Theta correspondence and simple factors in global Arthur parameters

Let $F$ be a number field and let $\mathbb{A}$ be its ring of adeles. Let $\pi$ be an irreducible cuspidal automorphic representation of a classical group $G$ defined over $F$. We also treat the case of metaplectic groups in this work. However to avoid excessive notation, we focus on the case of the symplectic groups $G=\operatorname{\mathrm{Sp}}(X)$ in this introduction where $X$ is a non-degenerate symplectic space over $F$. By Arthur’s theory of endoscopy [2], $\pi$ belongs to a global $A$-packet associated to an elliptic global $A$-parameter, which is of the form

where $\tau_{i}$ is an irreducible self-dual cuspidal automorphic representation of $\operatorname{\mathrm{GL}}_{n_{i}}(\mathbb{A})$ and $b_{i}$ is a positive integer which represents the unique $b_{i}$-dimensional irreducible representation of Arthur’s $\operatorname{\mathrm{SL}}_{2}(\mathbb{C})$. See Section 2, for more details.

In [17], Jiang proposed the $(\tau,b)$-theory. See, in particular, Principle 1.2 there. It is a conjecture that uses period integrals to link together automorphic representations in two global $A$-packets whose global $A$-parameters are “different” by a $(\tau,b)$-factor. We explain in more details. Let $\Pi_{\phi}$ denote the global $A$-packet with elliptic global $A$-parameter $\phi$. Let $\pi$ be an irreducible automorphic representation of $G(\mathbb{A})$ and let $\sigma$ be an irreducible automorphic representation of $H(\mathbb{A})$, where $H$ is a factor of an endoscopic group of $G$. Assume that $\pi$ (resp. $\sigma$) occurs in the discrete spectrum. Then it is expected that there exists some kernel function $\mathcal{K}$ depending on $G$, $H$ and $(\tau,b)$ only such that if $\pi$ and $\sigma$ satisfy a Gan–Gross–Prasad type of criterion, namely, that the period integral

is non-vanishing for some choice of $f_{\sigma}\in\sigma$ and $f_{\pi}\in\pi$, then $\pi$ is in the global $A$-packet $\Pi_{\phi}$ if and only if $\sigma$ is in the global $A$-packet $\Pi_{\phi_{2}}$ with $\phi=(\tau,b)\boxplus\phi_{2}$. Then [17, Section 5] proceeds to construct certain kernel functions and then using them, defines endoscopy transfer (by integrating over $H(F)\operatorname{\backslash}H(\mathbb{A})$ only in (0.1)) and endoscopy descent (by integrating over $G(F)\operatorname{\backslash}G(\mathbb{A})$ only in (0.1)). It is not yet known if these are the kernel functions making the statements of Principle 1.2 in [17] hold. As the kernel functions come from Bessel coefficients or Fourier–Jacobi coefficients as in [7, Section 23], we see the non-vanishing of this period integral is analogous to condition (i) in the global Gan–Gross–Prasad Conjecture [7, Conjecture 24.1].

Jiang suggested in [17, Section 7] that if $\tau$ is an automorphic character $\chi$, then the kernel function can be taken to be the theta kernel and endoscopy transfer and endoscopy descent are theta lifts. In this case, the span of

as $f_{\pi}$ runs over $\pi$ is the theta lift of $\pi$. This is an automorphic representation of $H(\mathbb{A})$. Lifting in the other direction is analogous. Assume that the theta lift of $\pi$ is non-zero. Write $\phi_{\pi}$ for the global $A$-parameter of $\pi$. Then [17, Principle 1.2] says that $\phi_{\pi}$ has a $(\chi,b)$-factor and that the global $A$-parameter of the theta lift of $\pi$ from $G$ to $H$ should be $\phi_{\pi}$ with the $(\chi,b)$-factor removed. Here $b$ should be of appropriate size relative to $G$ and $H$. Our work is one step in this direction.

