Algebraicity of critical values of adjoint L-functions for {\rm GSp}_4
aa r X i v : . [ m a t h . N T ] F e b ALGEBRAICITY OF CRITICAL VALUES OF ADJOINT L -FUNCTIONS FOR GSp SHIH-YU CHEN
Abstract.
We prove an algebraicity result for certain critical value of adjoint L -functions for GSp over atotally real number field in terms of the Petersson norm of normalized generic cuspidal newforms on GSp .This is a generalization of our previous result [CI19]. Introduction
Main result.
Let f be a normalized elliptic newform of weight κ ≥ ( N ). Denoteby L ( s, f, Ad) the completed adjoint L -function of f . By the result of Sturm [Stu89], the algebraicity of L (1 , f, Ad) is expressed in terms of the Petersson norm k f k = Z Γ ( N ) \ H | f ( τ ) | Im( τ ) κ − dτ. More precisely, we have σ (cid:18) L (1 , f, Ad) k f k (cid:19) = L (1 , σ f, Ad) k σ f k for all σ ∈ Aut( C ), as predicted by Deligne’s conjecture [Del79]. The purpose of this paper is to prove ananalogue of it for GSp .We give a description of our main result. Let Π = N v Π v be an irreducible globally generic cuspidalautomorphic representation of GSp ( A F ) with central character ω Π over a totally real number field F . Denoteby L ( s, Π , Ad) = Y v L ( s, Π v , Ad)the adjoint L -function of Π , where v runs through the places of F . Assume Π ∞ = N v |∞ Π v is a discreteseries representation of GSp ( F ∞ ). Then Π ∞ | Sp ( F ∞ ) = M v |∞ D ( λ ,v ,λ ,v ) ⊕ D ( − λ ,v , − λ ,v ) , where D ( λ ,v ,λ ,v ) is the discrete series representation of Sp ( F v ) ≃ Sp ( R ) with Blattner parameter ( λ ,v , λ ,v )such that 2 − λ ,v ≤ λ ,v ≤ − v . Here we follow [Mor04] for the choice of the Cartansubalgebra in sp ( R ) and the positive systems. Let f = N v f v ∈ Π be a non-zero cusp form satisfying thefollowing conditions: • f v is a paramodular newform of Π v for all finite places v ; • f v is a lowest weight vector of the minimal U(2)-type of D ( − λ ,v , − λ ,v ) for all real places v .These conditions characterize f ∈ Π up to scalars. Let W f be the Whittaker function of f with respect to ψ U defined by W f ( g ) = Z U ( F ) \ U ( A F ) f ( ug ) ψ U ( u ) du Tam . Here U is the standard maximal unipotent subgroup of GSp , ψ U is the standard non-degenerate characterof U ( F ) \ U ( A F ) (see § du Tam is the Tamagawa measure on U ( A F ). We maydecompose W f = Q v W v as a product of local Whittaker functions of Π v with respect to ψ U,v . We normalize f as follows: • for finite place v , we have W v (diag( ̟ − c v v , ̟ − c v v , , ̟ − c v v )) = 1 , where ̟ v is a uniformizer of F v and c v is the valuation of the different ideal of F v ; for real place v , we have W v (1) = e − π Z c + √− ∞ c −√− ∞ ds π √− Z c + √− ∞ c −√− ∞ ds π √− − s − s Γ R ( s + λ ,v + 1)Γ R ( − s − λ ,v ) × Γ R ( s + s + λ ,v − λ ,v + 2)Γ R ( s + s + λ ,v + λ ,v + 2) , (1.1) where Γ R ( s ) = π − s/ Γ( s ) and c , c ∈ R satisfy c + c + λ ,v + λ ,v + 2 > c + λ ,v + 1 > >c + λ ,v .We write f = f Π and call it the normalized newform of Π . We defined the normalized newform f Π ∨ of Π ∨ in a similar way. Let k f Π k be the Petersson norm of f Π defined by k f Π k = Z A × F GSp ( F ) \ GSp ( A F ) f Π ( g ) f Π ∨ ( g · diag( − , − , , ∞ ) dg Tam . Here dg Tam is the Tamagawa measure on A × F \ GSp ( A F ).For σ ∈ Aut( C ), let σ Π be the irreducible admissible representation of GSp ( A F ) defined by σ Π = σ Π ∞ ⊗ σ Π f , where σ Π f is the σ -conjugate of Π f = N v ∤ ∞ Π v and σ Π ∞ is the representation of GSp ( F ∞ ) so that its v -component is equal to Π σ − ◦ v . Assume further that Π is motivic, that is, there exists w ∈ Z such that | ω Π | = | | w A F and λ ,v − λ ,v ≡ w (mod 2)for all real places v . In Lemma 3.1 below, we show that σ Π is cuspidal automorphic and globally generic.The rationality field Q ( Π ) of Π is the fixed field of { σ ∈ Aut( C ) | σ Π = Π } and is a number field.The following theorem is our main result on the algebraicity of the critical adjoint L -value L (1 , Π , Ad) interms of the Petersson norm of the normalized newform of Π . Theorem 1.1.
Let Π be an irreducible motivic globally generic cuspidal automorphic representation of
GSp ( A F ) . For σ ∈ Aut( C ) , we have σ (cid:18) L (1 , Π , Ad) ζ F (2) ζ F (4) · k f Π k (cid:19) = L (1 , σ Π , Ad) ζ F (2) ζ F (4) · k f σ Π k . Here ζ F ( s ) is the completed Dedekind zeta function of F . In particular, we have L (1 , Π , Ad) ζ F (2) ζ F (4) · k f Π k ∈ Q ( Π ) . Remark 1.2.
In [CI19], we compute the ratio explicitly when ω Π is trivial and the paramodular conductorof Π is square-free. The theorem can be regarded as a generalization of [CI19]. Remark 1.3.
The Petersson norm k f Π k can be factorized into product of periods which are obtained bycomparing the rational structures via the Whittaker model and via the coherent cohomology (cf. [HK92]).We expect these periods to capture the transcendental part of the critical values of certain automorphic L -functions. Moreover, the expected period relation implies that Theorem 1.1 is compatible with Deligne’sconjecture for L (1 , Π , Ad). This is an ongoing project considered by the author.1.2.
An outline of the proof.
The first step is to show that for any non-trivial additive character ψ of F \ A F , we have k f Π k = C · L (1 , Π , Ad) ζ F (2) ζ F (4) · Y v C ψ v ( Π v )(1.2)for some non-zero constant C ψ v ( Π v ) depending only on Π v and ψ v , and some constant C ∈ Q × dependingonly on F and the type of Π (stable or endoscopic). This equality can be proved by proceeding exactly as inthe proof of [CI19, Proposition 5.4] (see also [CI19, § C ψ v ( Π v ). We have C ψ v ( Π v ) = 1 L (1 , Π v , Ad) · Z v (1 , Π v , F ψ v ) Z v ( , Π v , F ψ v ) , where F ψ v and F ψ v are sections in degenerate principal series representations of GSp ( F v ), and Z v ( s, Π v , F ψ v )and Z v ( s, Π v , F ψ v ) are local zeta integral for Π v × Π ∨ v and doubling local zeta integral for Π v defined and tudied by Jiang [Jia96] and Piatetski-Shapiro and Rallis [PSR87], respectively. We prove that the constantis well-defined by showing that Z v and Z v converge absolutely for Re( s ) ≥ s ) ≥ , respectively, andthere exists a local Siegel–Weil section F ψ v such that Z v ( , Π v , F ψ v ) is non-zero. The next step is to prove σ Y v ∤ ∞ C ψ v ( Π v ) = Y v ∤ ∞ C ψ v ( σ Π v )(1.3)for all σ ∈ Aut( C ). First we show that for all v ∤ ∞ and σ ∈ Aut( C ), we have σZ v ( , Π v , F ψ v ) = Z v ( , σ Π v , F σ ψ v ) , σ Z v (1 , Π v , F ψ v ) = Z v (1 , σ Π v , F σ ψ v ) , (1.4)which imply that σC ψ v ( Π v ) = C σ ψ v ( σ Π v ) . (1.5)For σ ∈ Aut( C ), in general N v ∤ ∞ σ ψ v is not the finite part of a non-trivial additive character of F \ A F .Nonetheless, since we have freedom to vary ψ in (1.2), we show that (1.3) holds by a global argumenttogether with (1.5). By (1.2) and (1.3), we then have σ L (1 , Π , Ad) ζ F (2) ζ F (4) · k f Π k · Y v |∞ C ψ v ( Π v ) = L (1 , σ Π , Ad) ζ F (2) ζ F (4) · k f σ Π k · Y v |∞ C ψ v ( Π v ) . (1.6)for all σ ∈ Aut( C ). Finally, we show that Y v |∞ C ψ v ( Π v ) ∈ Q × . (1.7)This is a local problem, but we address it by a global argument. Based on the Rallis inner product formula[GQT14] and archimedean computations, we prove that Theorem 1.1 holds when Π is endoscopic. Choosean endoscopic irreducible globally generic cuspidal automorphic representation Π ′ of GSp ( A F ) such that Π ′∞ = Π ∞ . We thus obtain (1.7) by comparing Theorem 1.1 with (1.6) for Π ′ . (Strictly speaking, we needonly to consider endoscopic lifts for F = Q .)This paper is organized as follows. In §
2, we recall the definition of the local zeta integrals Z v and Z v ,and state the precise form of (1.2) in Proposition 2.4. The proposition holds subject to Lemmas 2.1-(1) and2.3-(1) on the convergence of the local zeta integrals. In §
3, we prove (1.3) in Proposition 3.7 subject toLemmas 2.1-(2) and 2.3-(2) on the Galois equivariant property (1.4) of the local zeta integrals. In §
4, weprove in Theorem 4.8 that Theorem 1.1 holds when Π is endoscopic and F = Q . The proposition is provedbased on the Rallis inner product formula and the arithmeticity of global theta lifting in Proposition 4.5. Aswe sketched above, Theorem 1.1 follows from Propositions 2.4, 3.7, and Theorem 4.8. The context of § Notation.
