Periods and $(\chi,b)$-factors of Cuspidal Automorphic Forms of Metaplectic Groups

Let $G$ be a classical group over a number field $F$. Let $\mathbb{A}$ be the ring of adeles of $F$. Let $\sigma$ be an automorphic representation of $G(\mathbb{A})$ in the discrete spectrum, though later we consider only cuspidal automorphic representations. By the theory of endoscopic classification developed by [3] and extended by [31, 24], one attaches a global $A$-parameter to $\sigma$. These works depend on the stabilisation of the twisted trace formula which has been established by a series of works by Mœglin and Waldspurger. We refer the readers to their books [33, 34]. Via the Shimura–Waldspurger correspondence of the metaplectic group $\operatorname{\mathrm{Mp}}_{2n}(\mathbb{A})$, which is the non-trivial double cover of the symplectic group $\operatorname{\mathrm{Sp}}_{2n}(\mathbb{A})$, and odd special orthogonal groups, Gan and Ichino [9] also attached global $A$-parameters to genuine automorphic representations of $\operatorname{\mathrm{Mp}}_{2n}(\mathbb{A})$ in the discrete spectrum. This depends on the choice of an additive character $\psi$ that is used in the Shimura–Waldspurger correspondence. In this introduction, we also let $G$ denote $\operatorname{\mathrm{Mp}}_{2n}$ by abuse of notation and we suppress the dependence on $\psi$. The $A$-parameter $\phi(\sigma)$ of $\sigma$ is a formal sum

where $\tau_{i}$ is an irreducible self-dual cuspidal automorphic representation of some $\operatorname{\mathrm{GL}}_{n_{i}}(\mathbb{A})$, $b_{i}$ is a positive integer which represents the unique $b_{i}$-dimensional irreducible representation of Arthur’s $\operatorname{\mathrm{SL}}_{2}(\mathbb{C})$, together with some conditions on $(\tau_{i},b_{i})$ so that the type is compatible with the type of the dual group of $G$. When $G$ is a unitary group, then we need to introduce a quadratic field extension of $F$. As this case is not the focus of this article, we refer the readers to [42, Sec. 1.3] for details. The exact conditions for $\operatorname{\mathrm{Mp}}_{2n}$ are spelled out in Sec. 2. Here we have adopted the notation of [19] where the theory of $(\tau,b)$ was introduced.

One principle of the $(\tau,b)$-theory is that if $\phi(\sigma)$ has $(\tau,b)$ as a simple factor, then there should exist a kernel function constructed out of $(\tau,b)$ that transfers $\sigma$ to a certain automorphic representation $\mathcal{D}(\sigma)$ in the $A$-packet attached to ‘$\phi(\sigma)\boxminus(\tau,b)$’, i.e., $\phi(\sigma)$ with the factor $(\tau,b)$ removed. From the work of Rallis [36] and Kudla–Rallis [25], it is clear when $\tau$ is a character $\chi$, the kernel function is the theta kernel possibly twisted by $\chi$. (See Sec. 7 of [19].)

Now we focus on the case of the metaplectic groups. We will put the additive character $\psi$ back into the notation to emphasise, for example, that the $L$-functions etc. depend on it. Let $X$ be a $2n$-dimensional symplectic space. We will now write $\widetilde{G}(X)$ for $\operatorname{\mathrm{Mp}}_{2n}$ to match the notation in the body of the article.

The poles of the tensor product $L$-function $L_{\psi}(s,\sigma\times\tau)$ detect the existence of $(\tau,b)$-factors in the $A$-parameter $\phi_{\psi}(\sigma)$. As a first step, we consider the case when $\tau$ is a quadratic automorphic character $\chi$ of $\operatorname{\mathrm{GL}}_{1}(\mathbb{A})$. The $L$-function $L_{\psi}(s,\sigma\times\chi)$ has been well-studied. By the regularised Rallis inner product formula [25] which was proved using the regularised Siegel-Weil formula also proved there and the doubling method construction [35] (see also [29, 6, 43]), the partial $L$-function $L_{\psi}^{S}(s,\sigma\times\chi)$ detects whether the theta lifts of $\sigma$ to the odd orthogonal groups in certain Witt towers vanish or not. The complete theory that interprets the existence of poles or non-vanishing of the complete $L$-function as the obstruction to the local-global principle of theta correspondence was done in [43] and then extended to the ‘second term’ range by [11]. We will relate certain invariants of the theta lifts of $\sigma$ to the existence of $(\chi,b)$-factors in the $A$-parameter $\phi_{\psi}(\sigma)$.

