Theta correspondence and simple factors in global Arthur parameters

By using results on poles of $L$-functions and theta correspondence, we give a bound on $b$ for $(\chi,b)$-factors of the global Arthur parameter of a cuspidal automorphic representation $\pi$ of a classical group or a metaplectic group where $\chi$ is a conjugate self-dual automorphic character and $b$ is an integer which is the dimension of an irreducible representation of $\mathrm{SL}_{2}(\mathbf{CC})$. We derive a more precise relation when $\pi$ lies in a generic global $A$-packet.


Introduction
Let F be a number field and let ‫ށ‬ be its ring of adeles.Let π 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 = Sp(X ) in this introduction where X is a nondegenerate symplectic space over F. By Arthur's theory of endoscopy [2013], π belongs to a global A-packet associated to an elliptic global A-parameter, which is of the form where τ i is an irreducible self-dual cuspidal automorphic representation of GL n i ‫)ށ(‬ and b i is a positive integer which represents the unique b i -dimensional irreducible representation of Arthur's SL 2 ‫;)ރ(‬ see Section 2, for more details. Jiang [2014] proposed the (τ, 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 (τ, b)-factor.We explain in more details.Let φ denote the global A-packet with elliptic global A-parameter φ.Let π be an irreducible automorphic representation of G(‫)ށ‬ and let σ be an irreducible automorphic representation of H ‫,)ށ(‬ where H is a factor of an endoscopic group of G. Assume that π (resp.σ ) occurs in the discrete spectrum.Then it is expected that there exists some kernel function K depending on G, H and (τ, b) only such that if π and σ satisfy a Gan-Gross-Prasad type of criterion, namely, that the period integral 970 Chenyan Wu is nonvanishing for some choice of f σ ∈ σ and f π ∈ π, then π is in the global A-packet φ if and only if σ is in the global A-packet φ 2 with φ = (τ, b) ⊞ φ 2 .Then Jiang [2014, Section 5] proceeds to construct certain kernel functions and then using them, defines endoscopy transfer (by integrating over H (F) \ H ‫)ށ(‬ only in (0-1)) and endoscopy descent (by integrating over G(F) \ G(‫)ށ‬ only in (0-1)).It is not yet known if these are the kernel functions making the statements of Principle 1.2 in [Jiang 2014] hold.As the kernel functions come from Bessel coefficients or Fourier-Jacobi coefficients as in [Gan et al. 2012, Section 23], we see the nonvanishing of this period integral is analogous to condition (i) in the global Gan-Gross-Prasad conjecture [Gan et al. 2012, Conjecture 24.1].
Jiang [2014, Section 7] suggested that if τ is an automorphic character χ, 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 G(F) \ G(‫)ށ‬ K(h, g) f π (g) dg as f π runs over π is the theta lift of π.This is an automorphic representation of H ‫.)ށ(‬ Lifting in the other direction is analogous.Assume that the theta lift of π is nonzero.Write φ π for the global A-parameter of π.Then Jiang [2014, Principle 1.2] says that φ π has a (χ , b)-factor and that the global A-parameter of the theta lift of π from G to H should be φ π with the (χ , 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 (χ , b)-theory and to present the results of [Moeglin 1997;Ginzburg et al. 2009;Jiang and Wu 2016;2018;Wu 2022a;2022b] 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 emphasize 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 π.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 (χ , * ) 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 π v from being tempered for all places v of F, for classical groups and metaplectic groups.This should follow by generalizing the arguments in [Jiang and Liu 2018, 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 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 Jiang and Liu [2018, Section 5] found a Ramanujan bound for π by using the crucial results on the Ramanujan bound for GL 2 in [Kim 2003;Blomer and Brumley 2011].
We describe the idea of the proof of our result.First we relate the existence of a (τ, b)-factor in the elliptic global A-parameter of π to the existence of poles of partial L-functions L S (s, π × τ ∨ ); see Proposition 2.8.If the global A-parameter of π has a factor (τ, b) where b is maximal among all factors, we can show that the partial L-function L S (s, π × τ ∨ ) has a pole at s = 1 2 (b + 1).Thus studying the location of poles of L S (s, π ×τ ∨ ) for τ running through all self-dual cuspidal representations of GL n ‫ށ(‬ F ) can shed light on the size of the b i that occur in the global A-parameter of π.Then we specialize to the case where τ is a character χ and consider L S (s, π × χ ∨ ) in what follows.
Next we relate the poles of L S (s, π × χ ∨ ) to the poles of Eisenstein series attached to the cuspidal datum χ ⊠ π; see Section 3. In fact, in some cases, we use the nonvanishing of L S (s, π × χ ∨ ) at s = 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 π with respect to χ 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 π with respect to certain quadratic spaces to possibly nonmaximal and possibly negative poles of the Eisenstein series.