Fix a totally real number field F . Let o and D be the ring of integers and the absolutediscriminant of F , respectively. Let A = A F be the ring of adeles of F and A f be its finite part. We denote byˆ o the closure of o in A f . For a finite dimensional vector space V over F , let S ( V ( A )) be the space of Schwartzfunctions on V ( A ). We will write v for places of F and ∞ the archimedean place of Q .Let v be a place of F . If v is a finite place, let o v , ̟ v , and q v be the maximal compact subring of F v , agenerator of the maximal ideal of o v , and the cardinality of o v /̟ v o v , respectively. Let | | v be the absolutevalue on F v normalized so that | ̟ v | v = q − v . If v is a real place, let | | v = | | be the ordinary absolute valueon F v ≃ R . For a character χ of F × v , let e ( χ ) ∈ R be the exponent of χ defined so that | χ | = | | e ( χ ) v . When v is finite, let c ( χ ) ∈ Z ≥ be the smallest integer so that χ is trivial on 1 + ̟ c ( χ ) v o v .Let ζ ( s ) = ζ F ( s ) = Q v ζ v ( s ) be the completed Dedekind zeta function of F , where v ranges over the placesof F and ζ v ( s ) = ( (1 − q − sv ) − if v is finite ,π − s/ Γ( s/
2) if v is real . Here Γ( s ) is the gamma function. et ψ = N v ψ v, be the standard additive character of Q \ A Q defined so that ψ p, ( x ) = e − π √− x for x ∈ Z [ p − ] ,ψ ∞ , ( x ) = e π √− x for x ∈ R . Let ψ = N v ψ v be a non-trivial additive character of F \ A . We say ψ is standard if ψ = ψ ◦ tr F / Q . In thiscase, ψ v is called the standard additive character of F v . For a ∈ F × (resp. a ∈ F × v ), let ψ a (resp. ψ av ) be theadditive character of A (resp. F v ) defined by ψ a ( x ) = ψ ( ax ) (resp. ψ av ( x ) = ψ v ( ax )). For each finite place v ,let ̟ c v v o v be the largest fractional ideal of F v on which ψ v is trivial. The absolute conductor cond( ψ ) of ψ is the ˆ o -submodule of A f defined by cond( ψ ) = Y v ∤ ∞ ̟ | c v | v o v . We write v | cond( ψ ) if c v = 0.If S is a set, then we let I S be the characteristic function of S . Let M n,m be the matrix algebra of n by m matrices. Let GSp n and Sp n be the symplectic similitude group and symplectic group, respectively, definedby GSp n = (cid:26) g ∈ GL n (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) n − n (cid:19) t g = ν ( g ) (cid:18) n − n (cid:19) , ν ( g ) ∈ GL (cid:27) , Sp n = ker( ν ) . Let B = t ∗ ∗ ∗ t ∗ ∗ νt −
00 0 ∗ νt − ∈ GSp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t , t , ν ∈ GL be the standard Borel subgroup of GSp and U be its unipotent radical. Let T ⊂ B be the standard maximaltours of GSp . For a non-trivial additive character ψ of F \ A , let ψ U be the associated additive character of U ( F ) \ U ( A ) defined by ψ U x ∗ ∗ ∗ y − x = ψ ( − x − y ) . We call ψ U standard if ψ is standard. Similar notation apply to additive character ψ v of Q v . In GL , let B be the Borel subgroup consisting of upper triangular matrices, and put a ( ν ) = (cid:18) ν
00 1 (cid:19) , d ( ν ) = (cid:18) ν (cid:19) , m ( t ) = (cid:18) t t − (cid:19) , n ( x ) = (cid:18) x (cid:19) , w = (cid:18) −
11 0 (cid:19) for ν, t ∈ GL and x ∈ G a . Let v be a finite place of F and c ∈ Z ≥ . For n ≥ c , the quasi-paramodulargroup K ( ̟ nv ; c ) of level ̟ nv o v is the open compact subgroup of GSp ( F v ) consisting of g ∈ GSp ( F v ) suchthat ν ( g ) ∈ o × v and g ∈ o v o v ̟ − n + cv o v o v ̟ n − cv o v o v o v o v ̟ nv o v ̟ n − cv o v ̟ cv o v ̟ n − cv o v ̟ n − cv o v o v o v o v . Let σ ∈ Aut( C ). Define the σ -linear action on C ( X ), which is the field of formal Laurent series in variable X over C , as follows: σ P ( X ) = ∞ X n ≫−∞ σ ( a n ) X n for P ( X ) = P ∞ n ≫−∞ a n X n ∈ C ( X ). For a complex representation Π of a group G on the space V Π of Π ,let σ Π of Π be the representation of G defined σ Π ( g ) = t ◦ Π ( g ) ◦ t − , (1.8)where t : V Π → V Π is a σ -linear isomorphism. Note that the isomorphism class of σ Π is independent of thechoice of t . We call σ Π the σ -conjugate of Π . When v is a finite place and ϕ is a complex-valued functionon F nv or ( F × v ) n , we define σ ϕ ( x ) = σ ( ϕ ( x )) for x ∈ F nv or x ∈ ( F × v ) n . .4. Measures.
Let v be a place of F . If v is finite, we normalize the Haar measures on F v and F × v so thatvol( o v ) = 1 and vol( o × v ) = 1, respectively. If v is real, we normalize the Haar measures on F v ≃ R and F × v ≃ R × so that vol([1 , , m be a positive integer. Let dg v bethe Haar measure on GL m ( F v ) defined as follows: For φ ∈ L (GL m ( F v )), we have Z GL m ( F v ) φ ( g v ) dg v = Y ≤ i 1. The top differential form together with the Haar measure on F v determines a Haar measure on H ( F v )as explained in [Vos96, § H ( F v ). For any compact group K ,we take the Haar measure on K such that vol( K ) = 1.1.5. Weil representation. Let ( V, ( , )) be a non-degenerate quadratic space of even dimension m over F .Define the orthogonal similitude group GO( V ) byGO( V ) = { h ∈ GL( V ) | ( hx, hy ) = ν ( h )( x, y ) for x, y ∈ V } , here ν : GO( V ) → GL is the scale map. LetGSO( V ) = { h ∈ GO( V ) | det( h ) = ν ( h ) m/ } . Let O( V ) and SO( V ) be the orthogonal group and special orthogonal group defined byO( V ) = { h ∈ GO( V ) | ν ( h ) = 1 } , SO( V ) = { h ∈ GO(V) | det( h ) = ν ( h ) = 1 } . Let ψ = N v ψ v be a non-trivial additive character of F \ A . We denote by ω ψ,V,n = N v ω ψ v ,V,n the Weilrepresentation of Sp n ( A ) × O( V )( A ) on S ( V n ( A )) with respect to ψ (cf. [Kud94, § 5] and [Ich05, § S ( V n ( A )) be the subspace of S ( V n ( A )) consisting of functions which correspond topolynomials in the Fock model at the archimedean places. LetG(Sp n × O( V )) = { ( g, h ) ∈ GSp n × O( V ) | ν ( g ) = ν ( h ) } . We extend ω ψ,V,n to a representation of G(Sp n × O( V ))( A ) as follows: ω ψ,V,n ( g, h ) ϕ = ω ψ,V,n (cid:18) g (cid:18) n ν ( g ) − n (cid:19) , (cid:19) L ( h ) ϕ for ( g, h ) ∈ G(Sp n × O( V ))( A ) and ϕ ∈ S ( V n ( A )). Here L ( h ) ϕ ( x ) = | ν ( h ) | − nm/ A ϕ ( h − x ) . Adjoint L -functions. Let Π = N v Π v be an irreducible globally generic cuspidal automorphic repre-sentation of GSp ( A ). By [AS06] and [GT11a, Theorem 12.1], Π has a strong functorial lift Ψ to GL ( A ).By [GT11a, Theorem 12.1], either Ψ is cuspidal or Ψ = τ ⊞ τ for some irreducible cuspidal automorphicrepresentations τ and τ of GL ( A ) with equal central character such that τ = τ . We say that Π is stable(resp. endoscopic) if Ψ is cuspidal (resp. non-cuspidal).Recall that the dual group of GSp is GSp ( C ). Let Ad denote the adjoint representation of GSp ( C ) on pgsp ( C ), and std the composition of the projection GSp ( C ) → PGSp ( C ) with the standard representationof PGSp ( C ) ≃ SO ( C ) on C . Let S be a finite set of places of F including the archimedean places such hat, for v / ∈ S , Π v is unramified. Then the partial adjoint and standard L -functions of Π are defined as theEuler products L S ( s, Π , Ad) = Y v / ∈ S L ( s, Π v , Ad) , L S ( s, Π , std) = Y v / ∈ S L ( s, Π v , std)for s ∈ C , which are absolutely convergent for Re( s ) sufficiently large. Also, we have L S ( s, Ψ , Sym ⊗ ω − Π ) = L S ( s, Π , Ad) ,L S ( s, Ψ , ∧ ⊗ ω − Π ) = ζ S ( s ) L S ( s, Π , std) . In particular, L S ( s, Π , Ad) and L S ( s, Π , std) admit meromorphic continuations to C . (In a more general con-text, the meromorphic continuation of L S ( s, Π , std) was established by Piatetski-Shapiro and Rallis [PSR87]much earlier.) By [GT11a, Theorem 12.1], L S ( s, Ψ , ∧ ⊗ ω − Π ) has a simple (resp. double) pole at s = 1 if Π is stable (resp. endoscopic). Hence L S ( s, Π , std) is holomorphic and non-zero (resp. has a simple pole) at s = 1 if Π is stable (resp. endoscopic). In particular, L S ( s, Π , Ad) is holomorphic and non-zero at s = 1.For any place v of F , we denote by φ Π v : L F v → GSp ( C ) the local L -parameter attached to Π v by the localLanglands correspondence established by Gan and Takeda [GT11a] if v is finite and by Langlands [Lan89] if v is real. Here L F v is the Weil–Deligne group of F v if v is finite but the Weil group of F v if v is real. Since Ψ v is essentially unitary and generic (and hence “almost tempered”), the adjoint L -factor L ( s, Π v , Ad) = L ( s, Ad ◦ φ Π v )defined as in [Tat79, § 3] is holomorphic at s = 1. In fact, the same holds for any irreducible admissiblegeneric representation of GSp ( F v ) (see [GP92, Conjecture 2.6], [AS08], [GT11a], [GI16, Proposition B.1]).Hence the completed adjoint L -function L ( s, Π , Ad) is holomorphic and non-zero at s = 1.2. Formula for Petersson norms Let H = GSp and G = { ( g , g ) ∈ GSp × GSp | ν ( g ) = ν ( g ) } . Denote by Z H the center of H . We identify G with its image under the embedding G −→ H, (cid:18)(cid:18) a b c d (cid:19) , (cid:18) a b c d (cid:19)(cid:19) a b a − b c d − c d . (2.1)Let V , = F be the space of column vectors equipped with a non-degenerate symmetric bilinear form ( , )given by ( x, y ) = t x (cid:18) (cid:19) y for x, y ∈ V , . With respect to the standard basis of V , , we identify GO( V , ) with the split orthogonalsimilitude group GO , = (cid:26) h ∈ GL (cid:12)(cid:12)(cid:12)(cid:12) t h (cid:18) (cid:19) h = ν ( h ) (cid:18) (cid:19) , ν ( h ) ∈ GL (cid:27) . For a non-trivial additive character ψ v of F v , we write ω ψ v = ω ψ v ,V , , for the Weil representation ofSp ( F v ) × O , ( F v ) on S ( V , ( F v )) with respect to ψ v .Let Π = N v Π v be an irreducible globally generic cuspidal automorphic representation of GSp ( A ) withcentral character ω Π . We assume Π v | Sp ( F v ) = D ( λ ,v ,λ ,v ) ⊕ D ( − λ ,v , − λ ,v ) for each real place v , where D ( λ ,v ,λ ,v ) is the discrete series representation of Sp ( F v ) ≃ Sp ( R ) with Blattnerparameter ( λ ,v , λ ,v ) ∈ Z such that 2 − λ ,v ≤ λ ,v ≤ − 1. For each finite place v , by the newform theoryof Robert–Schmidt [RS07, Theorem 7.5.4] and Okazaki [Oka19, Main Theorem], there exists a smallest non-negative integer n v ≥ c ( ω Π v ) such that Π K ( ̟ nvv ; c ( ω Π v )) v = 0. In this case, we have dim Π K ( ̟ nvv ; c ( ω Π v )) v = 1. paramodular newform of Π v is a non-zero vector in this one-dimensional space. The paramodular conductorcond( Π ) of Π is the ˆ o -submodule of A f defined bycond( Π ) = Y v ∤ ∞ ̟ n v v o v . We write v | cond( Π ) if n v > 0. For a place v and a non-trivial additive character ψ v of F v , we denote by W ( Π v , ψ U,v ) the space of Whittaker functions of Π v with respect to ψ U,v .2.1. Doubling local zeta integrals. Let P be the standard Siegel parabolic subgroup of H defined by P = (cid:26) (cid:18) a ∗ ν t a − (cid:19) ∈ H (cid:12)(cid:12)(cid:12)(cid:12) a ∈ GL , ν ∈ GL (cid:27) . Let v be a place of F . Denote by I v ( s ) the degenerate principal series representation Ind H ( F v ) P ( F v ) ( δ s/ P ) of H ( F v ).Here δ P is the modulus character of P ( F v ) given by δ P (cid:18)(cid:18) a ∗ ν t a − (cid:19)(cid:19) = | det( a ) | v | ν | − v . Let φ v be a matrix coefficient of Π v and F v ∈ I v ( s ) be a holomorphic section. We define the local zetaintegral Z v ( s, φ v , F v ) = Z Sp ( F v ) F v ( δ ( g v , , s ) φ v ( g v ) dg v , (2.2)where δ = − − . Here dg v is a Haar measure on Sp ( F v ) normalized so that vol(Sp ( o v ) , dg v ) = 1 if v is finite and is the localTamagawa measure if v is real. Note that δ ( g, g ) δ − ∈ P ( F v )(2.3)for all g ∈ GSp ( F v ) and all places v .Let v be a finite place. Let φ v be a matrix coefficient of Π v and F v ∈ I ( s ). For σ ∈ Aut( C ), we define thematrix coefficient σ φ of σ Π v (cf. Lemma 5.9) and σ F by σ φ v ( g ) = σ ( φ v ( g )) , σ F v ( h, s ) = σ ( F v ( h, s ))for g ∈ GSp ( F v ) and h ∈ H ( F v ). Note that σ F v | s = n ∈ I v ( n ) for all odd integers n . Lemma 2.1. Let φ v be a matrix coefficient of Π v and F v ∈ I v ( s ) be a holomorphic section.(1) The integral Z v ( s, φ v , F v ) is absolutely convergent for Re( s ) ≥ .(2) Assume v is finite. For σ ∈ Aut( C ) , we have σZ v ( , φ v , F v ) = Z v ( , σ φ v , σ F v ) . Proof. The assertions will be proved in Proposition 5.10 below. (cid:3) We recall the local Siegel-Weil sections. Let ψ v be a non-trivial additive character of F v . Define a H ( F v )-intertwining map S ( V , ( F v )) −→ I v ( ) , ϕ F ψ v ( ϕ )(2.4)by F ψ v ( ϕ )( g, ) = ω ψ v ( g, h ) ϕ (0) , where ν ( g ) = ν ( h ). We extend F ψ v ( ϕ ) to a holomorphic section F ψ v ( ϕ ) of I v ( s ) such that its restriction to K v is independent of s , where K v is the maximal compact subgroup of H ( F v ) defined by K v = ( H ( o v ) if v is finite, H ( R ) ∩ O(2 n ) if v is real. emma 2.2. Let ψ v be a non-trivial additive character of F v . There exists ϕ ∈ S ( V , ( F v )) such that Z v ( , φ v , F ψ v ( ϕ )) = 0 . Proof. The assertion was proved for real v in [CI19, Lemma 5.3]. We assume v is finite. Let Π be anirreducible component of Π v | Sp ( F v ) . Fix a bilinear equivariant pairing h , i on Π × Π ∨ . For f ∈ Π and f ∈ Π ∨ , define a matrix coefficient φ f ⊗ f of Π by φ f ⊗ f ( g ) = h Π ( g ) f , f i . Then the local zeta integral Z v ( , φ f ⊗ f , F ) is absolutely convergent by Lemma 2.1-(1), and defines anSp ( F v ) × Sp ( F v )-intertwining map ℓ : I v ( ) −→ Π ∨ ⊗ Π , F v (cid:2) f ⊗ f Z v ( , φ f ⊗ f , F v ) (cid:3) . Let V be the quaternionic quadratic space of dimension 4 over F v . Let V (resp. V ) be the split quadraticspace (resp. quaternionic quadratic space) of dimension 6 over F v . For i = 1 , 2, let R ( V i ) be the image of theSp ( F v )-intertwining map S ( V i ( F v )) −→ I v ( ) , ϕ F ( i ) ψ v ( ϕ ) , where F ( i ) ψ v ( ϕ )( g, ) = ω ψ v ,V i , ( g, ϕ (0). By [KR92], I v ( ) = R ( V ) + R ( V ) , R ( V ) ≃ R ( V ) /R ( V ) ∩ R ( V ) . (2.5)By [KR94, Proposition 7.2.1], the intertwining map ℓ is non-zero. If ℓ | R ( V ) is zero, then it follows from (2.5)that ℓ | R ( V ) must be non-zero and defines a non-zero element inHom Sp ( F v ) × Sp ( F v ) ( R ( V ) , Π ∨ ⊗ Π ) . Therefore the local theta lift of Π to O( V )( F v ) is non-zero by [HKS96, Proposition 3.1]. This contradictsthe genericity of Π (cf. [GT11b, Corollary 4.2-(i)]). Hence ℓ | R ( V ) is non-zero. This completes the proof. (cid:3) Local zeta integrals for GSp × GSp . Let P be the parabolic subgroup of H defined by P = a ∗ ∗ ∗ a ′ ∗ b ′ ν t a − c ′ ∗ d ′ ∈ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∈ GL (cid:18) a ′ b ′ c ′ d ′ (cid:19) ∈ GL ν = a ′ d ′ − b ′ c ′ ∈ GL . Let v be a place of F and ψ v be a non-trivial additive character of F v . Denote by I v ( s ) the degenerateprincipal series representation Ind H ( F v ) P ( F v ) ( δ s/ P ) of H ( F v ). Here δ P is the modulus character of P ( F v ) given by δ P a ∗ ∗ ∗ a ′ ∗ b ′ ν t a − c ′ ∗ d ′ = | det( a ) | v | ν | − v . Let W ,v ∈ W ( Π v , ψ U,v ) and W ,v ∈ W ( Π ∨ v , ψ − U,v ) be Whittaker functions of Π v and Π ∨ v with respect to ψ U,v and ψ − U,v , respectively, and F v ∈ I v ( s ) be a holomorphic section. We define the local zeta integral Z v ( s, W ,v , W ,v , F v ) = Z Z H ( F v ) ˜ U ( F v ) \ G ( F v ) F v ( ηg v , s )( W ,v ⊗ W ,v )( g v ) d ¯ g v , (2.6)where ˜ U = u x 00 1 0 00 0 1 00 0 0 1 , u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ∈ U, x ∈ G a nd η = − − . Here d ¯ g v is the quotient measure normalized so that vol( Z H ( o v ) ˜ U ( o v ) \ G ( o v ) , d ¯ g v ) = 1 if v is finie and is thequotient measure defined by the local Tamagawa measures on Z H ( F v ) \ G ( F v ) and ˜ U ( F v ) if v is real.Let v be a finite place. Let W v ∈ W ( Π v , ψ U,v ) be a Whittaker function and F v ∈ I v ( s ). For σ ∈ Aut( C ),we define σ W v ∈ W ( σ Π v , σ ψ U,v ) and σ F v ∈ I v ( s ) by σ W v ( g ) = σ ( W v ( g )) , σ F v ( h, s ) = σ ( F v ( h, s ))for g ∈ GSp ( F v ) and h ∈ H ( F v ). Note that σ F v | s = m ∈ I v ( m ) for all odd integers m . Lemma 2.3. Let W ,v ∈ W ( Π v , ψ U,v ) and W ,v ∈ W ( Π ∨ v , ψ − U,v ) be Whittaker functions and F v ∈ I v ( s ) be aholomorphic section.(1) The integral Z v ( s, W ,v , W ,v , F v ) is absolutely convergent for Re( s ) ≥ .