We now describe our main results which are concerned with the metaplectic case, while pointing out the similarity to the statements in the symplectic case. Let $\operatorname{\mathrm{LO}}_{\psi,\chi}(\sigma)$ denote the lowest occurrence index of $\sigma$ in the theta correspondence to all Witt towers associated to the quadratic character $\chi$. (See Sec. 4 for the precise definition.) Let $\mathcal{A}_{\mathrm{cusp}}(\widetilde{G}(X))$ denote the set of all irreducible genuine cuspidal automorphic representations of $\widetilde{G}(X)(\mathbb{A})$. We note that there is no algebraic group $\widetilde{G}(X)$ and that the notation is used purely for aesthetic reason. Then we get the following constraint on $(\chi,b)$-factors of $\phi_{\psi}(\sigma)$ in terms of the invariant $\operatorname{\mathrm{LO}}_{\psi,\chi}(\sigma)$.

The symplectic analogue of our result on the existence of $(\chi,b)$-factors in the $A$-parameter of $\sigma$ is a key input to [20, Sec. 5] which gives a bound on the exponent that measures the departure from temperedness of the local components of cuspidal automorphic representations. In this sense, our results have bearings on the generalised Ramanujan conjecture as proposed in [38].

Following an idea of Mœglin’s in [30] which considered the even orthogonal and the symplectic case, we consider, instead of the partial $L$-function $L^{S}(s,\sigma\times\chi)$, the Eisenstein series $E(g,f_{s})$ attached to the cuspidal datum $\chi\otimes\sigma$ (c.f. Sec. 3). The theorem above is derived from our results on poles of $E(g,f_{s})$. We get much more precise relation between poles of the Eisenstein series and $\operatorname{\mathrm{LO}}_{\psi,\chi}(\sigma)$. This line of investigation has been taken up by [10] in the odd orthogonal group case and by [22, 42] in the unitary group case. We treat the metaplectic group case in this paper. Let $\mathcal{P}_{1}(\chi,\sigma)$ denote the set of all positive poles of the Eisenstein series we consider. Then we get:

It is known by Langlands’ theory [27] of Eisenstein series that the pole in the theorem is simple. (See also [32].) We note that the equality in the theorem is actually informed by considering periods of residues of the Eisenstein series. We get only the upper bound $2j+1$ by considering only poles of Eisenstein series. The odd orthogonal case was considered in [10], the symplectic case in [23] and the unitary case in [42]. Each case has its own technicalities. In this paper, we extend all results in the symplectic case to the metaplectic case to complete the other half of the picture. As pointed out in [6], unlike the orthogonal or symplectic case, the local factors change in the metaplectic case when taking contragredient. Thus it is important, in the present case, to employ complex conjugation in the definition of the global theta lift.

Let $\operatorname{\mathrm{FO}}_{\psi}^{Y}(\sigma)$ denote the first occurrence index of $\sigma$ in the Witt tower of $Y$ where $Y$ is a quadratic space (c.f. Sec. 4). We get a result on $\operatorname{\mathrm{FO}}_{\psi}^{Y}(\sigma)$ and the non-vanishing of periods of residues of the Eisenstein series (or distinction by subgroups of $G(X)$). Note that $\operatorname{\mathrm{FO}}_{\psi}^{Y}(\sigma)$ is a finer invariant than $\operatorname{\mathrm{LO}}_{\psi,\chi}(\sigma)$ while the non-vanishing of periods of residues provides more information than locations of simple poles do. The following theorem is part of Cor. 6.3. Let $\mathcal{E}_{s_{0}}(g,f_{s})$ denote the residue of the Eisenstein series $E(g,f_{s})$ at $s=s_{0}$ and let $\theta_{\psi,X_{1},Y}(g,1,\Phi)$ be the theta function attached to the Schwartz function $\Phi$. The space $X_{1}$ is the symplectic space formed by adjoining a hyperbolic plane to $X$ and similarly $Z_{1}$ is formed by adjoining the same hyperbolic place to $Z$. Thus $Z_{1}\subset X_{1}$. Let $L\subset Z\subset Z_{1}$ be a totally isotropic subspace. Then the group $J(Z_{1},L)$ is defined to be the subgroup of $G(Z_{1})$ that fixes $L$ element-wise.