More precise results can be derived if we assume that π has a generic global A-parameter.This is because we have a more precise result relating poles or nonvanishing of values of the complete L-functions to poles of the Eisenstein series supplied by [Jiang et al. 2013].Thus we get Theorem 0.2 (Theorem 6.7).Let π be a cuspidal member in a generic global A-packet of G(‫)ށ‬ = Sp(X ‫.)ށ()‬ Let χ be a self-dual automorphic character of GL 1 ‫.)ށ(‬Then the following are equivalent: (1) The global A-parameter φ π of π has a (χ , 1)-factor.
(3) The Eisenstein series E(g, f s ) has a pole at s = 1 for some choice of section f Here Q 1 is a parabolic subgroup of Sp(X 1 ) with Levi subgroup isomorphic to GL 1 × Sp(X ), where X 1 is the symplectic space formed from X by adjoining a hyperbolic plane.Roughly speaking, A Q 1 (s, χ ⊠π ) is a space of automorphic forms on Sp(X 1 ) induced from χ |•| s ⊠ π viewed as a representation of the parabolic subgroup Q 1 .We refer the reader to Section 3 for the precise definition of A Q 1 (s, χ ⊠ π ).We note that the lowest occurrence index LO χ X (π ) 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 nonvanishing of L(s, π × χ ∨ ) at s = 1 2 and the lowest occurrence index.We plan to improve this result in the future by studying a relation between nonvanishing of Bessel or Fourier-Jacobi periods and the lowest occurrence index.
We note that the L-function L(s, π × χ ∨ ) 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, π × χ ∨ ) at an appropriate point up to some ramified factors and some abelian L-functions.We refer the reader to [Yamana 2014] which is a culmination of many previous results.In our approach, the Eisenstein series E(g, f s ), which is not of Siegel type, is the key link between L(s, π × χ ∨ ) 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 (τ, b)-factors in an elliptic global A-parameter.In Section 3, we define Eisenstein series attached to the cuspidal datum χ ⊠ π 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 FO Y,χ X (π ) and the lowest occurrence index LO χ X (π ) of π 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 (χ , b)-factors of the global A-parameter of π.Finally in Section 6, we consider the case when π has a generic global A-parameter.We show that when L(s, π × χ ∨ ) has a pole at s = 1 (resp.L(s, π × χ ∨ ) is nonvanishing at s = 1 2 ), the lowest occurrence index is determined.

Notation
Let F be a number field and let E be either F or a quadratic field extension of F. Let ϱ ∈ Gal(E/F) be the trivial Galois element when E = F and the nontrivial Galois element when E ̸ = F.When E ̸ = F, write ε E/F for the quadratic character associated to E/F via Class field theory.Let G be an algebraic group over E. We write R E/F G for the restriction of scalars of Weil.This is an algebraic group over F.
Let ϵ be either 1 or −1.By an ϵ-skew Hermitian space, we mean an E-vector space X together with an F-bilinear pairing for all a, b ∈ E and x, y ∈ X .We consider the linear transformations of X to act from the right.We follow the notation from [Yamana 2014] closely and we intend to generalize the results here to the quaternionic unitary group case in our future work.
Let X be an ϵ-skew Hermitian space of finite dimension.Then the isometry group of X is one of the following: (1) The symplectic group Sp(X ) when E = F and ϵ = 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 ‫ށ‬ F (resp.‫ށ‬ E ) denote the ring of adeles of F (resp.E).Set ‫ށ‬ := ‫ށ‬ F .Write Mp(X )(F v ) (resp.Mp(X ‫ށ()‬ F )) for the metaplectic double cover of Sp(X )(F v ) (resp.Sp(X ‫ށ()‬ F )) defined by Weil [1964].We note that the functor Mp(X ) is not representable by an algebraic group.We will also need the ‫ރ‬ 1 -extension Mp(X )(F v ) × µ 2 ‫ރ‬ 1 of Sp(X )(F v ) and we denote it by Mp ‫ރ‬ 1 (F v ).Similarly we define Mp ‫ރ‬ 1 ‫ށ(‬ F ).
Let ψ be a nontrivial automorphic additive character of ‫ށ‬ F which will figure in the Weil representations as well as the global A-parameters for Mp(X ).
For an automorphic representation or admissible representation π, we write π ∨ for its contragredient.

Global Arthur parameters
First we recall the definition of elliptic global Arthur parameters (A-parameters) for classical groups as well as metaplectic groups; see [Arthur 2013] for the symplectic and the special orthogonal case and we adopt the formulation in [Atobe and Gan 2017] for the case of the (disconnected) orthogonal groups.For the unitary case, see [Mok 2015;Kaletha et al. 2014].For the metaplectic case, see [Gan and Ichino 2018].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 (τ, b)-factors in an elliptic global A-parameter for some positive integer r where (1) τ i is an irreducible conjugate self-dual cuspidal automorphic representation of GL n i ‫ށ(‬ E ); (2) b i is a positive integer which represents the unique b i -dimensional irreducible representation of Arthur's SL 2 ‫)ރ(‬ such that: Here d Ǧ is the degree of the standard representation of Ǧ which, explicitly, is Remark 2.1.We adopt the notation in [Jiang 2014] and hence we write (τ i , b i ) rather than τ i ⊠ ν b i as is more customary in the literature, so that the quantity b i , that we study, is more visible.
