}}(\widetilde{\varphi },\theta _\phi ,R)=\int _{{\mathrm {SL}}_2(F)\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\backslash V({\mathbb {A}})}\widetilde{\varphi }(g)\widetilde{\theta }_\phi (vg)R(vg)dvdg\ne 0. \end{aligned} \end{aligned}$$

This completes the proof. $\square$

Remark 6.3

For an $L^2$-automorphic form $\eta \in L^2(G_2(F)\backslash G_2({\mathbb {A}}))$, one can form the period

$$\begin{aligned} \eta _{\widetilde{\phi },\widetilde{\theta }_\phi }(g)=\int _{{\mathrm {SL}}_2(F)\backslash {\mathrm {SL}}_2({\mathbb {A}})}\int _{V(F)\backslash V({\mathbb {A}})}\widetilde{\varphi }(h)\widetilde{\theta }_\phi (vh)\eta (vhg)dvdh. \end{aligned}$$

Theorem 6.2 says that if $\eta \in {\mathcal {S}}(\frac{1}{2},\tau )$, then under the condition $Wd_\psi (\widetilde{\pi })=\chi \otimes \tau$, the period $\eta _{\widetilde{\varphi },\widetilde{\theta }_\phi }$ is non-vanishing for certain $\widetilde{\varphi }$ and $\phi$. For general $\eta$, one can ask under what conditions $\eta _{\widetilde{\varphi },\widetilde{\theta }_\phi }$ is not identically zero as $\widetilde{\varphi }$ varies in $\widetilde{\pi }$ and $\phi \in {\mathcal {S}}({\mathbb {A}})$. In the classical group case, this is the global Gan–Gross–Prasad conjecture for Fourier–Jacobi case, see [3]. It is natural to ask if it is possible to extend the GGP-conjecture to the $G_2$-case.