We hope our period integrals can inform problems in Arithmetic Geometry. For example, similar work in [10] has been used in [4]. (See also [5] and [14]). It should be pointed out that unlike the orthogonal or the unitary case, our period integrals may involve a Jacobi group $J(Z_{1},L)$ when $L$ is non-trivial (which occurs half of the time). This is to account for the lack of ‘odd symplectic group’.

The structure of the paper is as follows. Since we aim to generalise results of symplectic groups to metaplectic groups, we try not deviate too much from the structure of [23]. Many of the results there generalise readily, in which case, we put in a few words to explain why this is so. However at several points, new input is needed to get the proof through, in which case, we derive the results in great detail, noting all the subtleties. We hope in this way, this article makes the flow of arguments more evident than [23] in which we dealt with many technicalities.

In Sec. 1, we set up some common notation and in Sec. 2, we describe the global $A$-parameters of genuine automorphic representations of the metaplectic group following [9]. In Sec. 3, we introduce the Eisenstein series attached to the cuspidal datum $\chi\otimes\sigma$ and deduce results on their maximal poles. In Sec. 4, we define the two invariants, the first occurrence index and the lowest occurrence index of theta lifts. We deduce a preliminary result on the relation between the lowest occurrence index and the maximal pole of the Eisenstein series and hence get a characterisation of the locations of poles of the partial $L$-function. In this way, we are able to show that the lowest occurrence index poses constraints on the existence of $(\chi,b)$-factors in the $A$-parameter of $\sigma$. Then in Sec. 5, we turn to studying Fourier coefficients of theta lifts and we get results on vanishing and non-vanishing of certain periods on $\sigma$. Inspired by these period integrals, in Sec. 6, we study the augmented integrals which are periods of residues of Eisenstein series. We make use of the Arthur truncation to regularise the integrals. We derive results on first occurrence indices and the vanishing and the non-vanishing of periods of residues of Eisenstein series. To evaluate our period integral, we cut it into many pieces according to certain orbits in a flag variety in Sec. 6 and we devote Sec. 7 to the computation of each piece.

The author would like to thank her post-doctoral mentor, Dihua Jiang, for introducing her to the subject and leading her down this fun field of research.

Since this article aims at generalising results in [23] to the case of metaplectic groups, we will adopt similar notation to what was used there.

Let $F$ be a number field and $\mathbb{A}=\mathbb{A}_{F}$ be its ring of adeles. Let $(X,\langle{\ },{\ }\rangle_{X})$ be a non-degenerate symplectic space over $F$. For a non-negative integer $a$, let $\mathcal{H}_{a}$ be the direct sum of $a$ copies of the hyperbolic plane. We form the augmented symplectic space $X_{a}$ by adjoining $\mathcal{H}_{a}$ to $X$. Let $e_{1}^{+},\ldots,e_{a}^{+},e_{1}^{-},\ldots,e_{a}^{-}$ be a basis of $\mathcal{H}_{a}$ such that $\langle{e_{i}^{+}},{e_{j}^{-}}\rangle=\delta_{ij}$ and $\langle{e_{i}^{+}},{e_{j}^{+}}\rangle=\langle{e_{i}^{-}},{e_{j}^{-}}\rangle=0$, for $i,j=1,\ldots,a$, where $\delta_{ij}$ is the Kronecker delta. Let $\ell_{a}^{+}$ (resp. $\ell_{a}^{-}$) be the span of $e_{i}^{+}$’s (resp. $e_{i}^{-}$’s). Then $\mathcal{H}_{a}=\ell_{a}^{+}\oplus\ell_{a}^{-}$ is a polarisation of $\mathcal{H}_{a}$. Let $G(X_{a})$ be the isometry group of $X_{a}$ with the action on the right.