Remark 2.2.In the unitary case, we basically spell out what 2 (U(N ), ξ 1 ) in [Mok 2015, Definition 2.4.7]is.We have discarded the second factor ψ as it is determined by ψ N and ξ 1 in Mok's notation.
Remark 2.3.(1) For G = U(X ), we say that τ is conjugate self-dual of parity η if the Asai L-function L(s, τ, Asai η ) has a pole at s = 1.If η = +1, we also say that τ is conjugate orthogonal and if η = −1, we also say that τ is conjugate symplectic.The Asai representations come from the decomposition of the twisted tensor product representation of the L-group; see [Mok 2015, (2.2.9) and (2.5.9)] and [Goldberg 1994].
(2) For other cases, we mean self-dual when we write conjugate self-dual.We say that τ is self-dual of parity +1 or orthogonal, if L(s, τ, Sym 2 ) has a pole at s = 1; we say that τ is self-dual of parity −1 or symplectic, if L(s, τ, ∧ 2 ) has a pole at s = 1.
(3) The parity is uniquely determined for each irreducible conjugate self-dual cuspidal representation τ .
Let 2 (G) denote the set of elliptic global A-parameters of G. Let φ ∈ 2 (G).Via the local Langlands conjecture (which is proved for the general linear groups), at every place v of F, we localize φ to get an elliptic local A-parameter, Define the full near equivalence class L 2 φ,ψ (G) attached to the elliptic global A-parameter φ to be the Hilbert direct sum of all irreducible automorphic representations σ occurring in L 2 disc (G) such that for almost all v, the local L-parameter of σ v is ϕ φ v .We remark that in the Mp(X )-case, the parametrization of σ v is relative to ψ v since the local L-parameter of σ v is attached via the Shimura-Waldspurger correspondence which depends on ψ v .This is the only case in this article where L 2 φ,ψ (G) depends on ψ.Let A 2 (G) denote the dense subspace consisting of automorphic forms in L 2 disc (G).Similarly define A 2,φ,ψ (G) to be the dense subspace of L 2 φ,ψ (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 2 (G).
Remark 2.5.This crude form of Arthur's multiplicity formula has been proved for Sp(X ) and quasisplit O(X ) by Arthur [2013], for U(X ) by [Mok 2015;Kaletha et al. 2014] and for Mp(X ) by [Gan and Ichino 2018].This is also proved for nonquasisplit even orthogonal (and also unitary groups) in [Chen and Zou 2021] and for nonquasisplit odd orthogonal groups in [Ishimoto 2023].Thus for all cases needed in this paper, Theorem 2.4 is known.
We have some further remarks on the orthogonal and unitary cases.
Remark 2.6.Arthur's statements use SO(X ) rather than O(X ) and he needs to account for the outer automorphism of SO(X ) when dim X is even; see the paragraph below [Arthur 2013, Theorem 1.5.2].
The formulation for quasisplit even O(X ) is due to Atobe and Gan [2017, Theorem 7.1(1)].For odd O(X ), which is isomorphic to SO(X ) × µ 2 , the reformulation of Arthur's result is easy.Let T be a finite set of places of F. Assume that it has even cardinality.Let sgn T be the automorphic character of µ 2 ‫ށ(‬ F ) which is equal to the sign character at places in T and the trivial character at places outside T .These give all the automorphic characters of µ 2 ‫ށ(‬ F ). Then every irreducible automorphic representation π of O(X ‫ށ()‬ F ) is of the form π 0 ⊠ sgn T for some irreducible automorphic representation π 0 of SO(X ‫ށ()‬ F ) and some finite set T of places of even cardinality.A near equivalence class of O(X ‫ށ()‬ F ) then consists of all irreducible automorphic representations π 0 ⊠ sgn T for π 0 running over a near equivalence class of SO(X ‫ށ()‬ F ) and sgn T running over all automorphic characters of µ 2 ‫ށ(‬ F ).
Remark 2.7.For the U(X ) case, the global A-parameter depends on the choice of a sign and a conjugate self-dual character which determine an embedding of the L-group of U(X ) to the L-group of R E/F GL d where we recall that d := dim X .We refer the reader to [Mok 2015, Section 2.1], in particular (2.1.9)there, for details.In this work, we choose the +1 sign and the trivial character, which, in Mok's notation, means κ = 1 and χ κ = 1.Then this corresponds to the standard base change of U(X ) to R E/F GL d .We note that the L-functions we use below are such that for all places v, automorphic representations π of G(‫ށ‬ F ) and τ of R E/F GL a ‫ށ(‬ F ) where BC denotes the standard base change.
By Theorem 2.4, we get Proposition 2.8.Let π be an irreducible automorphic representation of G(‫ށ‬ F ) that occurs in A 2,φ,ψ (G). Then: (1) If φ has a (τ, b)-factor with b maximal among all factors, then the partial L-function L S (s, π × τ ∨ ) has a pole at s = 1 2 (b + 1) and this is its maximal pole.