One goal of this article is to expand on the $(\chi,b)$-theory and to present the results of [34], [10], [23], [46], [22], [48] for various cases in a uniform way. As different reductive dual pairs that occur in theta correspondence have their own peculiarities, the notation and techniques of these papers are adapted to the treatment of their own specific cases. We attempt to emphasise on the common traits of the results which are buried in lengthy and technical proofs in these papers.

After collecting the results on poles of $L$-functions, poles of Eisenstein series and theta correspondence, we derive a bound for $b$ when $b$ is maximal among all factors of the global $A$-parameter of $\pi$. In addition, we derive an implication on global $A$-packets. Of course, the heavy lifting was done by the papers mentioned above.

Another way of phrasing this is that we have a bound on the size of $b$ that can occur in a factor of type $(\chi,*)$ in the global $A$-parameter of a cuspidal automorphic representation. Thus our results have application in getting a Ramanujan bound, which measures the departure of the local components $\pi_{v}$ from being tempered for all places $v$ of $F$, for classical groups and metaplectic groups. This should follow by generalising the arguments in [18, Section 5] which treats the symplectic case. There they first established a bound for $b$ under some conditions on wave front sets. This enables them to control the contribution of $\operatorname{\mathrm{GL}}_{1}$-factors in the global $A$-parameter to the Ramanujan bound. Our result can supply this ingredient for classical groups and also metaplectic groups unconditionally. Then [18, Section 5] found a Ramanujan bound for $\pi$ by using the crucial results on the Ramanujan bound for $\operatorname{\mathrm{GL}}_{2}$ in [25] and [3].

We describe the idea of the proof of our result. First we relate the existence of a $(\tau,b)$-factor in the elliptic global $A$-parameter of $\pi$ to the existence of poles of partial $L$-functions $L^{S}(s,\pi\times\tau^{\vee})$. See Proposition 2.8. If the global $A$-parameter of $\pi$ has a factor $(\tau,b)$ where $b$ is maximal among all factors, we can show that the partial $L$-function $L^{S}(s,\pi\times\tau^{\vee})$ has a pole at $s=\frac{1}{2}(b+1)$. Thus studying the location of poles of $L^{S}(s,\pi\times\tau^{\vee})$ for $\tau$ running through all self-dual cuspidal representations of $\operatorname{\mathrm{GL}}_{n}(\mathbb{A}_{F})$ can shed light on the size of $b_{i}$’s that occur in the global $A$-parameter of $\pi$. Then we specialise to the case where $\tau$ is a character $\chi$ and consider $L^{S}(s,\pi\times\chi^{\vee})$ in what follows.

Next we relate the poles of $L^{S}(s,\pi\times\chi^{\vee})$ to the poles of Eisenstein series attached to the cuspidal datum $\chi\boxtimes\pi$. See Section 3. In fact, in some cases, we use the non-vanishing of $L^{S}(s,\pi\times\chi^{\vee})$ at $s=\frac{1}{2}$ instead. See Proposition 3.1. Then we recall in Theorem 3.5 that the maximal positive pole of the Eisenstein series has a bound which is supplied by the study of global theta lifts. This is enough for showing Corollary 5.3, though we have a more precise result that the maximal positive pole corresponds to the invariant called the lowest occurrence index of $\pi$ with respect to $\chi$ in Theorem 4.4. The lowest occurrence index is the minimum of the first occurrence indices over some Witt towers. For the precise definition see (4.2). We also have a less precise result (Theorem 4.1) relating the first occurrence index of $\pi$ with respect to certain quadratic spaces to possibly non-maximal and possibly negative poles of the Eisenstein series.