Let $L$ be an isotropic subspace of $X$. Let $Q(X,L)$ denote the parabolic subgroup of $G(X)$ that stabilises $L$ and let $N(X,L)$ denote its unipotent radical. We write $Q_{a}$ (resp. $N_{a}$) for $Q(X_{a},\ell_{a}^{-})$ (resp. $N(X_{a},\ell_{a}^{-})$) as shorthand. We also write $M_{a}$ for the Levi subgroup of $Q_{a}$ such that with respect to the ‘basis’ $\ell_{a}^{+},X,\ell_{a}^{-}$, $M_{a}$ consists of elements of the form

for $x\in\operatorname{\mathrm{GL}}_{a}$ and $h\in G(X)$ where $x^{*}$ is the adjoint of $x$. Fix a maximal compact subgroup $K_{G(X),v}$ of $G(X)(F_{v})$ for each place $v$ of $F$ and set $K_{G(X)}=\prod_{v}K_{G(X),v}$. We require that $K_{G(X)}$ is good in the sense that the Iwasawa decomposition holds and that it is compatible with the Levi decomposition (c.f. [32, I.1.4]). Most often we abbreviate $K_{G(X_{a})}$ as $K_{a}$.

Next we describe the metaplectic groups. Let $v$ be a place of $F$. Let $\widetilde{G}(X_{a})(F_{v})$ denote the metaplectic double cover of $G(X_{a})(F_{v})$. It sits in the unique non-trivial central extension

unless $F_{v}=\mathbb{C}$, in which case, the double cover splits. Let $\widetilde{G}(X_{a})(\mathbb{A})$ be the double cover of $G(X_{a})(\mathbb{A})$ defined as the restricted product of $\widetilde{G}(X_{a})(F_{v})$ over all places $v$ modulo the group

We note that there is a canonical lift of $G(X)(F)$ to $\widetilde{G}(X)(\mathbb{A})$ and that $\widetilde{G}(X)(F_{v})$ splits uniquely over any unipotent subgroup of $G(X)(F_{v})$.

Fix a non-trivial additive character $\psi:F\operatorname{\backslash}\mathbb{A}\rightarrow\mathbb{C}^{1}$. It will be used in the construction of the Weil representation that underlies the theta correspondence and the definition of Arthur parameters for cuspidal automorphic representations of the metaplectic group. For any subgroup $H$ of $G(X_{a})(\mathbb{A})$ (resp. $G(X_{a})(F_{v})$) we write $\widetilde{H}$ for its preimage in $\widetilde{G}(X_{a})(\mathbb{A})$ (resp. $\widetilde{G}(X_{a})(F_{v})$). We note that $\operatorname{\mathrm{GL}}_{a}(F_{v})$ occurs as a direct factor of $M_{a}(F_{v})$ and on $\widetilde{\operatorname{\mathrm{GL}}}_{a}(F_{v})$ multiplication is given by

which has a Hilbert symbol twist when multiplying the $\mu_{2}$-part. Analogous to the notation $m(x,h)$ above, we write

for $x\in\operatorname{\mathrm{GL}}_{a}(\mathbb{A})$, $h\in G(X)(\mathbb{A})$ and $\zeta\in\mu_{2}$. There is also a local version.

As we integrate over automorphic quotients often, we write $[H]$ for $H(F)\operatorname{\backslash}H(\mathbb{A})$ if $H$ is an algebraic group and $[\widetilde{H}]$ for $H(F)\operatorname{\backslash}\widetilde{H}(\mathbb{A})$ if $\widetilde{H}(\mathbb{A})$ covers a subgroup $H(\mathbb{A})$ of a metaplectic group.

Let $\mathcal{A}_{\mathrm{cusp}}(\widetilde{G}(X))$ denote the set of all irreducible genuine cuspidal automorphic representations of $\widetilde{G}(X)(\mathbb{A})$. We note that there is no algebraic group $\widetilde{G}(X)$ and that it is used purely for aesthetic reason.

We recall the description of $A$-parameters attached to genuine irreducible cuspidal automorphic representations of metaplectic groups from [9]. The $A$-parameters are defined via those attached to irreducible automorphic representations of odd special orthogonal groups via Shimura–Waldspurger correspondence. In this section, we write $\operatorname{\mathrm{Mp}}_{2n}$ rather than $\widetilde{G}(X)$.