(2) Assume v is finite. For σ ∈ Aut( C ) , we have σ Z v (1 , W ,v , W ,v , F v ) = Z v (1 , σ W ,v , σ W ,v , σ F v ) . Proof. The assertions will be proved in Proposition 5.14 below. (cid:3) Let v be a place of F and ψ v a non-trivial additive character of F v . When v is finite, let ̟ d v v o v be thelargest fractional ideal of F v on which ψ v is trivial. Let K ′ v and K ′ v be the maximal compact subgroups ofGO , ( F v ) and O , ( F v ), respectively, defined by K ′ v = ̟ − d v v ! GO , ( o v ) ̟ d v v ! if v is finite,GO , ( R ) ∩ O(6) if v is real .K ′ v = K ′ v ∩ O , ( F v ) . Note that K ′ v and K ′ v depend on the additive character ψ v when v is finite. Let f ◦ ψ v ∈ Ind GO , ( F v ) P ′ ( F v ) ( δ s/ P ′ )be the K ′ v -invariant section such that f ◦ ψ v (1 , s ) = 1, where P ′ is the standard Siegel parabolic subgroup ofGO , . Define the partial Fourier transform S ( V , ( F v )) −→ S (M , ( F v )) , ϕ ˆ ϕ, where ˆ ϕ ( u, v ) = Z M , ( F v ) ϕ (cid:18) xu (cid:19) ψ v (tr( v t x )) dx (2.7)for u, v ∈ M , ( F v ) and the Haar measure on M , ( F v ) ≃ F v is the product measure of the Haar measure on F v defined in § ω ψ v be the representation of G(Sp × O , )( F v ) on S (M , ( F v )) defined byˆ ω ψ v ( g, h ) ˆ ϕ = ( ω ψ v ( g, h ) ϕ )ˆ . Define a H ( F v )-intertwining map S ( V , ( F v )) −→ I v ( s ) , ϕ ψ v ( ϕ )(2.8)by F ψ v ( ϕ )( g, s ) = Z GL ( F v ) Z K ′ v ˆ ω ψ v ( g, k v h ) ˆ ϕ (0 × , t a v , × ) f ◦ ψ v ( k v h, s ) | det( a v ) | s +3 v dk v da v , here ν ( h ) = ν ( g ). The Haar measure da v on GL ( F v ) is normalized as in (1.9). Note that the integral isabsolutely convergent for Re( s ) > − s ∈ C (cf. [GJ72]). Therefore, F ψ v ( ϕ ) defines a meromorphic section of I v ( s ) which is holomorphic for Re( s ) > − Petersson norms and adjoint L -values. Let f Π = N v f v ∈ Π and f Π ∨ = N v f ∨ v ∈ Π ∨ be thenormalized newforms of Π and Π ∨ . Choose local bilinear equivariant pairings h , i v on Π v × Π ∨ v such that k f Π k = Y v |∞ h f v , Π ∨ v (diag( − , − , , f ∨ v i v · Y v ∤ ∞ h f v , f ∨ v i v , and define a matrix coefficient φ v of Π v by φ v ( g ) = h Π v ( g ) f v , f ∨ v i v h f v , f ∨ v i v if v is finite , h Π v ( g ) f v , Π ∨ v (diag( − , − , , f ∨ v i v h f v , Π ∨ v (diag( − , − , , f ∨ v i v if v is real . Let v be a place of F and ψ v a non-trivial additive character of F v . Let W ψ v ∈ W ( Π v , ψ U,v ) be a non-zeroWhittaker function satisfying the following condition: • W ψ v is a paramodular newform if v is finite; • W ψ v is a lowest weight vector of the minimal U(2)-type of D ( − λ ,v , − λ ,v ) if v is real.The condition characterize W ψ v up to scalars, we normalize it so that • W ψ v (diag( a v , a v , , a v )) = 1 if v is finite; • W ψ v ((diag( a v , a v , , a v )) is normalized as in (1.1) if v is real.Here a v ∈ F × v is chosen so that • a v o v is the largest fractional ideal of F v on which ψ v is trivial if v is finite; • ψ a v v is the standard additive character of F v if v is real.Note that W ψ v (diag( a v , a v , , a v )) = 0 for finite place v by the results of Robert–Schmidt [RS07] and Okazaki[Oka19], hence the normalization is valid. We call W ψ v the normalized Whittaker newform of Π v with respectto ψ U,v . By definition, for a ∈ F × v , we have W ψ av ( g ) = W ψ v (diag( a , a , , a ) g )for g ∈ GSp ( F v ). We define the normalized Whittaker newform W ∨ ψ v of Π ∨ v with respect to ψ v in a similarway. Proposition 2.4. Let ψ be a non-trivial additive character of F \ A . We have k f Π k = 2 c · D − · L (1 , Π , Ad) ζ F (2) ζ F (4) · Y v C ψ v ( Π v ) . Here c = ( if Π is stable , if Π is endoscopic , and C ψ v ( Π v ) is a non-zero constant depending only on Π v and ψ v given by C ψ v ( Π v ) = ζ v (1) − ζ v (3) − ζ v (4) L (1 , Π v , Ad) − × ζ v (2) − ζ v (4) − · Z v (1 , W ψ v , W ∨ ψ − v , F ψ v ( ϕ v )) Z v (cid:0) , φ v , F ψ v ( ϕ v ) (cid:1) if v is finite , Z v (1 , W ψ v , Π ∨ v (diag( − , − , , W ∨ ψ − v , F ψ v ( ϕ v )) Z v (cid:0) , φ v , F ψ v ( ϕ v ) (cid:1) if v is real , (2.9) where ϕ v ∈ S ( V , ( F v )) is any Schwartz function such that Z v (cid:0) , φ v , F ψ v ( ϕ v ) (cid:1) = 0 . Moreover, if v is a finiteplace such that Π v and ψ v are unramified, then we have C ψ v ( Π v ) = 1 . roof. The assertion can be proved by proceeding exactly as in the proof of [CI19, Proposition 5.4] withLemmas 4.2, 4.4, and 5.3 in [CI19] replaced by Lemmas 2.1-(1), 2.3-(1), and 2.2, respectively. Note that theconstant C ψ v ( Π v ) is well-defined, since L ( s, Π v , Ad) is holomorphic and non-zero at s = 1 by the genericity of Π v . Also note that the factors D − and ζ v (2) − ζ v (4) − for finite places v comparing with [CI19, Proposition5.4] are due to different normalization of Haar measures used to define the local zeta integrals Z v and Z v forfinite places v , and the partial Fourier transform ˆ ϕ v . (cid:3) Proof of Main Theorem We keep the notation of § 2. Assume further that Π is motivic, that is, there exists w ∈ Z such that | ω Π | = | | w A and λ ,v − λ ,v ≡ w (mod 2)for all real places v . In other words, we have ω Π v = sgn w | | w v for all real places v . In the following lemma, weshow that being globally generic is an arithmetic property of an irreducible cuspidal automorphic represen-tation of GSp ( A ) of motivic discrete type at the archimedean places. Lemma 3.1. Assume Π is motivic. For σ ∈ Aut( C ) , the representation σ Π is an irreducible motivic globallygeneric cuspidal automorphic representation of GSp ( A ) .Proof. Fix σ ∈ Aut( C ). We denote by Ψ = N v Ψ v the strong functorial lift of Π to GL ( A ). By [GT11a,Theorem 12.1], the automorphic representation Ψ ⊠ ω Π is equal to the global theta lift of Π from GSp ( A )to GSO , ( A ), where we identify GSO , with(GL × GL ) / { ( a , a − ) | a ∈ GL } . For a real place v , by the assumption on Π v , we have Ψ v ≃ Ind GL ( R ) P , ( R ) ( D ( λ ,v + λ ,v ) ⊠ D ( λ ,v − λ ,v )) ⊗ | | w / v , where P , is the standard maximal parabolic subgroup of GL of type (2 , 2) and D ( κ ) denotes the discreteseries representation of GL ( R ) with minimal weight κ ≥ κ . In particular,we see that Ψ is regular algebraic in the sense of Clozel [Clo90, § σ Ψ = σ Ψ ∞ ⊗ σ Ψ f is automorphic by [Clo90, Th´eor`eme 3.13], where σ Ψ f is the σ -conjugate of Ψ f = N v ∤ ∞ Ψ v and σ Ψ ∞ is the representation of GL ( F ∞ ) so that its v -component is equal to Ψ σ − ◦ v . We claim that thereexists an irreducible globally generic cuspidal automorphic representation Π ♯ = N v Π ♯v of GSp ( A ) withcentral character σ ω Π = ω Π ∞ · σ ω Π f such that σ Ψ is the strong functorial lift of Π ♯ . Assume the claim holds.For any finite place v , σ Ψ v ⊠ σ ω Π v is the local theta lift of both σ Π v and Π ♯v from GSp ( F v ) to GSO , ( F v ).Indeed, since Ψ v ⊠ ω Π v is the local theta lift of Π v from GSp ( F v ) to GSO , ( F v ), it follows from [Rob01,Proposition 1.9] and [Mor12, Proposition 5.7] that σ Ψ v ⊠ σ ω Π v is the local theta lift of σ Π v from GSp ( F v )to GSO , ( F v ). Also note that σ Ψ ∞ ⊠ ω Π ∞ is the local theta lift of both σ Π ∞ and Π ♯ ∞ from GSp ( F ∞ ) toGSO , ( F ∞ ) (cf. [Pau05]). By the Howe duality principle, we have σ Π v ≃ Π ♯v , σ Π ∞ ≃ Π ♯ ∞ for all finite places v . We conclude that Π σ ≃ Π ♯ is an irreducible globally generic cuspidal automorphicrepresentation of GSp ( A ).It remains to verify the claim. Firstly we consider the case when Π is stable. In this case, σ Ψ is cuspidalby [Clo90, Theorem 3.13] and L ( s, Ψ , ∧ ⊗ ω − Π ) has a pole at s = 1. By [GT11a, Theorem 12.1], the claimholds if and only if L ( s, σ Ψ , ∧ ⊗ σ ω − Π ) has a pole at s = 1. The last assertion was proved by Gan in [GR14,Theorem 3.6.2] based on the result on functorial lifts from GSpin ( A ) to GL ( A ). Now we assume Π isendoscopic. In this case, we write Ψ = τ ⊞ τ for some non-isomorphic irreducible cuspidal automorphicrepresentations τ and τ of GL ( A ) with equal central character ω Π such that τ ,v = D ( λ ,v + λ ,v ) ⊗ | | w / v , τ ,v = D ( λ ,v − λ ,v ) ⊗ | | w / v for all real places v . In particular, we see that τ and τ are regular algebraic. Therefore, σ Ψ = σ τ ⊞ σ τ is anisobaric automorphic representation of GL ( A ). We then take Π ♯ be the global theta lift of σ τ ⊠ σ τ ∨ fromGSO , ( A ) to GSp ( A ), where we identify GSO , with(GL × GL ) / { ( a , a ) | a ∈ GL } s in § Π ♯ is globally generic cuspidal and σ Ψ is the strong functorial lift of Π ♯ by [GT11a, Theorem12.1]. This completes the proof. (cid:3) Lemma 3.2. The rationality field Q ( Π ) of Π is equal to the fixed field of { σ ∈ Aut( C ) | σ Π f = Π