Remark 2.9.In the Mp(X ) case, the L-function depends on ψ, but we suppress it from notation here.
Assume that where S is a finite set of places of F outside of which all data are unramified.
Assume that φ has a (τ, b)-factor with b maximal among all factors, then by the properties of the Rankin-Selberg L-functions, we see that L S (s, π × τ ∨ ) has a pole at s = 1 2 (b + 1) and it is maximal.Next assume that the partial L-function L S (s, π × τ ∨ ) has a pole at s = 1 2 (b ′ + 1).If φ has no (τ, c)-factor for any c ∈ ‫ޚ‬ >0 , then L S (s, π × τ ∨ ) is holomorphic for all s ∈ ‫ރ‬ and we get a contradiction.
Thus φ has a (τ, b)-factor.We take b maximal among all factors of the form (τ, * ) in φ.As b may not be maximal among all simple factors of φ, we can only conclude that L S (s, π × τ ∨ ) is holomorphic for Re Given an irreducible cuspidal automorphic representation π , write φ π for the global A-parameter of π .By studying poles of L S (s, π × τ ∨ ) for varying τ , we can detect the existence of (τ, b)-factors with maximal b in φ π .We would also like to construct an irreducible cuspidal automorphic representation with global A-parameter φ π ⊟ (τ, b) which means removing the (τ, b)-factor from φ π if φ π has a (τ, 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, π × τ ∨ ) where τ is a conjugate self-dual irreducible cuspidal automorphic representation of R E/F GL 1 ‫.)ށ(‬Now we write χ for τ to emphasize 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, π × χ ∨ ) 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 [Kudla and Rallis 1994;Wu 2017;Gan et al. 2014;Yamana 2014] for details.One of the key steps is the regularized 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, π × χ ∨ ).

Eisenstein series attached to χ ⊠ π
In this section we deviate slightly from the notation in Section 2. We use G(X ) to denote one of Sp(X ), O(X ) and U(X ).We let G(X ) be a cover group of G(X ), which means We adopt similar notation to that in [Moeglin and Waldspurger 1995].We define Eisenstein series on a larger group of the same type as G(X ) and collect some results on their maximal positive poles.
Let π be an irreducible cuspidal automorphic representation of G(X ‫.)ށ()‬We always assume that π is genuine when Let a be a positive integer.Let X a be the ϵ-skew Hermitian space over E that is formed from X by adjoining a-copies of the hyperbolic plane.More precisely, let ℓ + a (resp.ℓ − a ) be a totally isotropic a-dimensional E-vector space spanned by e + 1 , . . ., e + a (resp.e − 1 , . . ., e − a ) such that ⟨e + i , e − j ⟩ = δ i j where δ i j is the Kronecker symbol.Then Let G(X a ) be the isometry group of X a .Let Q a be the parabolic subgroup of G(X a ) that stabilizes ℓ − 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 R E/F GL a with R E/F GL(ℓ + a ).Let ρ Q a be the half sum of the positive roots in N a , which can be viewed as an element in a * M a := Rat(M a ) ⊗ ‫ޚ‬ ‫ޒ‬ where Rat(M a ) is the group of rational characters of M a .We note that as Q a is a maximal parabolic subgroup, a * M a is one-dimensional.Via the Shahidi normalization [2010], we identify a * M a with ‫ޒ‬ and thus may regard ρ Q a as the real number (1) f is right K a -finite.
(2) For any x ∈ R E/F GL a ‫)ށ(‬ and g ∈ G(X a ‫)ށ()‬ we have (3) For any fixed k ∈ K a , the function h → f (m(I, h)k) on G(X ‫)ށ()‬ is in the space of π.Now let G(X ) = Mp(X ).This case depends on ψ.Let GL 1 (F v ) be the double cover of GL 1 (F v ) defined as follows.As a set it is GL 1 (F v ) × µ 2 and the multiplication is given by which has a Hilbert symbol twist when multiplying the µ 2 -part.Analogously we define the double cover GL 1 ‫)ށ(‬ of GL 1 ‫.)ށ(‬Let χ ψ,v denote the genuine character of GL 1 (F v ) defined by ) is a fourth root of unity defined via the Weil index.It is the same one as in [Gan and Ichino 2014, page 521] except that we have put in the subscripts v. Then is a genuine automorphic character of GL 1 ‫.)ށ(‬Let K a denote the preimage of K a under the projection Mp(X a ‫)ށ()‬ → Sp(X a ‫.)ށ()‬We will also use to denote the preimages of other subgroups of Sp(X a ‫.)ށ()‬Let m be the isomorphism We keep writing det for the nongenuine homomorphism Given a nongenuine representation τ of GL a ‫,)ށ(‬ we can twist it by χ ψ • det to get a genuine representation which we denote by τ χ ψ .We remark that there are canonical embeddings of N a ‫)ށ(‬ and Sp(X a )(F) to Mp(X a ‫,)ށ()‬ so we may regard them as subgroups of G(X a ‫.)ށ()‬Let A Q a ψ (s, χ ⊠π ) denote the space of ‫-ރ‬valued smooth functions f on N a (‫)ށ‬M a (F) \ G(X a ‫)ށ()‬ such that: (1) f is right K a -finite.