The global elliptic $A$-parameters $\phi$ for $\operatorname{\mathrm{Mp}}_{2n}$ are of the form:

where

• •

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

• •

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

such that

• •

  $\sum_{i}n_{i}b_{i}=2n$;

• •

  if $b_{i}$ is odd, then $\tau_{i}$ is symplectic, that is, $L(s,\tau_{i},\wedge^{2})$ has a pole at $s=1$;

• •

  if $b_{i}$ is even, then $\tau_{i}$ is orthogonal, that is, $L(s,\tau_{i},\operatorname{\mathrm{Sym}}^{2})$ has a pole at $s=1$;

• •

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

Given a global elliptic $A$-parameter $\phi$ for $\operatorname{\mathrm{Mp}}_{2n}$, for each place $v$ of $F$, we can attach a local $A$-parameter

where $\phi_{i,v}$ is the $L$-parameter of $\tau_{i,v}$ given by the local Langlands correspondence [28, 15, 13, 39] for $\operatorname{\mathrm{GL}}_{n_{i}}$. We can write the local $A$-parameter $\phi_{v}$ as a homomorphism

where $L_{F_{v}}$ is the local Langlands group, which is the Weil–Deligne group of $F_{v}$ for $v$ non-archimedean and is the Weil group of $F_{v}$ for $v$ archimedean. We associate to it the $L$-parameter

Let $L^{2}(\operatorname{\mathrm{Mp}}_{2n})$ denote the subspace of $L^{2}(\operatorname{\mathrm{Sp}}_{2n}(F)\operatorname{\backslash}\operatorname{\mathrm{Mp}}_{2n}(\mathbb{A}))$ on which $\mu_{2}$ acts as the non-trivial character. Let $L^{2}_{\operatorname{\mathrm{disc}}}(\operatorname{\mathrm{Mp}}_{2n})$ denote the subspace of $L^{2}(\operatorname{\mathrm{Mp}}_{2n})$ that is the direct sum of all irreducible sub-representations of $L^{2}(\operatorname{\mathrm{Mp}}_{2n})$ under the action of $\operatorname{\mathrm{Mp}}_{2n}(\mathbb{A})$. Let $L^{2}_{\phi,\psi}(\operatorname{\mathrm{Mp}}_{2n})$ denote subspace of $L_{\operatorname{\mathrm{disc}}}^{2}(\operatorname{\mathrm{Mp}}_{2n})$ generated by the full near equivalence class of $\pi\subset L_{\operatorname{\mathrm{disc}}}^{2}(\operatorname{\mathrm{Mp}}_{2n})$ such that the $L$-parameter of $\pi_{v}$ with respect to $\psi$ is $\varphi_{\phi_{v}}$ for almost all $v$. We note here that for a genuine irreducible automorphic representation of the metaplectic group, its $L$-parameter depends on the choice of an additive character $\psi$. If one changes $\psi$ to $\psi_{a}:=\psi(a\cdot)$, the $L$-parameter changes in the way prescribed in [9, Remark 1.3]. See also [12]. Then we have the decomposition:

Thus we see that the $A$-parameter of $\sigma\in\mathcal{A}_{\mathrm{cusp}}(\operatorname{\mathrm{Mp}}_{2n})$ has the factor $(\chi,b)$ where $\chi$ is a quadratic automorphic character of $\operatorname{\mathrm{GL}}_{1}(\mathbb{A})$ and $b$ maximal among all factors if and only if the partial $L$-function $L^{S}_{\psi}(s,\sigma\times\chi)$ has its rightmost pole at $s=(b+1)/2$. The additive character $\psi$ used in the definition of the $L$-function is required to agree with the one used in defining the $A$-parameter of $\sigma$.

For $\sigma\in\mathcal{A}_{\mathrm{cusp}}(\operatorname{\mathrm{Mp}}_{2n})$ and a quadratic automorphic character $\chi$ of $\mathbb{A}^{\times}$, the $L$-function $L_{\psi}(s,\sigma\times\chi)$ appears in the constant term of the Eisenstein series on $\widetilde{G}(X_{a})(\mathbb{A})$ attached to the maximal parabolic subgroup $Q_{a}$. We study the poles of these Eisenstein series in this section.