f } and is anumber field.Proof. Let Ψ = N v Ψ v be the strong functorial lift of Π to GL ( A ). The rationality field Q ( Ψ ) of Ψ is thefixed field of { σ ∈ Aut( C ) | σ Ψ = Ψ } . By the result of Clozel [Clo90, Th´eor`eme 3.13], Q ( Ψ ) is a number field.It follows from the strong multiplicity one theorem for isobaric automorphic representations [JS81, Theorem4.4] that Q ( Ψ ) is equal to the fixed field of { σ ∈ Aut( C ) | σ Ψ f = Ψ f } .Let σ ∈ Aut( C ). For each place v , we have explained in the proof of Lemma 3.1 that σ Ψ v is the functoriallift of σ Π v to GL ( F v ) via the local theta correspondence. It then follows from the Howe duality principlethat σ Ψ v = Ψ v if and only if σ Π v = Π v . Therefore, we conclude that Q ( Π ) = Q ( Ψ ) and is equal to the fixedfield of { σ ∈ Aut( C ) | σ Π f = Π f } . This completes the proof. (cid:3) In the following lemma, we prove the Galois equivariant property of the local adjoint L -functions. Theargument is standard and we give a proof for convenience of the reader (cf. [Rag10, Proposition 3.17] and[Mor12, Proposition 5.4]). Lemma 3.3. Let v be a finite place. For σ ∈ Aut( C ) , we have σL (1 , Π v , Ad) = L (1 , σ Π v , Ad) . Proof. For n ≥ 1, let Irr(GL n ) be the set of isomorphism classes of irreducible admissible representationsof GL n ( F v ) and Φ(GL n ) the set of equivalence classes of admissible n -dimensional representation of theWeil–Deligne group L F v of F v . Let Irr(GL n ) −→ Φ(GL n ) , Ψ φ Ψ be the local Langlands correspondence established in [HT01] and [Hen99]. Let σ ∈ Aut( C ). By [Clo90,Lemme 4.6] and [Hen01, Propri´et´e 3, § σ L ( s + n − , φ Ψ ) = L ( s + n − , φ σ Ψ ) , σ φ Ψ = φ σ Ψ ⊗ χ n − σ . Here we regard the L -functions as rational functions in q − sv and the σ -linear action is defined as in § χ σ is the quadratic character of L F v associated to the quadratic character σ ( | | / v ) · | | − / v of F × v .Let Ψ v ∈ Irr(GL ) be the local functorial lift of Π v to GL ( F v ). We see from the proof of Lemma 3.1 that σ Ψ v is the local functorial lift of σ Π v to GL ( F v ). Let Sym : GL ( C ) → GL ( C ) be the symmetric squarerepresentation. It is easy to verify that σ (Sym ◦ φ ) = Sym ◦ σ φ, Sym ◦ ( φ ⊗ χ ) = (Sym ◦ φ ) ⊗ χ for any φ ∈ Φ(GL ) and character χ of L F v . In particular, we have σ (Sym ◦ φ Ψ v ) = Sym ◦ φ σ Ψ v . Therefore, we deduce that σ L ( s, Π v , Ad) = σ L ( s, Sym ◦ φ Ψ v )= σ L ( s + , (Sym ◦ φ Ψ v ) ⊗ | | − / v )= L ( s + , σ (Sym ◦ φ Ψ v ) ⊗ σ ( | | − / v ) χ σ )= L ( s + , (Sym ◦ φ σ Ψ v ) ⊗ | | − / v )= L ( s, Sym ◦ φ σ Ψ v )= L ( s, σ Π v , Ad) . We obtain the assertion by evaluating at s = 1. This completes the proof. (cid:3) emma 3.4. Let ψ v be a non-trivial additive character of F v and ϕ ∈ S ( V , ( F v )) .(1) Let a ∈ F × v so that a ∈ o × v if v is finite. We have F ψ a v ( ϕ ) = | a | v · F ψ v ( ϕ ′ ) , F ψ a v ( ϕ ) = | a | − s − v · F ψ v ( ϕ ′ ) , where ϕ ′ = ω ψ v (cid:18)(cid:18) a − a (cid:19) , (cid:19) ϕ. (2) Assume v is finite. For σ ∈ Aut( C ) , we have σ F ψ v ( ϕ )( g, ) = F σ ψ v ( σ ϕ )( g, ) , σ F ψ v ( ϕ )( g, 1) = F σ ψ v ( σ ϕ )( g, . Proof. Note that we have ω ψ a v = ω ψ v (cid:18)(cid:18) a a − (cid:19) , (cid:19) ◦ ω ψ v ◦ ω ψ v (cid:18)(cid:18) a − a (cid:19) , (cid:19) . (3.1)Therefore, one can easily verify thatˆ ω ψ a v ( g, h ) ˆ ϕ ( u, v ) = ˆ ω ψ v ( g, h ) ˆ ϕ ′ ( au, av )(3.2)for ( g, h ) ∈ G(Sp × O , )( F v ) and u, v ∈ M , ( F v ). The first assertion for F follows immediately from (3.1).By definition, we have f ◦ ψ v = f ◦ ψ a v if v is real. The assumption a ∈ o × v if v is finite implies that f ◦ ψ v = f ◦ ψ a v .The first assertion for F thus follows from (3.2).Assume v is finite. Let m be a positive integer and φ ∈ S (M m,m ( F v )). The integral Z ( s, φ ) = Z GL m ( F v ) φ ( a ) | det( a ) | sv da is absolutely convergent for Re( s ) > m − 1. Moreover, we can deduce from the proof of [GJ72, Proposition1.1] that Z ( s, φ ) defines a rational function in q − sv and satisfies the Galois equivariant property σZ ( s, φ ) = Z ( s, σ φ ) . (3.3)By the explicit formula for the Weil representation, we have σ ( ω ψ v ( g, h ) ϕ ) = ω σ ψ v ( g, h ) σ ϕ, σ (ˆ ω ψ v ( g, h ) ˆ ϕ ) = ˆ ω σ ψ v ( g, h ) σ ˆ ϕ (3.4)for ( g, h ) ∈ G(Sp × O( V , ))( F v ). Also note that σ f ◦ ψ v ( g, 1) = f ◦ σ ψ v ( g, 1) by definition. The second assertionthen follows immediately from (3.3) and (3.4). This completes the proof. (cid:3) Lemma 3.5. Let ψ v be a non-trivial additive character of F v and a ∈ F × v . Assume either v is finite and Π v is unramified, or v is real and a > . Then we have C ψ av ( Π v ) = | a | − v · C ψ v ( Π v ) . Proof. First assume v is finite and Π v is unramified. Let F ◦ v ∈ I v ( s ) and F ◦ v ∈ I v ( s ) be the H ( o v )-invariantsections such that F ◦ v (1 , s ) = F ◦ v (1 , s ) = 1. Let ̟ d v v o v be the largest fractional ideal of F v on which ψ v istrivial. For a ∈ F × v , define ϕ = ϕ a ∈ S ( V , ( F v )) by ϕ (cid:18) xy (cid:19) = I M , ( a − ̟ dvv o v ) ( x ) I M , ( o v ) ( y ) . Then the partial Fourier transform (2.7) of ϕ with respect to ψ av ◦ tr is equal toˆ ϕ = | a | − v q − d v v · I M , ( o v ) . Note that ω ψ av ( k, k ′ ) ϕ = ϕ for ( k, k ′ ) ∈ ( H ( o v ) × K ′ v ) ∩ G(Sp × O , )( F v ), where K ′ v = (cid:18) a̟ − d v v (cid:19) GO , ( o v ) (cid:18) a − ̟ d v v (cid:19) . Therefore, we have F ψ av ( ϕ ) = F ψ av ( ϕ )(1 , ) · F ◦ v , F ψ av ( ϕ ) = F ψ av ( ϕ )(1 , s ) · F ◦ v . ote that F ψ av ( ϕ )(1 , ) = ϕ (0) = 1 and F ψ av ( ϕ )(1 , s ) = | a | − v q − d v v Z GL ( F v ) I M , ( o v ) ( t ) | det( t ) | s +3 v dt = | a | − v q − d v v ζ v ( s + 1) ζ v ( s + 2) ζ v ( s + 3) . On the other hand, a change of variables g (diag( a , a , , a ) , diag( a , a , , a )) − g shows that Z v ( s, W ψ av , W ∨ ψ − av , F ) = | a | − s/ / v Z v ( s, W ψ v , W ∨ ψ − v , F ) . for any holomorphic section F of I v ( s ). We conclude that Z v ( s, φ v , F ψ av ( ϕ )) = Z v ( s, φ v , F ◦ v ) , Z v ( s, W ψ av , W ∨ ψ − av , F ψ av ( ϕ )) = | a | − s/ − / v q − d v v ζ v ( s + 1) ζ v ( s + 2) ζ v ( s + 3) Z v ( s, W ψ v , W ∨ ψ − v , F ◦ v ) . The assertion then follows immediately.Now we assume v is real and a > 0. Put f W ξ v = Π ∨ v (diag( − , − , , W ∨ ξ v for any additive character ξ v of F v . Let ϕ ∈ S ( V , ( F v )) such that Z v (cid:0) , φ v , F ψ av ( ϕ ) (cid:1) = 0. By Lemma 3.4-(1), we have F ψ av ( ϕ ) = | a | v · F ψ v ( ϕ ′ ) , F ψ av ( ϕ ) = | a | − s/ − / v · F ψ v ( ϕ ′ ) , where ϕ ′ = ω ψ v (cid:18)(cid:18) √ a − √ a (cid:19) , (cid:19) ϕ. Similarly, a change of variables shows that Z v ( s, W ψ av , f W ψ − av , F ) = | a | − s/ / v Z v ( s, W ψ v , f W ψ − v , F )for any holomorphic section F of I v ( s ). We conclude that Z v ( s, φ v , F ψ av ( ϕ )) = | a | v Z v ( s, φ v , F ψ v ( ϕ ′ )) , Z v ( s, W ψ av , f W ψ − av , F ψ av ( ϕ )) = | a | − s +1 v Z v ( s, W ψ v , f W ψ − v , F ψ v ( ϕ ′ )) . This completes the proof. (cid:3) Lemma 3.6. Let ψ be a non-trivial additive character of F \ A . Let S be any finite set of finite places of F containing the divisors of cond( Π ) . For a ∈ T v ∈ S o × v ∩ F × > , we have Y v ∈ S C ψ av ( Π v ) = Y v ∈ S C ψ v ( Π v ) . Here F × > denotes the set of totally positive elements in F .Proof. By Proposition 2.4, we have Y v C ψ av ( Π v ) = Y v C ψ v ( Π v ) . On the other hand, since a is totally positive, by Lemma 3.5 we have Y v / ∈ S C ψ av ( Π v ) = Y v / ∈ S | a | − v · C ψ v ( Π v ) . Since a ∈ T v ∈ S o × v ∩ F × , we have Q v / ∈ S | a | v = | a | A = 1. This completes the proof. (cid:3) In the following theorem we prove the Galois equivariance property of the product of local constant C ψ v ( Π v ) over finite places. Proposition 3.7. Let ψ be a non-trivial additive character of F \ A . For σ ∈ Aut( C ) , we have σ Y v ∤ ∞ C ψ v ( Π v ) = Y v ∤ ∞ C ψ v ( σ Π v ) . roof. Let v be a place of F and ξ v a non-trivial additive character of F v . For ϕ v ∈ S ( V , ( F v )) such that Z v ( , φ v , F ξ v ( ϕ v )) = 0, define C ξ v ( Π v , ϕ v ) = ζ v (1) − ζ v (3) − ζ v (4) L (1 , Π v , Ad) − × ζ v (2) − ζ v (4) − · Z v (1 , W ξ v , W ∨ ξ − v , F ξ v ( ϕ v )) Z v (cid:0) , φ v , F ξ v ( ϕ v ) (cid:1) if v is finite , Z v (1 , W ξ v , Π ∨ v (diag( − , − , , W ∨ ξ − v , F ξ v ( ϕ v )) Z v (cid:0) , φ v , F ξ v ( ϕ v ) (cid:1) if v is real . When ξ v is the local component at v of a non-trivial additive character of F \ A , it follows from Proposition2.4 that C ξ v ( Π v , ϕ v ) = C ξ v ( Π v ) is the non-zero constant in (2.9) and does not depend on the choice of ϕ v .Let σ ∈ Aut( C ). Let u ∈ b Z × be the unique element such that σ ( ψ ( x )) = ψ ( ux ) for x ∈ A f and any non-trivial additive character ψ of F \ A . For each finite place v lying over a rational prime p and any non-trivialadditive character ξ v of F v , define the σ -linear isomorphism t σ,v : W ( Π v , ξ U,v ) −→ W ( σ Π v , ξ U,v ) ,t σ,v W ( g ) = σ W (diag( u − p , u − p , , u − p ) g ) . Note that σ ξ v = ξ u p v . Then t σ,v W ξ v is the normalized Whittaker newform of σ Π v with respect to ξ U,v (cf. § σ -linear isomorphism t σ,v : W ( Π ∨ v , ξ U,v ) → W ( σ Π ∨ v , ξ U,v ) in the same way. By the Chineseremainder theorem, there exists a ∈ T p | N F / Q (cond( Π ) · cond( ψ )) Z × p ∩ Q × > such that au p = s p for some s p ∈ Z × p for p | N F / Q (cond( Π ) · cond( ψ )). Let v | cond( Π ) · cond( ψ ) lying over a prime p . We have σC ψ av ( Π v ) C ψ aupv ( σ Π v , σ ϕ v ) = L (1 , σ Π v , Ad) σL (1 , Π v , Ad) · σ Z v (1 , W ψ av , W ∨ ψ − av , F ψ av ( ϕ v )) Z v (1 , t σ,v W ψ aupv , t σ,v W ∨ ψ − aupv , F ψ aupv ( σ ϕ v )) × Z v (cid:16) , σ φ v , F ψ aupv ( σ ϕ v ) (cid:17) σZ v (cid:0) , φ v , F ψ av ( ϕ v ) (cid:1) . Note that σ W ψ av = t σ,v W ψ aupv by definition. It follows from Lemmas 2.1-(2), 2.3-(2), 3.3, and 3.4-(2) that theabove ratio is equal to 1. Similarly, we also have C ψ aupv ( σ Π v , σ ϕ v ) = C ψ v ( σ Π v , ϕ ′ v ) = C ψ v ( σ Π v ) , where ϕ ′ v = ω ψ v (cid:18)(cid:18) s − p s p (cid:19) , (cid:19) σ ϕ v . Indeed, since au p ∈ ( Z × p ) , by Lemma 3.4-(1) we have C ψ aupv ( σ Π v , σ ϕ v ) C ψ v ( σ Π v , ϕ ′ v ) = Z v (1 , t σ,v W ψ aupv , t σ,v W ∨ ψ − aupv , F ψ v ( ϕ ′ v )) Z v (1 , t σ,v W ψ v , t σ,v W ∨ ψ − v , F ψ v ( ϕ ′ v )) . Note that t σ,v W ψ aupv ( g ) = t σ,v W ψ v (diag( s p , , s − p , s − p ) g )and F ψ v ( ϕ ′ v )( η (diag( s p , , s − p , s − p ) , diag( s p , , s − p , s − p )) g, s ) = F ψ v ( ϕ ′ v )( ηg, s )for g ∈ G ( F v ). We see that the above ratio is also equal to 1. We conclude that σ Y v | cond( Π ) · cond( ψ ) C ψ av ( Π v ) = Y v | cond( Π ) · cond( ψ ) C ψ v ( σ Π v ) . It then follows from Theorem 3.6 that σ Y v | cond( Π ) · cond( ψ ) C ψ v ( Π v ) = Y v | cond( Π ) · cond( ψ ) C ψ v ( σ Π v ) . inally, note that for v ∤ ∞ · cond( Π ) · cond( ψ ), we have C ψ v ( Π v ) = C ψ v ( σ Π v ) = 1. This completes theproof. (cid:3) By Propositions 2.4 and 3.7, our main result Theorem 1.1 then follows from the following result. Theorem 3.8. Let v be a real place and ψ v the standard additive character of F v . We have C ψ v ( Π v ) ∈ Q × . Proof. Applying [Shi12, Theorem 5.7] to GSO , ( A Q ), there exist cohomological irreducible cuspidal auto-morphic representations τ and τ of GL ( A Q ) with central character inverse to each other such that τ , ∞ = D ( λ ,v − λ ,v ) ⊗ | | w / , τ , ∞ = D ( λ ,v + λ ,v ) ⊗ | | − w / and τ = τ ∨ . We regard τ ⊠ τ as an irreducible cuspidal automorphic representation of GSO , ( A Q ) andconsider its global theta lift Π ′ = θ ( τ ⊠ τ ) to GSp ( A Q ). Then Π ′ is an irreducible motivic globally genericcuspidal automorphic representation of GSp ( A Q ) such that Π ′∞ = Π v . Let ψ be the standard additive character of Q \ A Q . Thus ψ ∞ , = ψ v . By Propositions 2.4 and 3.7 appliedto ψ and Π ′ , we have σ (cid:18) L (1 , Π ′ , Ad) ζ Q (2) ζ Q (4) · k f Π ′ k · C ψ ∞ , ( Π ′∞ ) (cid:19) = L (1 , σ Π ′ , Ad) ζ Q (2) ζ Q (4) · k f σ Π ′ k · C ψ ∞ , ( Π ′∞ )for all σ ∈ Aut( C ). On the other hand, we proved in Theorem 4.8 below that Theorem 1.1 holds for Π ′ . Itfollows that σC ψ ∞ , ( Π ′∞ ) = C ψ ∞ , ( Π ′∞ )for all σ ∈ Aut( C ). Hence C ψ v ( Π v ) = C ψ ∞ , ( Π ′∞ ) ∈ Q . This completes the proof. (cid:3) Petersson norms of endoscopic lifts The purpose of this section is to prove Theorem 1.1 for endoscopic lifts based on Rallis inner productformula. For simplicity, we assume F = Q in this section.4.1. Cohomological cusp forms on GL . Let τ = N v τ v be a cohomological irreducible cuspidal auto-morphic representation of GL ( A ). There exist κ ∈ Z ≥ and w ∈ Z with κ ≡ w (mod 2) such that τ ∞ = D ( κ ) ⊗ | | w / ∞ . Here D ( κ ) denotes the discrete series representation of GL ( R ) with minimal weight κ and central charactersgn κ . Let τ + (resp. τ − ) be the space of holomorphic (resp. anti-holomorphic) cusp forms in τ . For a non-trivial additive character ψ v of Q v , let W ( τ v , ψ v ) be the space of Whittaker functions of τ v with respect to ψ v . When v = p is finite, for σ ∈ Aut( C ), we define the σ -linear isomorphism t σ,p : W ( τ p , ψ p ) −→ W ( σ τ p , ψ p ) ,t σ,p W ( g ) = σ W (diag( u − p , g ) , where u p ∈ Z × p is the unique element such that σ ( ψ p ( x )) = ψ p ( u p x ) for x ∈ Q p . When v = ∞ and ψ ∞ = ψ a ∞ , ,let W ( ± κ ; w ) ,ψ ∞ ∈ W ( τ ∞ , ψ ∞ ) be the weight ± κ Whittaker function given by W ( ± κ ; w ) ,ψ ∞ ( z n ( x ) a ( y ) k θ ) = z w ( ± ay ) ( κ + w ) / e π √− a ( x ±√− y ) ±√− κθ · I R × > ( ± ay )for x ∈ R , y, z ∈ R × , and k θ = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) ∈ SO(2). Let ψ be a non-trivial additive character of Q \ A .For f ∈ τ , let W f,ψ be the Whittaker function of f with respect to ψ defined by W f,ψ ( g ) = Z Q \ A f ( n ( x ) g ) ψ ( x ) dx Tam . Here dx Tam is the Tamagawa measure on A . For f ∈ τ ± , let W ( ∞ ) f,ψ ∈ N p W ( τ p , ψ p ) be the unique Whittakerfunction such that W f,ψ = W ( ± κ ; w ) ,ψ ∞ · W ( ∞ ) f,ψ . hen we obtain the GL ( A f )-equivariant isomorphism τ ± −→ O p W ( τ p , ψ p ) , f W ( ∞ ) f,ψ . For σ ∈ Aut( C ), it is well-known that the irreducible admissible representation σ τ = τ ∞ ⊗ σ τ f of GL ( A ) is automorphic and cuspidal, where τ f = N p τ p . Let τ ± −→ σ τ ± , f σ f (4.1)be the σ -linear GL ( A f )-equivariant isomorphism defined such that the diagram τ ± σ τ ± N p W ( τ p , ψ p ) N p W ( σ τ p , ψ p ) N p t σ,p commutes. In other words, we have W ( ∞ ) σ f,ψ = O p t σ,p W ( ∞ ) f,ψ . Arithmeticity of global theta lifting. Let ( V, Q ) be the quadratic space over Q defined by V = M , and Q [ x ] = det( x ). Let ι be the main involution on M , defined by (cid:18) x x x x (cid:19) ι = (cid:18) x − x − x x (cid:19) . The associated symmetric bilinear form is given by ( x, y ) = tr( xy ι ). We have an exact sequence1 −→ GL −→ (GL × GL ) ρ −→ GSO( V ) −→ , (4.2)where ∆( a ) = ( a , a ) and ρ ( h , h ) x = h xh − for a ∈ GL , h , h ∈ GL , and x ∈ V . For h , h ∈ GL ,we write ρ ( h , h ) = [ h , h ]. Note that ν ([ h , h ]) = det( h h − ). For a non-trivial additive character ψ v of Q v , we write ω ψ v = ω ψ v ,V, for the Weil representation of Sp ( Q v ) × O( V )( Q v ) on S ( V ( Q v )) with respectto ψ v . Let ψ be a non-trivial additive character of Q \ A . For ϕ ∈ S ( V ( A )), the theta function associated to ϕ with respect to ψ is defined by Θ ψ ( g, h ; ϕ ) = X x ∈ V ( Q ) ω ψ ( g, h ) ϕ ( x )for ( g, h ) ∈ G(Sp × O( V ))( A ). Let f be a cusp form on GSO( V )( A ) and let ϕ ∈ S ( V ( A )). For g ∈ GSp ( A ),choose h ∈ GSO( V )( A ) such that ν ( h ) = ν ( g ), and put θ ψ ( f, ϕ )( g ) = Z SO( V )( Q ) \ SO( V )( A ) f ( h h )Θ ψ ( g, h h ; ϕ ) dh Tam1 . Here dh Tam1 is the Tamagawa measure on SO( V )( A ). Then θ ψ ( f, ϕ ) is an automorphic form on GSp ( A ).Let τ and τ be cohomological irreducible cuspidal automorphic representations with central characterinverse to each other. There exist κ , κ ∈ Z ≥ and w ∈ Z with κ ≡ κ ≡ w (mod 2) such that τ , ∞ = D ( κ ) ⊗ | | w / ∞ , τ , ∞ = D ( κ ) ⊗ | | − w / ∞ . We assume κ ≥ κ and regard τ ⊠ τ as an automorphic representation of GSO( V )( A ) via the exact sequence(4.2). We assume further that τ = τ ∨ . Then the global theta lift θ ( τ ⊠ τ ) = (cid:8) θ ψ ( f, ϕ ) (cid:12)(cid:12) f ∈ τ ⊠ τ , ϕ ∈ S ( V ( A )) (cid:9) is an irreducible motivic globally generic cuspidal automorphic representation of GSp ( A ) (cf. [GT11a, The-orem 12.1]). As the notation suggests, the global theta