(2) For any x ∈ GL a ‫)ށ(‬ and g ∈ G(X a ‫)ށ()‬ we have (3) For any fixed k ∈ K a , the function h → f ( m(I, h)k) on G(X ‫)ށ()‬ is in the space of π .
To unify notation, we will also write A Q a ψ (s, χ ⊠ π ) for A Q a (s, χ ⊠ π ) in the nonmetaplectic case.It should be clear from the context whether we are treating the Sp(X ) case or the Mp(X ) case.Now return to the general case, so G(X ) is one of Sp(X ), O(X ), U(X ) and Mp(X ).Let f s be a holomorphic section of A Q a ψ (s, χ ⊠ π ).We associate to it the Eisenstein series Note that the series is over γ ∈ Q a (F) \ Sp(X a )(F) when G(X ) = Mp(X ).By Langlands' theory of Eisenstein series [Moeglin and Waldspurger 1995, IV.1], this series is absolutely convergent for Re(s) > ρ 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 a * M a with ‫ޒ‬ and the fact that χ 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 Re(s) > 0.
We give the setup for any positive integer a, though we will only need a = 1 in the statements of our results.However the proofs require "going up the Witt tower" to G(X a ) for a large enough.Since we plan to prove analogous results for quaternionic unitary groups in the future, we keep the setup for general a.
There is a relation between poles of L-functions and the Eisenstein series.
Proposition 3.1.(1) Assume that the partial L-function L S ψ (s, π × χ ∨ ) has its rightmost positive pole at s = s 0 .Then E Q 1 ψ (g, f s ) has a pole at s = s 0 .
where j ∈ ‫ޚ‬ such that where r X denotes the Witt index of X .
Remark 3.6.The middle quantities in the inequalities of (3-3) are, in fact, the lowest occurrence index of π in the global theta lift which depends on χ and ψ; see Theorem 4.4.In some cases, the lowest occurrence index turns out to be independent of ψ.
Remark 3.7.To derive the inequalities r X ≤ • • • , we already need to make use of properties of the global theta correspondence.The other parts of the statements can be derived by relating our Eisenstein series to Siegel-Eisenstein series whose poles are completely known.We note that via the Siegel-Weil formula, Siegel-Eisenstein series are related to global theta correspondence.

Theta correspondence
We keep the notation of Section 3. First we define the theta lifts and the two invariants called the first occurrence index and the lowest occurrence index.Then we relate the invariants to poles of our Eisenstein series.
Recall that we have taken an ϵ-skew Hermitian space X over E. Let Y be an ϵ-Hermitian space equipped with the form ⟨ for all a, b ∈ E and y 1 , y 2 ∈ Y .Let G(Y ) be its isometry group.We note that G(X ) acts on X from the right while G(Y ) acts on Y from the left.Let W be the vector space R E/F (Y ⊗ E X ) over F and equip it with the symplectic form given by With this set-up, G(X ) and G(Y ) form a reductive dual pair inside Sp(W ).
realized on the space of Schwartz functions S(W + (F v )).The Weil representation depends on the additive character ψ v , but we suppress it from notation.When v is archimedean, we actually take the Fock model [Howe 1989] rather than the full Schwartz space and it is a (sp(W )(F v ), K Sp(W ),v )-module but we abuse language and call it a representation of Mp ‫ރ‬ 1 (W )(F v ).When neither G(X ) or G(Y ) is an odd orthogonal group, by [Kudla 1994] there exists a homomorphism When G(X ) is an odd orthogonal group, we take G(Y )(F v ) to be the metaplectic double cover of G(Y )(F v ) and set G(X ) = G(X ).Then by [Kudla 1994] there exists a homomorphism The case is analogous when G(Y ) is an odd orthogonal group.In any case, we get a homomorphism It should be clear from the context when G(X ) (resp.G(Y )) refers to a cover group and when it is not truly a cover.In the unitary case, there are many choices of ι v .Once we fix χ and an additional character χ 2 , then ι v is fixed.This is worked out in great details in [Harris et al. 1996, Section 1].Our (χ , χ 2 ) matches (χ 1 , χ 2 ) in [Harris et al. 1996, (0.2)].We note that Y should be compatible with χ and χ determines the embedding of G(X ‫)ށ()‬ into Mp ‫ރ‬ 1 ‫)ށ(‬ whereas X should be compatible with χ 2 and χ 2 determines the embedding of G(Y ‫)ށ()‬ into Mp ‫ރ‬ 1 ‫.)ށ(‬By "compatible", we mean ϵ χ ≡ dim Y (mod 2) (resp.ϵ χ 2 ≡ dim X (mod 2)); see [Kudla 1994] for more details.We pull back ω v to G(X )(F v )×G(Y )(F v ) via ι v and still denote the representation by ω v .