Let $\mathfrak{a}_{M_{a}}^{*}=\operatorname{\mathrm{Rat}}(M_{a})\otimes_{\mathbb{Z}}\mathbb{R}$ and $\mathfrak{a}_{M_{a},\mathbb{C}}^{*}=\operatorname{\mathrm{Rat}}(M_{a})\otimes_{\mathbb{Z}}\mathbb{C}$ where $\operatorname{\mathrm{Rat}}(M_{a})$ denotes the group of rational characters of $M_{a}$. Let $\mathfrak{a}_{M_{a}}=\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(\operatorname{\mathrm{Rat}}(M_{a}),\mathbb{R})$. As $Q_{a}$ is a maximal parabolic subgroup, $\mathfrak{a}_{M_{a}}^{*}\cong\mathbb{R}$ and we identify $\mathfrak{a}_{M_{a},\mathbb{C}}^{*}$ with $\mathbb{C}$ via $s\mapsto s(\frac{\dim X+a+1}{2})^{-1}\rho_{Q_{a}}$, following [40], where $\rho_{Q_{a}}$ is the half sum of the positive roots in $N_{a}$. Thus we may regard $\rho_{Q_{a}}$ as the number $(\dim X+a+1)/2$. In fact, it is clear that $\rho_{Q_{a}}(m(x,h))=|\det x|_{\mathbb{A}}^{(\dim X+a+1)/2}$ for $m(x,h)\in M_{a}(\mathbb{A})$.

Let $H_{a}$ be the homomorphism $M_{a}(\mathbb{A})\rightarrow\mathfrak{a}_{M_{a}}$ such that for $m\in M_{a}(\mathbb{A})$ and $\xi\in\mathfrak{a}_{M_{a}}^{*}$ we have $\exp(\langle{H_{a}(m)},{\xi}\rangle)=\prod_{v}|\xi(m_{v})|_{v}$. We may think of $H_{a}(m(x,h))\in\mathfrak{a}_{M_{a}}\cong\mathbb{R}$ as $\log(|\det x|_{\mathbb{A}})$. We extend $H_{a}$ to $G(X_{a})(\mathbb{A})$ via the Iwasawa decomposition and then to $\widetilde{G}(X_{a})(\mathbb{A})$ via projection.

Let $\chi_{\psi}$ be the genuine character of $\widetilde{\operatorname{\mathrm{GL}}}_{1}(F_{v})$ defined by

where $\gamma(\cdot,\psi_{1/2})$, valued in the 4-th roots of unity, is defined via the Weil index. We note that the notation here agrees with that in [8, page 521]. Then via the determinant map, we get a genuine character of $\widetilde{\operatorname{\mathrm{GL}}}_{a}(F_{v})$ which we also denote by $\chi_{\psi}$.

Let $\sigma$ be a genuine cuspidal automorphic representation of $\widetilde{G}(X)(\mathbb{A})$. Let $\chi$ be a quadratic automorphic character of $\operatorname{\mathrm{GL}}_{a}(\mathbb{A})$. Then $\chi\chi_{\psi}:(g,\zeta)\mapsto\chi(g)\chi_{\psi}((g,\zeta))$ gives a genuine automorphic character of $\widetilde{\operatorname{\mathrm{GL}}}_{a}(\mathbb{A})$. Let $\mathcal{A}_{a,\psi}(s,\chi,\sigma)$ denote the space of $\mathbb{C}$-valued smooth functions $f$ on $N_{a}(\mathbb{A})M_{a}(F)\operatorname{\backslash}\widetilde{G}(X_{a})(\mathbb{A})$ that satisfy the following properties:

1. (1)

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

2. (2)

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

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

3. (3)

   for any fixed $k\in\widetilde{K}_{a}$, the function $h\mapsto f(\widetilde{m}(I_{a},h,\zeta)k)$ on $\widetilde{G}(X)(\mathbb{A})$ is a smooth right $\widetilde{K}_{G(X)}$-finite vector in the space of $\sigma$.

Let $f\in\mathcal{A}_{a,\psi}(0,\chi,\sigma)$ and $s\in\mathbb{C}$. Since the metaplectic group splits over $G(X_{a})(F)$, we can form the Eisenstein series on the metaplectic group as in the case of non-cover groups:

where $f_{s}(g)=\exp(\langle{H_{a}(g)},{s}\rangle)f(g)\in\mathcal{A}_{a,\psi}(s,\chi,\sigma)$. By the general theory of Eisenstein series [32, IV.1], it is absolute convergent for $\operatorname{\mathrm{Re}}s>\rho_{Q_{a}}$ and has meromorphic continuation to the whole $s$-plane with finitely many poles in the half plane $\operatorname{\mathrm{Re}}s>0$, which are all real as we identify $\mathfrak{a}_{M_{a}}^{*}$ with $\mathbb{R}$.