lift does not depend on the choice of ψ (cf. [Rob01,Proposition 1.9]). Write Π = θ ( τ ⊠ τ ) . ote that ω Π = ω τ = ω − τ . Moreover, Π ∞ is a generic discrete sereis representation of GSp ( R ) with Π ∞ | Sp ( R ) = D ( λ ,λ ) ⊕ D ( − λ , − λ ) and ( λ , λ ) = ( κ + κ , κ − κ ) . Let Π mot be the space of cusp forms in Π such that the archimedean component is a lowest weight vectorof the minimal U(2)-type of D ( − λ , − λ ) . For σ ∈ Aut( C ) and ψ p a non-trivial additive character of Q p , wedefine the σ -linear isomorphism t σ,p : W ( Π p , ψ U,p ) −→ W ( σ Π p , ψ U,p ) ,t σ,p W ( g ) = σ W (diag( u − p , u − p , , u − p ) g ) , where u p ∈ Z × p is the unique element such that σ ( ψ p ( x )) = ψ p ( u p x ) for x ∈ Q p . For v = ∞ and ψ ∞ the standard additive character of R , let W ( λ ,λ ; w ) ,ψ U, ∞ ∈ W ( Π ∞ , ψ U, ∞ ) be the lowest weight Whittakerfunction of the minimal U(2)-type of D ( − λ , − λ ) with respect to ψ U, ∞ normalized as in (1.1). For a ∈ R × ,we then define W ( λ ,λ ; w ) ,ψ aU, ∞ ∈ W ( Π ∞ , ψ aU, ∞ ) by W ( λ ,λ ; w ) ,ψ aU, ∞ ( g ) = W ( λ ,λ ; w ) ,ψ U, ∞ (diag( a , a , , a ) g ) . Let ψ be a non-trivial additive character of Q \ A . For f ∈ Π , let W f,ψ U be the Whittaker function of f withrespect to ψ U defined by W f,ψ U ( g ) = Z U ( Q ) \ U ( A ) f ( ug ) ψ U ( u ) du Tam . Here du Tam is the Tamagawa measure on U ( A ). For f ∈ Π mot , let W ( ∞ ) f,ψ U ∈ N p W ( Π p , ψ U,p ) be the uniqueWhittaker function such that W f,ψ U = W ( λ ,λ ; w ) ,ψ U, ∞ · W ( ∞ ) f,ψ U . Then we obtain the GSp ( A f )-equivariant isomorphism Π mot −→ O p W ( Π p , ψ U,p ) , f W ( ∞ ) f,ψ U . For σ ∈ Aut( C ), the irreducible admissible representation σ Π = Π ∞ ⊗ σ Π f of GSp ( A ) is automorphic, globally generic, and cuspidal by Lemma 3.1, where Π f = N p Π p . Let Π mot −→ σ Π mot , f σ f (4.3)be the σ -linear GSp ( A f )-equivariant isomorphism defined such that the diagram Π mot σ Π mot N p W ( Π p , ψ U,p ) N p W ( σ Π p , ψ U,p ) N p t σ,p commutes. In other words, we have W ( ∞ ) σ f,ψ U = O p t σ,p W ( ∞ ) f,ψ U . (4.4)The main result of this section is Proposition 4.5. Roughly speaking, we show that global theta liftingcommutes with the Galois actions (4.1) and (4.3).We begin with the following formula for Whittaker functions of global theta lifts. For each place v of Q ,let dh ,v be the Haar measure on SO( V )( Q v ) defined as in [CI19, § Lemma 4.1. Let ψ be a non-trivial additive character of Q \ A . Let ϕ = N v ϕ v ∈ S ( V ( A )) and f = f ⊗ f ∈ τ ⊠ τ with W f ,ψ = Q v W ,v and W f ,ψ = Q v W ,v . Then we have W θ ψ ( f,ϕ ) ,ψ U = ζ (2) − Y v W ψ v ( W ,v ⊗ W ,v , ϕ v ) . ere W ψ v ( W ,v ⊗ W ,v , ϕ v )( g v ) = Z ∆ N ( Q v ) \ SO( V )( Q v ) ( W ,v ⊗ W ,v )( h ,v h v ) ω ψ v ( g v , h ,v h v ) ϕ v ( x , y ) dh ,v for ( g v , h v ) ∈ G(Sp × O( V ))( Q v ) with h v ∈ GSO( V )( Q v ) , x = (cid:18) − 10 0 (cid:19) , y = a ( − , ∆ N = { [ n ( x ) , n ( − x )] ∈ SO( V ) | x ∈ G a } , and dh ,v is the quotient measure defined by the Haar measureson ∆ N ( Q v ) and SO( V )( Q v ) .Proof. This is [CI19, Lemma 7.1]. The factor ζ (2) − is due to the comparison between quotient of Tamagawameasures on ∆ N ( A ) and SO( V )( A ) with Q v dh ,v . (cid:3) For v = p , let ϕ ◦ p ∈ S ( V ( Q p )) defined by ϕ ◦ p = I V ( Z p ) . Note that for a ∈ Q × p and ψ p the standard additive character of Q p , we have ω ψ ap ( k, [ k , k ]) ϕ ◦ p = ϕ ◦ p (4.5)for k ∈ diag(1 , , a, a ) GSp ( Z p )diag(1 , , a, a ) − and ( k , k ) ∈ GL ( Z p ) × GL ( Z p ) such that ν ( k ) =det( k k − ). When τ ,p and τ ,p are both unramified, let W ◦ p,a ∈ W ( τ ,p , ψ ap ) ⊗ W ( τ ,p , ψ ap ) be the GL ( Z p ) × GL ( Z p )-invariant Whittaker function normalized so that W ◦ p,a (( a ( a − ) , a ( a − ))) = 1 . By (4.5), the Whittaker function W ψ ap ( W ◦ p,a , ϕ ◦ p ) ∈ W ( Π p , ψ aU,p ) is diag(1 , , a, a ) GSp ( Z p )diag(1 , , a, a ) − -invariant. Lemma 4.2. Let ψ p be the standard additive character of Q p and a ∈ Q × p . Assume both τ ,p and τ ,p areunramified. We have W ψ ap ( W ◦ p,a , ϕ ◦ p )(diag( a − , a − , a − , a − )) = ω Π p ( a ) − | a | − p . Proof. The computation is similar to [CI19, Lemma 7.3] and we leave the detail to the readers. (cid:3) For v = ∞ , let ϕ ±∞ ∈ S ( V ( R )) defined by ϕ + ∞ ( x, y ) = 2 λ +4 ( −√− x − x − x + √− x ) λ ( y + √− y − √− y + y ) − λ e − π tr( x t x + y t y ) ,ϕ −∞ ( x, y ) = 2 λ +4 ( √− x − x + x + √− x ) λ ( − y + √− y + √− y + y ) − λ e − π tr( x t x + y t y ) . Let ψ ∞ be the standard additive character of R , then we have ω ψ ∞ (1 , [ k θ , k θ ]) ϕ + ∞ = e −√− κ θ + κ θ ) ϕ + ∞ , ω ψ − ∞ (1 , [ k θ , k θ ]) ϕ −∞ = e −√− κ θ − κ θ ) ϕ −∞ for k θ , k θ ∈ SO(2), and ω ψ ±∞ ( Z, · ϕ ±∞ = − λ · ϕ ±∞ , ω ψ ±∞ ( Z ′ , · ϕ ±∞ = − λ · ϕ ±∞ , ω ψ ±∞ ( N − , · ϕ ±∞ = 0 . (4.6)Here Z, Z ′ , N − are elements in sp ( R ) ⊗ R C defined by Z = −√− − , Z ′ = −√− − , N − = √− − √− −√− −√− − . Define W ±∞ ∈ W ( τ , ∞ , ψ ±∞ ) ⊗ W ( τ , ∞ , ψ ±∞ ) by W + ∞ = W ( κ ; w ) ,ψ ∞ ⊗ W ( κ ; − w ) ,ψ ∞ , W −∞ = W ( κ ; w ) ,ψ − ∞ ⊗ W ( − κ ; − w ) ,ψ − ∞ . By (4.6), the Whittaker function W ψ ±∞ ( W ±∞ , ϕ ±∞ ) ∈ W ( Π ∞ , ψ ± U, ∞ ) is a lowest weight Whittaker function ofthe minimal U(2)-type of D ( − λ , − λ ) with respect to ψ ± U, ∞ . emma 4.3. Let ψ ∞ be the standard additive character of R . We have W ψ ±∞ ( W ±∞ , ϕ ±∞ ) = W ( λ ,λ ; w ) ,ψ ± U, ∞ . Proof. The assertion for W ψ ∞ ( W + ∞ , ϕ + ∞ ) was proved in [CI19, Lemma 7.7]. The computation for the othercase is similar and we leave the detail to the readers. (cid:3) In the following lemma, we establish an explicit relation between W ψ v and W ψ a v . Lemma 4.4. Let ψ v be a non-trivial additive character of Q v and a ∈ Q × v . For ϕ ∈ S ( V ( Q v )) and W ∈ W ( τ ,v , ψ v ) ⊗ W ( τ ,v , ψ v ) , we have W ψ a v ( ℓ (( a ( a ) , a ( a ))) W, ϕ ) (cid:0) diag( a − , a − , , a − ) g (cid:1) = ω Π v ( a ) − | a | − v · W ψ v (cid:18) W, ω ψ v (cid:18)(cid:18) a − a (cid:19) , (cid:19) ϕ (cid:19) ( g ) for g ∈ GSp ( Q v ) . Here ℓ (( a ( a ) , a ( a ))) W ∈ W ( τ ,v , ψ a v ) ⊗ W ( τ ,v , ψ a v ) is defined by ℓ (( a ( a ) , a ( a ))) W ( h ) = W (( a ( a ) , a ( a )) h ) . Proof. First note that ω ψ a v = ω ψ v (cid:18)(cid:18) a a − (cid:19) , (cid:19) ◦ ω ψ v ◦ ω ψ v (cid:18)(cid:18) a − a (cid:19) , (cid:19) and ω ψ v (cid:0) diag( b − , , b, , [ a ( b − ) , a ( b − )] (cid:1) ϕ ′ ( x , y ) = | b | − v ϕ ′ ( x , y )for any ϕ ′ ∈ S ( V ( Q v )) and b ∈ Q × v . Let ( g, h ) ∈ G(Sp × O( V ))( Q v ) with h ∈ GSO( V )( Q v ). Thus we have W ψ a v ( ℓ (( a ( a ) , a ( a ))) W, ϕ ) (cid:0) diag( a − , a − , , a − ) g (cid:1) = ω Π v ( a ) − · W ψ a v ( ℓ (( a ( a ) , a ( a ))) W, ϕ ) (cid:0) diag( a − , a − , a , a ) g (cid:1) = ω Π v ( a ) − | a | v Z ∆ N ( Q v ) \ SO( V )( Q v ) W ( h h ) ω ψ a v (cid:0) diag( a − , a − , a , a ) g, [ a ( a − ) , a ( a − )] h h (cid:1) ϕ ( x , y ) dh = ω Π v ( a ) − | a | v × Z ∆ N ( Q v ) \ SO( V )( Q v ) W ( h h ) ω ψ v (cid:0) diag( a − , , a , g diag( a − , a − , a, a ) , [ a ( a − ) , a ( a − )] h h (cid:1) ϕ ( x , y ) dh = ω Π v ( a ) − | a | − v · W ψ v (cid:18) W, ω ψ v (cid:18)(cid:18) a − a (cid:19) , (cid:19) ϕ (cid:19) ( g ) . Here the factor | a | v in the second equality is due to the change of variables h [ a ( a − ) , a ( a − )] h . Thiscompletes the proof. (cid:3) Let ψ be a non-trivial additive character of Q \ A and S a finite set of finite places of Q . We write τ i,S = O p ∈ S τ i,p , Π S = O p ∈ S Π p , Q S = Y p ∈ S Q p , | | S = Y p ∈ S | | p , ψ S = O p ∈ S ψ p , ω ψ S = O p ∈ S ω ψ p for i = 1 , 2. Assume ψ is standard. For W S ∈ W ( τ ,S , ψ ± S ) ⊗ W ( τ ,S , ψ ± S ), let f W S ∈ τ +1 ⊠ τ ± defined so that W ( ∞ ) f WS ,ψ ± ⊠ ψ ± = W S · Y p/ ∈ S W ◦ p, ± . For a ∈ Q × , by the definition of global Whittaker function, we easily see that W ( ∞ ) f WS ,ψ ± a ⊠ ψ ± a = ℓ (( a ( a ) , a ( a ))) W S · Y p/ ∈ S W ◦ p, ± a . or σ ∈ Aut( C ), let u σ ∈ b Z × be the unique element such that σ ( ψ ( x )) = ψ ( u σ x ) for all x ∈ A f . We havethe σ -linear SO( V )( A f )-equivariant isomorphism τ +1 ⊠ τ ± −→ σ τ +1 ⊠ σ τ ± , f σ f (4.7)defined by the σ -linear isomorphisms in (4.1) for τ +1 and τ ± . Proposition 4.5. Let ψ be the standard