More precise results can be derived if we assume that $\pi$ has a generic global $A$-parameter. This is because we have a more precise result relating poles or non-vanishing of values of the complete $L$-functions to poles of the Eisenstein series supplied by [19]. Thus we get

Here $Q_{1}$ is a parabolic subgroup of $\operatorname{\mathrm{Sp}}(X_{1})$ with Levi subgroup isomorphic to $\operatorname{\mathrm{GL}}_{1}\times\operatorname{\mathrm{Sp}}(X)$, where $X_{1}$ is the symplectic space formed from $X$ by adjoining a hyperbolic plane. Roughly speaking, $\mathcal{A}^{Q_{1}}(s,\chi\boxtimes\pi)$ is a space of automorphic forms on $\operatorname{\mathrm{Sp}}(X_{1})$ induced from $\chi|\ |^{s}\boxtimes\pi$ viewed as a representation of the parabolic subgroup $Q_{1}$. We refer the reader to Section 3 for the precise definition of $\mathcal{A}^{Q_{1}}(s,\chi\boxtimes\pi)$. We note that the lowest occurrence index $\operatorname{\mathrm{LO}}_{X}^{\chi}(\pi)$ is an invariant in the theory of theta correspondence related to the invariant called the first occurrence index. See Section 4 for their definitions. We also include a result (Theorem 6.3) that concerns the non-vanishing of $L(s,\pi\times\chi^{\vee})$ at $s=\frac{1}{2}$ and the lowest occurrence index. We plan to improve this result in the future by studying a relation between non-vanishing of Bessel or Fourier–Jacobi periods and the lowest occurrence index.

We note that the $L$-function $L(s,\pi\times\chi^{\vee})$ has been well-studied and is intricately entwined with the study of theta correspondence, most prominently in the Rallis inner product formula which says that the inner product of two theta lifts is equal to the residue or value of $L(s,\pi\times\chi^{\vee})$ at an appropriate point up to some ramified factors and some abelian $L$-functions. We refer the reader to [49] which is a culmination of many previous results. See also the references in [49]. In our approach, the Eisenstein series $E(g,f_{s})$, which is not of Siegel type, is the key link between $L(s,\pi\times\chi^{\vee})$ and the theta lifts.

Now we describe the structure of this article. In Section 1, we set up some basic notation. In Section 2, we define elliptic global $A$-parameters for classical groups and metaplectic groups and also the global $A$-packet associated to an elliptic global $A$-parameter. We show how poles of partial $L$-functions detect $(\tau,b)$-factors in an elliptic global $A$-parameter. In Section 3, we define Eisenstein series attached to the cuspidal datum $\chi\boxtimes\pi$ and recall some results on the possible locations of their maximal positive poles. In Section 4, we introduce two invariants of theta correspondence. They are the first occurrence index $\operatorname{\mathrm{FO}}_{X}^{Y,\chi}(\pi)$ and the lowest occurrence index $\operatorname{\mathrm{LO}}_{X}^{\chi}(\pi)$ of $\pi$ with respect to some data. We relate them to poles of Eisenstein series. Results in Sections 3 and 4 are not new. Our aim is to present the results in a uniform way for easier access. In Section 5, we show a bound for $b$ in $(\chi,b)$-factors of the global $A$-parameter of $\pi$. Finally in Section 6, we consider the case when $\pi$ has a generic global $A$-parameter. We show that when $L(s,\pi\times\chi^{\vee})$ has a pole at $s=1$ (resp. $L(s,\pi\times\chi^{\vee})$ is non-vanishing at $s=\frac{1}{2}$), the lowest occurrence index is determined.

The author would like to thank Professor Dihua Jiang for suggesting the topic to her. The author would also like to thank the referee for reading the manuscript carefully and for informing her of some recent developments.

Let $F$ be a number field and let $E$ be either $F$ or a quadratic field extension of $F$. Let $\varrho\in\operatorname{\mathrm{Gal}}(E/F)$ be the trivial Galois element when $E=F$ and the non-trivial Galois element when $E\neq F$. When $E\neq F$, write $\varepsilon_{E/F}$ for the quadratic character associated to $E/F$ via Class field theory. Let $G$ be an algebraic group over $E$. We write $\operatorname{\mathrm{R}}_{E/F}G$ for the restriction of scalars of Weil. This is an algebraic group over $F$.