Let $\mathcal{P}_{a,\psi}(\chi,\sigma)$ denote the set of positive poles of $E(g,s,f)$ for $f$ running over $\mathcal{A}_{a,\psi}(0,\chi,\sigma)$.

Next we relate the Eisenstein series $E(s,g,f)=E(g,f_{s})$ to Siegel Eisenstein series on the ‘doubled group’ to glean more information on the locations of poles.

Let $X^{\prime}$ be the symplectic space with the same underlying vector space as $X$ but with the negative symplectic form $\langle{\ },{\ }\rangle_{X^{\prime}}=-\langle{\ },{\ }\rangle_{X}$. Let $W=X\oplus X^{\prime}$ and form $W_{a}=\ell_{a}^{+}\oplus W\oplus\ell_{a}^{-}$. Let $X^{\Delta}=\{(x,x)\in W|x\in X\}$ and $X^{\nabla}=\{(x,-x)\in W|x\in X\}$. Then $W_{a}$ has the polarisation $(X^{\Delta}\oplus\ell_{a}^{+})\oplus(X^{\nabla}\oplus\ell_{a}^{-})$. Let $P_{a}$ be the Siegel parabolic subgroup of $G(W_{a})$ that stabilises $X^{\nabla}\oplus\ell_{a}^{-}$. For a $\widetilde{K}_{G(W_{a})}$-finite section $\mathbb{F}_{s}$ in $\operatorname{\mathrm{Ind}}_{\widetilde{P}_{a}(\mathbb{A})}^{\widetilde{G}(W_{a})(\mathbb{A})}\chi\chi_{\psi}|\ |_{\mathbb{A}}^{s}$, we form the Siegel Eisenstein series $E^{P_{a}}(\cdot,\mathbb{F}_{s})$.

Since $G(X_{a})$ acts on $\ell_{a}^{+}\oplus X\oplus\ell_{a}^{-}$ and $G(X)$ acts on $X^{\prime}$, we have the obvious embedding $\iota:G(X_{a})(\mathbb{A})\times G(X)(\mathbb{A})\rightarrow G(W_{a})(\mathbb{A})$ which induces a homomorphism $\widetilde{\iota}:\widetilde{G}(X_{a})(\mathbb{A})\times_{\mu_{2}}\widetilde{G}(X)(\mathbb{A})\rightarrow\widetilde{G}(W_{a})(\mathbb{A})$. The cocycles for the cover groups are compatible by [37, Theorem 4.1].

The following is the metaplectic version of [23, Proposition 2.3] whose proof generalises immediately. We note that the integrand is non-genuine, and thus descends to a function on $G(X)(\mathbb{A})$.

Now we list the locations of possible poles of the Siegel Eisenstein series obtained in [17, 18].

Combining the above propositions, we get:

The locations of poles of the Eisenstein series defined in Sec. 3 are intimately related to the invariants, called the lowest occurrences of theta lifts, attached to $\sigma\in\mathcal{A}_{\mathrm{cusp}}(\widetilde{G}(X))$. The lowest occurrences are defined via the more familiar notion of first occurrences. In Sec. 6, we will derive a relation between periods of residues of Eisenstein series and first occurrences.

First we review briefly the definition of theta lift. For each automorphic additive character $\psi$ of $\mathbb{A}$, there is a Weil representation $\omega_{\psi}$ of the metaplectic group $\operatorname{\mathrm{Mp}}_{2n}(\mathbb{A})$ unique up to isomorphism. In our case, $X/F$ is a symplectic space and $Y/F$ is a quadratic space. From them, we get the symplectic space $Y\otimes X$. We consider the Weil representation of $\widetilde{G}(Y\otimes X)(\mathbb{A})$ realised on the Schwartz space $\mathcal{S}((Y\otimes X)^{+}(\mathbb{A}))$ where $(Y\otimes X)^{+}$ denotes any maximal isotropic subspace of $Y\otimes X$. Different choices of maximal isotropic subspaces give different Schwartz spaces which are intertwined by Fourier transforms. Thus the choice is not essential and sometimes we simply write $\mathcal{S}_{X,Y}(\mathbb{A})$ for $\mathcal{S}((Y\otimes X)^{+}(\mathbb{A}))$.