Denote by ι the adelic analogue of ι v .We also have the (global) Weil representation ω of Mp ‫ރ‬ 1 ‫)ށ(‬ on the Schwartz space S(W + ‫))ށ(‬ and its pullback via Then we can define the theta function which will be used as a kernel function.Let This is an automorphic form on G(Y ‫.)ށ()‬It depends on χ and χ 2 in the unitary case and when we want to emphasize the dependency, we will write θ Y,χ X,χ 2 ( f, ).Let Y X (π ) denote the space of functions spanned by the θ Y X ( f, ) and let Y,χ X,,χ 2 (π ) denote the space of functions spanned by the θ Y,χ X,χ 2 ( f, ) in the unitary case.From now on assume that Y is anisotropic (possibly zero), so that it sits at the bottom of its Witt tower.Define Y r to be the ϵ-Hermitian space formed by adjoining r -copies of the hyperbolic plane to Y .These Y r form the Witt tower of Y .By the tower property [Rallis 1984;Wu 2013], if the theta lift to G(Y r ) is nonzero then the theta lift to G(Y r ′ ) is also nonzero for all r ′ ≥ r .Define the first occurrence index of π in the Witt tower of Y to be (4-1) Note that it depends on χ but not on χ 2 in the unitary case as changing χ 2 to another compatible one produces only a character twist on where ( • , • ) is the Hilbert symbol.
Define the lowest occurrence index to be (4-3) Define the lowest occurrence index to be We have the following relations of the first occurrence (resp.the lowest occurrence) and the poles (resp.the maximal positive pole) of the Eisenstein series; see [Jiang and Wu 2018, Corollary 3.9, Theorem 3.10] for the symplectic case, [Wu 2022a, Corollary 6.3, Theorem 6.4] for the metaplectic case, [Jiang and Wu 2016, Corollaries 3.5 and 3.7] for the unitary case and [Ginzburg et al. 2009, Theorems 5.1 and 1.3] for the orthogonal case.
Theorem 4.1.Let π be an irreducible cuspidal automorphic representation of G(X ‫.)ށ()‬ Let χ be a conjugate self-dual automorphic character of R E/F GL 1 ‫.)ށ(‬Let Y be an anisotropic ϵ-Hermitian space that is compatible with χ in the sense of (4-3).Assume that FO Y,χ (4-5) Remark 4.2.Using the notation from Section 2. The quantity s 0 in (4-5) can be written uniformly as Remark 4.3.Note that we always have r ≤ dim X .The extra condition when G(X ) = O(X ) is to avoid treating period integrals over the orthogonal groups of split binary quadratic forms, as our methods cannot deal with the technicality.Theorem 4.1 allows negative s 0 .It is possible to detect nonmaximal poles and negative poles of the Eisenstein series by the first occurrence indices.
Theorem 4.4.Let π be an irreducible cuspidal automorphic representation of G(X ‫.)ށ()‬ Let χ be a conjugate self-dual automorphic character of R E/F GL 1 ‫.)ށ(‬Then the maximal positive pole of E Q 1 ψ (g, f s ) for f s running over (4-6) Remark 4.5.Theorem 4.4 does not allow negative s 0 .
In Remark 3.7, we mentioned that the part r X ≤ • • • in Theorem 3.5 is proved by using theta correspondence.What we used is that we always have LO (5-1) where r X denotes the Witt index of X .
Proof.If b is not maximal among all factors (τ, b) appearing in φ π , then b < 1 2 d G(X ) ∨ .Then it is clear that b satisfies (5-1).Now we assume that b is maximal among all factors appearing in φ π .By Proposition 2.8, L S (s, π × χ −1 ) has its rightmost pole at s = 1 2 (b + 1).Then by Proposition 3.1, E Q 1 ψ (g, f s ) has a pole at s = 1 2 (b + 1) for some choice of f s .Assume that s = 1 2 (b 1 + 1) is the rightmost pole of the Eisenstein series with b 1 ≥ b.By Theorem 3.5, or Mp(X ); or in other words, b 1 is less than or equal to the quantity on the RHS of (5-1).Using the fact that b ≤ b 1 , we get the desired bound for b. □ Remark 5.2.Our result generalizes [Jiang and Liu 2018, Theorem 3.1] for symplectic groups to classical groups and metaplectic groups.In addition, we do not require the assumption on the wave front set in When π is globally generic, b = 1 for every factor (τ, b) in the global A-parameter φ π .This is because the Langlands functorial lift of π is an isobaric sum of conjugate self-dual cuspidal representations of some R E/F GL n ‫;)ށ(‬ see Theorem 11.2 of [Ginzburg et al. 2011]. By [Jiang et al. 2013], there is a more precise relation on the poles of L-functions and the poles of Eisenstein series.The set of possible poles of the normalized Eisenstein series is determined by the complete L-function L(s, π ×χ ∨ ).From the assumption that π is globally generic, in the right half-plane, L(s, π ×χ ∨ ) has at most a simple pole at s = 1.In fact we only need [loc.cit., Proposition 4.1] rather than the full strength of [loc.cit., Theorem 1.2] which allows the induction datum to be a Speh representation on the general linear group factor of the Levi.By [Jiang and Zhang 2020, Theorem 5.1], [Jiang et al. 2013, Proposition 4.1] can be strengthened to include the case where π is a cuspidal member in a generic global A-packet of G(X ‫)ށ()‬ where G(X ) = Sp(X ), O(X ), U(X ) does not have to be quasisplit.We rephrase [Jiang et al. 2013, Proposition 4.1] in our context as Theorem 6.3.