Let $\epsilon$ be either $1$ or $-1$. By an $\epsilon$-skew Hermitian space, we mean an $E$-vector space $X$ together with an $F$-bilinear pairing

such that

for all $a,b\in E$ and $x,y\in X$. We consider the linear transformations of $X$ to act from the right. We follow [49]’s notation closely and we intend to generalise the results here to the quaternionic unitary group case in our future work.

Let $X$ be an $\epsilon$-skew Hermitian space of finite dimension. Then the isometry group of $X$ is one of the following:

1. (1)

   the symplectic group $\operatorname{\mathrm{Sp}}(X)$ when $E=F$ and $\epsilon=1$;

2. (2)

   the orthogonal group $\mathrm{O}(X)$ when $E=F$ and $\epsilon=-1$;

3. (3)

   the unitary group $U(X)$ when $E\neq F$ and $\epsilon=\pm 1$.

We will also consider the metaplectic group. Let $v$ be a place of $F$ and let $F_{v}$ denote the completion of $F$ at $v$. Let $\mathbb{A}_{F}$ (resp. $\mathbb{A}_{E}$) denote the ring of adeles of $F$ (resp. $E$). Set $\mathbb{A}:=\mathbb{A}_{F}$. Write $\operatorname{\mathrm{Mp}}(X)(F_{v})$ (resp. $\operatorname{\mathrm{Mp}}(X)(\mathbb{A}_{F})$) for the metaplectic double cover of $\operatorname{\mathrm{Sp}}(X)(F_{v})$ (resp. $\operatorname{\mathrm{Sp}}(X)(\mathbb{A}_{F})$) defined by Weil [43]. We note that the functor $\operatorname{\mathrm{Mp}}(X)$ is not representable by an algebraic group. We will also need the $\mathbb{C}^{1}$-extension $\operatorname{\mathrm{Mp}}(X)(F_{v})\times_{\mu_{2}}\mathbb{C}^{1}$ of $\operatorname{\mathrm{Sp}}(X)(F_{v})$ and we denote it by $\operatorname{\mathrm{Mp}}^{\mathbb{C}^{1}}(F_{v})$. Similarly we define $\operatorname{\mathrm{Mp}}^{\mathbb{C}^{1}}(\mathbb{A}_{F})$.

Let $\psi$ be a non-trivial automorphic additive character of $\mathbb{A}_{F}$ which will figure in the Weil representations as well as the global $A$-parameters for $\operatorname{\mathrm{Mp}}(X)$.

For an automorphic representation or admissible representation $\pi$, we write $\pi^{\vee}$ for its contragredient.

First we recall the definition of elliptic global Arthur parameters ($A$-parameters) for classical groups as well as metaplectic groups. See [2] for the symplectic and the special orthogonal case and we adopt the formulation in [1] for the case of the (disconnected) orthogonal groups. For the unitary case, see [35, 26]. For the metaplectic case, see [9]. Then we focus on simple factors of global Arthur parameters and relate their presence to poles of partial $L$-functions. This is a crude first step for detecting $(\tau,b)$-factors in an elliptic global $A$-parameter according to the ‘$(\tau,b)$-theory’ proposed in [17].

Let $\mathbf{G}$ be $\operatorname{\mathrm{U}}(X)$, $\mathrm{O}(X)$, $\operatorname{\mathrm{Sp}}(X)$ or $\operatorname{\mathrm{Mp}}(X)$. Let $d$ denote the dimension of $X$. Set $\mathbf{G}^{\circ}=\operatorname{\mathrm{SO}}(X)$ when $\mathbf{G}=\mathrm{O}(X)$. Set $\mathbf{G}^{\circ}=\mathbf{G}$ otherwise. Write $\check{\mathbf{G}}$ for the (complex) dual group of $\mathbf{G}^{\circ}$. Then

An elliptic global $A$-parameter for $\mathbf{G}$ is a finite formal sum of the form

where

1. (1)

   $\tau_{i}$ is an irreducible conjugate self-dual cuspidal automorphic representation of $\operatorname{\mathrm{GL}}_{n_{i}}(\mathbb{A}_{E})$;

2. (2)

   $b_{i}$ is a positive integer which represents the unique $b_{i}$-dimensional irreducible representation of Arthur’s $\operatorname{\mathrm{SL}}_{2}(\mathbb{C})$

such that

• •

  $\sum_{i}n_{i}b_{i}=d_{\check{\mathbf{G}}}$;

• •

  $\tau_{i}$ is conjugate self-dual of parity $(-1)^{N_{\check{\mathbf{G}}}+b_{i}}$ (see Remark 2.3);

• •

  the factors $(\tau_{i},b_{i})$ are pairwise distinct.