First we set up some notation and outline the method for extending [loc.cit., Proposition 4.1] to the case of Mp(X ).Let (6-1) and (6-2) The results of [loc.cit.] do not cover the metaplectic case, but the method should generalize without difficulty.We explain the strategy.First the poles of the Eisenstein series are related to those of the intertwining operators where τ is a conjugate self-dual cuspidal automorphic representation of GL a ‫ށ(‬ E ) and w 0 is the longest Weyl element in Q a \ G(X a )/Q a .Then define the normalized intertwining operator The proof of [loc.cit., Proposition 4.1] relies on the key result that the normalized intertwining operator is holomorphic and nonzero for Re s ≥ 1 2 .Then it boils down to finding the poles of the normalizing factors or equivalently Once we have the key result available, we expect to have a version of [loc.cit., Proposition 4.1] for the metaplectic groups.Note that our ρ ± defined in (6-1) and (6-2) is different from the ρ and ρ − in [loc.cit.].
Then by using an inductive formula, we expect to be able to prove [loc.cit., Theorem 1.2] as well.We hope to supply the details in a future work.
Next we allow G(X ) to be non-quasisplit.We assume that π is a cuspidal member in a generic global A-packet of G(X ).Then by [Jiang and Zhang 2020, Theorem 5.1], (6-3) is holomorphic and nonzero for Re s ≥ 1 2 when G(X ) = Sp(X ), O(X ), U(X ).Then the proof of [Jiang et al. 2013, Proposition 4.1] goes through verbatim for such π .The proof of [Jiang and Zhang 2020, Theorem 5.1] does not generalize readily to the case of Mp(X ) as the relevant results for Mp(X ) are not available.
Remark 6.2.This is shown to be true by [Jiang and Zhang 2020, Theorem 5.1] when π is a cuspidal member in a generic global Thus this is only a condition when G(X ) = Mp(X ).
Theorem 6.3.Assume Assumption 6.1.Let π be a cuspidal member in a generic global A-packet of G(X ‫.)ށ()‬ Let χ be a conjugate self-dual automorphic character of R E/F GL 1 ‫.)ށ(‬ (1) Assume G(X ) = U(X ) with ϵ χ ̸ ≡ dim X (mod 2), O(X ) with dim X even or Sp(X ).Then L(s, π×χ ∨ ) has a pole at s = 1 if and only if E Q 1 (g, f s ) has a pole at s = 1 and it is its maximal pole.
(2) Assume G(X ) = U(X ) with ϵ χ ≡ dim X (mod 2), O(X ) with dim X odd or Mp(X ).Then L(s, π×χ ∨ ) is nonvanishing at s = 1 2 if and only if E Q 1 ψ (g, f s ) has a pole at s = 1 2 and it is its maximal pole.
Remark 6.4.The result of [Jiang et al. 2013] involves normalized Eisenstein series, but the normalization has no impact on the positive poles.The following remarks use the notation in [loc.cit.].We only need the case b = 1 in [loc.cit.] which is Proposition 4.1 there.Furthermore we only apply it in the case where τ is a character.The condition that L(s, τ, ρ) has a pole at s = 1 is automatically satisfied by the requirement on our χ that it is conjugate self-dual of parity (−1) N G(X ) ∨ +1 ; see Section 2, especially Remark 2.3.
The global A-parameter φ π can possibly have a (χ , 1)-factor only when χ satisfies the condition that L(s, χ , ρ + ) has a pole at s = 1.Due to the parity condition on factors of an elliptic global A-parameter, in some cases, φ π cannot have a (χ , 1)-factor.
Combining our result (Theorem 4.4) on poles of Eisenstein series and lowest occurrence indices with Theorem 6.3 which gives a precise relation between poles of the complete L-function and those of the Eisenstein series, we get Theorem 6.5.Assume Assumption 6.1.Let π be a cuspidal member in a generic global A-packet of G(X ‫.)ށ()‬ Let χ be a conjugate self-dual automorphic character of R E/F GL 1 ‫.)ށ(‬In each of the following statements, we consider only those G(X ) that are listed: (1) Assume that L(s, π × χ ∨ ) has a pole at s = 1.Then (2) Assume that L(s, π × χ ∨ ) does not have a pole at s = 1.Then ) is nonvanishing.Thus in the case of items (2), (4) and G(X ) ̸ = O(X ), the collection {Z [v] } v for v running over all places of F is always incoherent, i.e., there does not exist an ϵ-Hermitian space Z over E such that the localization Z v is isomorphic to Z [v] for all v.
In the case of items (2), (4) and G(X ) = O(X ), we have a nontrivial theta lift of ) for η [v] being the trivial character or the sign character for each place v of F, but the collection {η [v] } v is incoherent, i.e., there does not exist an automorphic character η of ‫ށ‬ × F such that the localization η v is equal to η [v] for all v; see the definitions of first occurrence (4-1) and lowest occurrence (4-4) for O(X ) for why we have a (χ v • υ v )-twist.We also note that when π is an irreducible cuspidal automorphic representation and L 1 2 , π × χ ∨ = 0, it is conjectured that there is an arithmetic version of the Rallis inner product formula which says that the conjectural Beilinson-Bloch height pairing of arithmetic theta lifts (which are cycles on Shimura varieties constructed from an incoherent collection of ϵ-Hermitian spaces) gives the derivative L ′ 1 2 , π × χ ∨ up to some ramified factors and some abelian L-functions.The low rank cases have been proved in [Kudla et al. 2006;Liu 2011a;2011b].More recently, the cases of unitary groups of higher rank have been proved in [Li and Liu 2021;2022], conditional on hypothesis of the modularity of Kudla's generating functions of special cycles.
(2) The complete L-function L(s, π × χ ∨ ) has a pole at s = 1 (and this is its maximal pole).
(3) The Eisenstein series E Q 1 (g, f s ) has a pole at s = 1 for some choice of f s ∈ A Q 1 (s, χ ⊠ π ) (and this is its maximal pole).
(4) The lowest occurrence index Remark 6.8.The statements that the poles are maximal are automatic since π lies in a generic global A-packet.We note that when G(X ) = U(X ) with ϵ χ ≡ dim X (mod 2), G(X ) = O(X ) with dim X odd or G(X ) = Mp(X ), φ π cannot have a (χ , 1)-factor as the parity condition is not satisfied.
Theorem 0.1 (Corollary 5.3).The global A-packet attached to the elliptic global A-parameter φ cannot have a cuspidal member if φ has a (χ , b)-factor with b Remark 3.3.The allowed G(X ) in item (2) are those for which we have theta dichotomy and epsilon dichotomy (in the local nonarchimedean setting); see [Gan and Ichino 2014, Corollary 9.2, Theorem 11.1].Remark 3.4.When π is a cuspidal member in a generic global A-packet of G(X ‫,)ށ()‬ there is a more precise result; see Theorem 6.3 which was proved in [Jiang et al. 2013] and strengthened in [Jiang and Zhang 2020].We summarize the results on the maximal positive pole of E Q 1 ψ (g, f s ) from [Ginzburg et al. 2009, Theorem 3.1; Jiang and Wu 2016, Theorem 3.1; 2018, Theorem 2,8; Wu 2022a, Theorem 4.2].Theorem 3.5.The maximal positive pole of E Q 1 Y r ,χ X,χ 2 (π ).For more details, see [Wu 2022b, (1-1)].In the orthogonal Theta correspondence and simple factors in global Arthur parameters 983 case, we twist π by χ • υ where υ denotes the spinor norm.If G(X ) = Sp(X ) or Mp(X ), we require that χ Y = χ where χ Y is the quadratic automorphic character of GL 1 ‫)ށ(‬ associated to Y given by

χX
(π ) ≥ r X by the stable range condition[Rallis  1984, Theorem I.2.1].5.Application to global Arthur packetsWe have derived relations among (χ , b)-factors of global A-parameters, poles of partial L-functions, poles of Eisenstein series and lowest occurrence indices of global theta lifts.Combining these, we have the following implication on global A-packets.Theorem 5.1.Let π be an irreducible cuspidal automorphic representation of G(X ‫.)ށ()‬ Let φ π be its global A-parameter.Let χ be a conjugate self-dual automorphic character of R E/F GL 1 ‫.)ށ(‬ Assume that φ π has a (χ, b)-factor for some positive integer b.Then b 2 if G(X ) = U(X ) and ϵ χ ≡ dim X (mod 2); dim X + 1 if G(X ) = O(X ) with dim X odd; dim X + 3 if G(X ) = Mp(X ).Remark 6.6.By the conservation relation for local theta correspondence [Sun and Zhu 2015], there always exists an ϵ-Hermitian space Z [v] over E v of dimension given by the RHS of the equalities in items (1), (3) such that the local theta lift of π v to G(Z[v] This is [Jiang and Wu 2018, Proposition 2.2] in the symplectic case, [Wu 2022a, Proposition 3.2] in the metaplectic case, [Jiang and Wu 2016, Proposition 2.2] in the unitary case and [Moeglin 1997, Remarque 2] and [Jiang and Wu 2016, Proposition 2.2] in the orthogonal case.