Here $d_{\check{\mathbf{G}}}$ is the degree of the standard representation of $\check{\mathbf{G}}$ which, explicitly, is

and

Let $\Psi_{2}(\mathbf{G})$ denote the set of elliptic global $A$-parameters of $\mathbf{G}$. Let $\phi\in\Psi_{2}(\mathbf{G})$. Via the local Langlands conjecture (which is proved for the general linear groups), at every place $v$ of $F$, we localise $\phi$ to get an elliptic local $A$-parameter,

where $W_{F_{v}}$ is the Weil group of $F_{v}$ and $L_{F_{v}}$ is $W_{F_{v}}$ if $v$ is archimedean and the Weil–Deligne group $W_{F_{v}}\times\operatorname{\mathrm{SL}}_{2}(\mathbb{C})$ if $v$ is non-archimedean. To $\phi_{v}$ we associate the local $L$-parameter $\varphi_{\phi_{v}}:L_{F_{v}}\rightarrow\check{\mathbf{G}}\rtimes W_{F_{v}}$ given by

Let $L^{2}_{\mathrm{disc}}(\mathbf{G})$ denote the discrete part of $L^{2}(\mathbf{G}(F)\operatorname{\backslash}\mathbf{G}(\mathbb{A}_{F}))$ when $\mathbf{G}\neq\operatorname{\mathrm{Mp}}(X)$ and the genuine discrete part of $L^{2}(\operatorname{\mathrm{Sp}}(F)\operatorname{\backslash}\operatorname{\mathrm{Mp}}(\mathbb{A}_{F}))$ for $\mathbf{G}=\operatorname{\mathrm{Mp}}(X)$. Define the full near equivalence class $L^{2}_{\phi,\psi}(\mathbf{G})$ attached to the elliptic global $A$-parameter $\phi$ to be the Hilbert direct sum of all irreducible automorphic representations $\sigma$ occurring in $L^{2}_{\mathrm{disc}}(\mathbf{G})$ such that for almost all $v$, the local $L$-parameter of $\sigma_{v}$ is $\varphi_{\phi_{v}}$. We remark that in the $\operatorname{\mathrm{Mp}}(X)$-case, the parametrisation of $\sigma_{v}$ is relative to $\psi_{v}$ since the local $L$-parameter of $\sigma_{v}$ is attached via the Shimura–Waldspurger correspondence which depends on $\psi_{v}$. This is the only case in this article where $L^{2}_{\phi,\psi}(\mathbf{G})$ depends on $\psi$.

Let $\mathcal{A}_{2}(\mathbf{G})$ denote the dense subspace consisting of automorphic forms in $L^{2}_{\mathrm{disc}}(\mathbf{G})$. Similarly define $\mathcal{A}_{2,\phi,\psi}(\mathbf{G})$ to be the dense subspace of $L^{2}_{\phi,\psi}(\mathbf{G})$ consisting of automorphic forms. Then we have a crude form of Arthur’s multiplicity formula which decomposes the $L^{2}$-discrete spectrum into near equivalence classes indexed by $\Psi_{2}(\mathbf{G})$.

We have some further remarks on the orthogonal and unitary cases.

By Theorem 2.4, we get

Given an irreducible cuspidal automorphic representation $\pi$, write $\phi_{\pi}$ for the global $A$-parameter of $\pi$. By studying poles of $L^{S}(s,\pi\times\tau^{\vee})$ for varying $\tau$’s, we can detect the existence of $(\tau,b)$-factors with maximal $b$ in $\phi_{\pi}$. We would also like to construct an irreducible cuspidal automorphic representation with global $A$-parameter $\phi_{\pi}\boxminus(\tau,b)$ which means removing the $(\tau,b)$-factor from $\phi_{\pi}$ if $\phi_{\pi}$ has a $(\tau,b)$-factor. Doing this recursively, we will be able to compute the global $A$-parameter of a given irreducible cuspidal automorphic representation. In reverse, the construction should produce concrete examples of cuspidal automorphic representations in a given global $A$-packet with an elliptic global $A$-parameter. This will be investigated in our future work.

In this article, we focus our attention on the study of poles of $L^{S}(s,\pi\times\tau^{\vee})$ where $\tau$ is a conjugate self-dual irreducible cuspidal automorphic representation of $\operatorname{\mathrm{R}_{E/F}\operatorname{\mathrm{GL}}}_{1}(\mathbb{A})$. Now we write $\chi$ for $\tau$ to emphasise that we are considering the case of twisting by characters. This case has been well-studied and it is known that the poles of $L^{S}(s,\pi\times\chi^{\vee})$ are intricately related to invariants of theta correspondence via the Rallis inner product formula which relates the inner product of two theta lifts to a residue or a value of the $L$-function. We refer the readers to [27, 45, 12, 49] for details. One of the key steps is the regularised Siegel–Weil formula which relates a theta integral to a residue or a value of a Siegel Eisenstein series. Our work considers an Eisenstein series which is not of Siegel type, but which is closely related to $L(s,\pi\times\chi^{\vee})$.

In this section we deviate slightly from the notation in Section 2. We use $G(X)$ to denote one of $\operatorname{\mathrm{Sp}}(X)$, $\mathrm{O}(X)$ and $\operatorname{\mathrm{U}}(X)$. We let $\mathbf{G}(X)$ be a cover group of $G(X)$, which means $\mathbf{G}(X)=\operatorname{\mathrm{Sp}}(X)$ or $\operatorname{\mathrm{Mp}}(X)$ if $G(X)=\operatorname{\mathrm{Sp}}(X)$, $\mathbf{G}(X)=\mathrm{O}(X)$ if $G(X)=\mathrm{O}(X)$ and $\mathbf{G}(X)=\operatorname{\mathrm{U}}(X)$ if $G(X)=\operatorname{\mathrm{U}}(X)$. We adopt similar notation to that in [36]. We define Eisenstein series on a larger group of the same type as $\mathbf{G}(X)$ and collect some results on their maximal positive poles.

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathbf{G}(X)(\mathbb{A})$. We always assume that $\pi$ is genuine when $\mathbf{G}(X)=\operatorname{\mathrm{Mp}}(X)$. Let $\chi$ be a conjugate self-dual automorphic character of $\operatorname{\mathrm{R}}_{E/F}\operatorname{\mathrm{GL}}_{1}(\mathbb{A})=\mathbb{A}_{E}^{\times}$. When $E\neq F$, we define

Let $a$ be a positive integer. Let $X_{a}$ be the $\epsilon$-skew Hermitian space over $E$ that is formed from $X$ by adjoining $a$-copies of the hyperbolic plane. More precisely, let $\ell_{a}^{+}$ (resp. $\ell_{a}^{-}$) be a totally isotropic $a$-dimensional $E$-vector space spanned by $e_{1}^{+},\ldots,e_{a}^{+}$ (resp. $e_{1}^{-},\ldots,e_{a}^{-}$) such that $\langle{e_{i}^{+}},{e_{j}^{-}}\rangle=\delta_{ij}$ where $\delta_{ij}$ is the Kronecker symbol. Then $X_{a}=\ell_{a}^{+}\oplus X\oplus\ell_{a}^{-}$ with $X$ orthogonal to $\ell_{a}^{+}\oplus\ell_{a}^{-}$.

Let $G(X_{a})$ be the isometry group of $X_{a}$. Let $Q_{a}$ be the parabolic subgroup of $G(X_{a})$ that stabilises $\ell_{a}^{-}$. Write $Q_{a}=M_{a}N_{a}$ in the Levi decomposition with $N_{a}$ being the unipotent radical and $M_{a}$ the standard Levi subgroup. We have an isomorphism

where we identify $\operatorname{\mathrm{R}_{E/F}\operatorname{\mathrm{GL}}}_{a}$ with $\operatorname{\mathrm{R}_{E/F}\operatorname{\mathrm{GL}}}(\ell_{a}^{+})$. Let $\rho_{Q_{a}}$ be the half sum of the positive roots in $N_{a}$, which can be viewed as an element in $\mathfrak{a}_{M_{a}}^{*}:=\operatorname{\mathrm{Rat}}(M_{a})\otimes_{\mathbb{Z}}\mathbb{R}$ where $\operatorname{\mathrm{Rat}}(M_{a})$ is the group of rational characters of $M_{a}$. We note that as $Q_{a}$ is a maximal parabolic subgroup, $\mathfrak{a}_{M_{a}}^{*}$ is one-dimensional. Via the Shahidi normalisation [41], we identify $\mathfrak{a}_{M_{a}}^{*}$ with $\mathbb{R}$ and thus may regard $\rho_{Q_{a}}$ as the real number

Let $K_{a,v}$ be a good maximal compact subgroup of $G(X_{a})(F_{v})$ in the sense that the Iwasawa decomposition holds and set $K_{a}=\prod_{v}K_{a,v}$.

Let $\mathcal{A}^{Q_{a}}(s,\chi\boxtimes\pi)$ denote the space of $\mathbb{C}$-valued smooth functions $f$ on $N_{a}(\mathbb{A})M_{a}(F)\operatorname{\backslash}G(X_{a})(\mathbb{A})$ such that

1. (1)

   $f$ is right $K_{a}$-finite;

2. (2)

   for any $x\in\operatorname{\mathrm{R}_{E/F}\operatorname{\mathrm{GL}}}_{a}(\mathbb{A})$ and $g\in G(X_{a})(\mathbb{A})$ we have

   $$f(m(x,I)g)=\chi(\det(x))|\det(x)|_{\mathbb{A}_{E}}^{s+\rho_{Q_{a}}}f(g);$$

3. (3)

   for any fixed $k\in K_{a}$, the function $h\mapsto f(m(I,h)k)$ on $G(X)(\mathbb{A})$ is in the space of $\pi$.

Now let $\mathbf{G}(X)=\operatorname{\mathrm{Mp}}(X)$. This case depends on $\psi$. Let $\widetilde{\operatorname{\mathrm{GL}}}_{1}(F_{v})$ be the double cover of $\operatorname{\mathrm{GL}}_{1}(F_{v})$ defined as follows. As a set it is $\operatorname{\mathrm{GL}}_{1}(F_{v})\times\mu_{2}$ and the multiplication is given by

which has a Hilbert symbol twist when multiplying the $\mu_{2}$-part. Analogously we define the double cover $\widetilde{\operatorname{\mathrm{GL}}}_{1}(\mathbb{A})$ of $\operatorname{\mathrm{GL}}_{1}(\mathbb{A})$. Let $\chi_{\psi,v}$ denote the genuine character of $\widetilde{\operatorname{\mathrm{GL}}}_{1}(F_{v})$ defined by

where $\gamma_{v}(\cdot,\psi_{\frac{1}{2},v})$ is a $4$-th root of unity defined via the Weil index. It is the same one as in [8, page 521] except that we have put in the subscripts $v$. Then

is a genuine automorphic character of $\widetilde{\operatorname{\mathrm{GL}}}_{1}(\mathbb{A})$. Let $\widetilde{K}_{a}$ denote the preimage of $K_{a}$ under the projection $\operatorname{\mathrm{Mp}}(X_{a})(\mathbb{A})\rightarrow\operatorname{\mathrm{Sp}}(X_{a})(\mathbb{A})$. We will also use $\widetilde{\ }$ to denote the preimages of other subgroups of $\operatorname{\mathrm{Sp}}(X_{a})(\mathbb{A})$. Let $\widetilde{m}$ be the isomorphism

that lifts $m:\operatorname{\mathrm{GL}}_{a}(\mathbb{A})\times G(X)(\mathbb{A})\rightarrow M_{a}(\mathbb{A})$. Let $\widetilde{\det}$ be the homomorphism

We keep writing $\det$ for the non-genuine homomorphism

Given a non-genuine representation $\tau$ of $\widetilde{\operatorname{\mathrm{GL}}}_{a}(\mathbb{A})$, we can twist it by $\chi_{\psi}\circ\widetilde{\det}$ to get a genuine representation which we denote by $\tau\chi_{\psi}$.

We remark that there are canonical embeddings of $N_{a}(\mathbb{A})$ and $\operatorname{\mathrm{Sp}}(X_{a})(F)$ to $\operatorname{\mathrm{Mp}}(X_{a})(\mathbb{A})$, so we may regard them as subgroups of $\mathbf{G}(X_{a})(\mathbb{A})$. Let $\mathcal{A}^{Q_{a}}_{\psi}(s,\chi\boxtimes\pi)$ denote the space of $\mathbb{C}$-valued smooth functions $f$ on $N_{a}(\mathbb{A})M_{a}(F)\operatorname{\backslash}\mathbf{G}(X_{a})(\mathbb{A})$ such that

1. (1)

   $f$ is right $\widetilde{K}_{a}$-finite;

2. (2)

   for any $x\in\widetilde{\operatorname{\mathrm{GL}}}_{a}(\mathbb{A})$ and $g\in\mathbf{G}(X_{a})(\mathbb{A})$ we have

   $$f(\widetilde{m}(x,I)g)=\chi\chi_{\psi}(\widetilde{\det}(x))|\det(x)|_{\mathbb{A}_{E}}^{s+\rho_{Q_{a}}}f(g);$$

3. (3)

   for any fixed $k\in\widetilde{K}_{a}$, the function $h\mapsto f(\widetilde{m}(I,h)k)$ on $\mathbf{G}(X)(\mathbb{A})$ is in the space of $\pi$.

To unify notation, we will also write $\mathcal{A}^{Q_{a}}_{\psi}(s,\chi\boxtimes\pi)$ for $\mathcal{A}^{Q_{a}}(s,\chi\boxtimes\pi)$ in the non-metaplectic case. It should be clear from the context whether we are treating the $\operatorname{\mathrm{Sp}}(X)$ case or the $\operatorname{\mathrm{Mp}}(X)$ case.

Now return to the general case, so $\mathbf{G}(X)$ is one of $\operatorname{\mathrm{Sp}}(X)$, $\mathrm{O}(X)$, $\operatorname{\mathrm{U}}(X)$ and $\operatorname{\mathrm{Mp}}(X)$. Let $f_{s}$ be a holomorphic section of $\mathcal{A}^{Q_{a}}_{\psi}(s,\chi\boxtimes\pi)$. We associate to it the Eisenstein series

Note that the series is over $\gamma\in Q_{a}(F)\operatorname{\backslash}\operatorname{\mathrm{Sp}}(X_{a})(F)$ when $\mathbf{G}(X)=\operatorname{\mathrm{Mp}}(X)$. By Langlands’ theory of Eisenstein series [36, IV.1], this series is absolutely convergent for $\operatorname{\mathrm{Re}}(s)>\rho_{Q_{a}}$, has meromorphic continuation to the whole $s$-plane, its poles lie on root hyperplanes and there are only finitely many poles in the positive Weyl chamber. By our identification of $\mathfrak{a}_{M_{a}}^{*}$ with $\mathbb{R}$ and the fact that $\chi$ is conjugate self-dual, the statements on poles mean that the poles are all real and that there are finitely many poles in the half plane $\operatorname{\mathrm{Re}}(s)>0$.