additive character of Q \ A and S a finite set of finite places of Q containing the prime divisors of cond( Π ) . Let σ ∈ Aut( C ) and a ∈ Q × > . Assume u σ,S = at for some t ∈ Q × S . For ϕ S ∈ S ( V ( Q S )) and W S ∈ W ( τ ,S , ψ ± S ) ⊗ W ( τ ,S , ψ ± S ) , we have σ Π S ∪{∞} (diag(1 , , a, a )) σ θ ψ ± (cid:0) f W S , ϕ ±∞ ⊗ ϕ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p ) (cid:1) = ζ (2) σ ( ζ (2)) · ω Π ∞ ( √ a ) − · σ ω Π S ( a t ) − × θ ψ ± a (cid:18) σ f W S , ω ψ ±∞ (cid:18)(cid:18) √ a √ a − (cid:19) , (cid:19) ϕ ±∞ ⊗ ω ψ ± S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p ) (cid:19) . (4.8) Here σ Π S ∪{∞} = N v / ∈ S ∪{∞} σ Π v and the Galois action on the cusp form on the left-hand side (resp. right-hand side) is defined in (4.3) (resp. (4.7)).Proof. To prove the assertion, it suffices to show that the Whittaker functions of both sides with respect to ψ ± U are equal. By Lemmas 4.3 and 4.4, we have W ψ ± b ∞ ℓ (( a ( b ) , a ( b ))) W ±∞ , ω ψ ±∞ √ b √ b − ! , ! ϕ ±∞ ! (diag( b − , b − , , b − ) g )= ω Π ∞ ( √ b ) − | b | − v · W ( λ ,λ ; w ) ,ψ ± U, ∞ (4.9)for all b > 0. We have W f WS ,ψ ± ⊠ ψ ± = W ±∞ · W S · Y p/ ∈ S W ◦ p, ± . By Lemma 4.1 for the additive character ψ ± , (4.4), and (4.9) with b = 1, we see that the Whittaker functionof the global theta lift on the left-hand side of (4.8) with respect to ψ ± U is equal to σ ( ζ (2)) − · W ( λ ,λ ; w ) ,ψ ± U, ∞ ( g ∞ ) · t σ,S W ψ ± S ( W S , ϕ S ) ( g S ) · Y p/ ∈ S t σ,p W ψ ± p ( W ◦ p, ± a , ϕ ◦ p )( g p · diag(1 , , a, a ))(4.10)for g ∈ GSp ( A ). Let p / ∈ S . By Lemma 4.2, we have W ψ ± p ( W ◦ p, ± , ϕ ◦ p )( g p · diag(1 , , a, a )) = ω Π p ( a ) | a | p · W ψ ± ap ( W ◦ p, ± a , ϕ ◦ p )(diag( a − , a − , , a − ) g p )for g p ∈ GSp ( Q p ). Both t σ,p W ψ ± p ( W ◦ p, ± , ϕ ◦ p ) and W ψ ± p ( t σ,p W ◦ p, ± , ϕ ◦ p ) are GSp ( Z p )-invariant Whittakerfunctions of σ Π p with respect to ψ ± U,p . By evaluating these Whittaker functions at 1, we see that they mustbe equal. Thus we have t σ,p W ψ ± p ( W ◦ p, ± , ϕ ◦ p )( g p · diag(1 , , a, a ))= σ ω Π p ( a ) | a | p · W ψ ± ap ( t σ,p W ◦ p, ± a , ϕ ◦ p )(diag( a − , a − , , a − ) g p )(4.11)for g p ∈ GSp ( Q p ). Let p ∈ S . By the explicit formula for the Weil representation, we have σ ( ω ξ S ( g, h ) ϕ ′ S ) = ω σ ξ S ( g, h ) σ ϕ ′ S for any non-trivial additive character ξ S of Q S and ϕ ′ S ∈ S ( V ( Q S )). It follows that σ W ψ ± S ( W S , ϕ S ) = W ψ ± at S ( σ W S , σ ϕ S ) . herefore, by Lemma 4.4, we have t σ,S W ψ ± S ( W S , ϕ S ) ( g S )= σ W ψ ± S ( W S , ϕ S ) (cid:16) diag( u − σ,S , u − σ,S , , u − σ,S ) g S (cid:17) = W ψ ± at S ( σ W S , σ ϕ S ) (cid:16) diag( u − σ,S , u − σ,S , , u − σ,S ) g S (cid:17) = σ ω Π S ( t ) − | t | − S · W ψ ± aS (cid:18) ℓ (( a ( t − ) , a ( t − ))) σ W S , ω ψ ± S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S (cid:19) (cid:0) diag( a − , a − , , a − ) g S (cid:1) = σ ω Π S ( t ) − | t | − S · W ψ ± aS (cid:18) ℓ (( a ( a ) , a ( a ))) t σ,S W S , ω ψ ± S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S (cid:19) (cid:0) diag( a − , a − , , a − ) g S (cid:1) (4.12)for g S ∈ GSp ( Q S ). Note that | t | − S Y p/ ∈ S σ ω Π p ( a ) | a | p = ω Π ∞ ( a ) − | a | − ∞ σ ω Π S ( a ) − by the product formula and the automorphy of σ ω Π = ω Π ∞ · σ ω Π f . By (4.11) and (4.12), we conclude that(4.10) is equal to σ ( ζ (2)) − · ω Π ∞ ( a ) − | a | − ∞ · σ ω Π S ( a t ) − · W ( λ ,λ ; w ) ,ψ ± U, ∞ ( g ∞ ) × W ψ ± aS (cid:18) ℓ (( a ( a ) , a ( a ))) t σ,S W S , ω ψ ± S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S (cid:19) (cid:0) diag( a − , a − , , a − ) g S (cid:1) × Y p/ ∈ S W ψ ± ap ( t σ,p W ◦ p, ± a , ϕ ◦ p ) (cid:0) diag( a − , a − , , a − ) g p (cid:1) (4.13)for g ∈ GSp ( A ). Note that for f ∈ Π , we have W f,ψ ± U ( g ) = W f,ψ ± aU (diag( a − , a − , , a − ) g ) . Since W σ f WS ,ψ ± a ⊠ ψ ± a = ℓ (( a ( a ) , a ( a ))) W ±∞ · ℓ (( a ( a ) , a ( a ))) t σ,S W S · Y p/ ∈ S t σ,p W ◦ p, ± a , by Lemma 4.1 for the additive character ψ ± a and (4.9) with b = a , we deduce that (4.13) is equal to theWhittaker function of the global theta lift on the right-hand side of (4.8) with respect to ψ ± U . This completesthe proof. (cid:3) Petersson norms of endoscopic lifts. Let h , i SO( V ) : ( τ ⊠ τ ) × ( τ ∨ ⊠ τ ∨ ) → C and h , i GSp : Π × Π ∨ → C be the Petersson bilinears pairing defined by h f, f ′ i SO( V ) = Z SO( V )( Q ) \ SO( V )( A ) f ( h ) f ′ ( h ) dh Tam1 , h f , f i GSp = Z A × GSp ( Q ) \ GSp ( A ) f ( g ) f ( g ) dg Tam . Here dh Tam1 and dg Tam are the Tamagawa measures on SO( V )( A ) and A × \ GSp ( A ), respectively. For eachplace v of Q , we fix a non-zero SO( V )( Q v )-equivariant bilinear pairing h , i v : ( τ ,v ⊠ τ ,v ) × ( τ ∨ ,v ⊠ τ ∨ ,v ) −→ C . We assume the pairings are chosen so that if f = N v f v ∈ τ ⊠ τ and f ′ = N v f ′ v ∈ τ ∨ ⊠ τ ∨ , then h f v , f ′ v i v = 1for almost all v and h f, f ′ i SO( V ) = Y v h f v , f ′ v i v . Let B v : S ( V ( Q v )) × S ( V ( Q v )) → C be the bilinear pairing defined by B v ( ϕ v , ϕ ′ v ) = Z V ( Q v ) ϕ v ( x v ) ϕ ′ v ( x v ) dx v , here dx v is defined by the Haar measure on Q v in § B v is equivariant under the Weilrepresentation ω ψ v ⊗ ω ψ − v for any non-trivial additive character ψ v of Q v . For f v ∈ τ ,v ⊠ τ ,v , f ′ v ∈ τ ∨ ,v ⊠ τ ∨ ,v ,and ϕ v , ϕ ′ v ∈ S ( V ( Q v )), we define the local zeta integral Z v ( f v , f ′ v , ϕ v , ϕ ′ v ) = ζ v (2) ζ v (4) L (1 , τ ,v × τ ,v ) · Z SO( V )( Q v ) B v ( h ,v · ϕ v , ϕ ′ v ) h ( τ ,v ⊠ τ ,v )( h ,v ) f v , f ′ v i v dh ,v . Here L ( s, τ ,v × τ ,v ) is the Rankin–Selberg L -function of τ ,v × τ ,v , dh ,v is the Haar measure on SO( V )( Q v )defined as in [CI19, § h ,v · ϕ v ( x ) = ϕ v ( h − ,v · x ). Note that the integral converges absolutely (cf. [GI11,Lemma 7.7]). We recall the Rallis inner product formula in the following theorem. Let L ( s, τ × τ ) = Y v L ( s, τ ,v × τ ,v )be the Rankin–Selberg L -function of τ × τ . Theorem 4.6 (Rallis inner product formula) . Let ψ be a non-trivial additive character of Q \ A . For f ∈ τ ⊠ τ , f ′ ∈ τ ∨ ⊠ τ ∨ , and ϕ, ϕ ′ ∈ S ( V ( A )) with f = N v f v , f ′ = N v f ′ v , ϕ = N v ϕ v , ϕ ′ = N v ϕ ′ v , we have h θ ψ ( f, ϕ ) , θ ψ − ( f ′ , ϕ ′ ) i GSp = 2 ζ (2) − · L (1 , τ × τ ) ζ (2) ζ (4) · Y v Z v ( f v , f ′ v , ϕ v , ϕ ′ v ) . Proof. This is a special case of the Rallis inner product formula proved in [GQT14, Theorem 11.3] (seealso [GI11, § ζ (2) − is the ratio between the Tamagawa measure and the product measure Q v dh ,v on SO( V )( A ). (cid:3) For each rational prime p and σ ∈ Aut( C ), we fix σ -linear isomorphisms T σ,p : τ ,p ⊠ τ ,p −→ σ τ ,p ⊠ σ τ ,p , T ∨ σ,p : τ ∨ ,p ⊠ τ ∨ ,p −→ σ τ ∨ ,p ⊠ σ τ ∨ ,p . Let h , i σ,p : ( σ τ ,p ⊠ σ τ ,p ) × ( σ τ ∨ ,p ⊠ σ τ ∨ ,p ) −→ C be the bilinear pairing defined by h T σ,p f p , T ∨ σ,p f ′ p i σ,p = σ h f p , f ′ p i p for f p ∈ τ ,p ⊠ τ ,p and f ′ p ∈ τ ∨ ,p ⊠ τ ∨ ,p . It is easy to verify that h , i σ,p is SO( V )( Q p )-equivariant (cf. Lemma5.9). Lemma 4.7. Let f p ∈ τ ,p ⊠ τ ,p , f ′ p ∈ τ ∨ ,p ⊠ τ ∨ ,p , and ϕ p , ϕ ′ p ∈ S ( V ( Q p )) .(1) If p ∤ cond( Π ) , f p and f ′ p are SO( V )( Z p ) -invariant, and ϕ p = ϕ ′ p = ϕ ◦ p , then Z p ( f p , f ′ p , ϕ p , ϕ ′ p ) = h f p , f ′ p i p . (2) We have σZ p ( f p , f ′ p , ϕ p , ϕ ′ p ) = Z p ( T σ,p f p , T ∨ σ,p f ′ p , σ ϕ p , σ ϕ ′ p ) for all σ ∈ Aut( C ) .Proof. The first assertion is a special case of [PSR87, Proposition 6.2] (see also [LR05, Proposition 3]). Let σ ∈ Aut( C ). An argument similarly to the proof of Lemma 3.3 shows that σL (1 , τ ,p × τ ,p ) = L (1 , σ τ ,p × σ τ ,p ) . Note that the integral defining Z p is actually a doubling local zeta integral (cf. [GI11, § σ Z SO( V )( Q p ) B p ( h ,p · ϕ p , ϕ ′ p ) h ( τ ,p ⊠ τ ,p )( h ,p ) f p , f ′ p i p dh ,p ! = Z SO( V )( Q p ) σ B p ( h ,p · ϕ p , ϕ ′ p ) σ h ( τ ,p ⊠ τ ,p )( h ,p ) f p , f ′ p i p dh ,p . Since ϕ p and ϕ ′ p have compact support, B p ( h ,p · ϕ p , ϕ ′ p ) is a finite sum involving ϕ p and ϕ ′ p . Thus we have σ B p ( h ,p · ϕ p , ϕ ′ p ) = B p ( h ,p · σ ϕ p , σ ϕ ′ p ) . Also note that σ h ( τ ,p ⊠ τ ,p )( h ,p ) f p , f ′ p i p = h ( σ τ ,p ⊠ σ τ ,p )( h ,p ) T σ,p f p , T ∨ σ,p f ′ p i σ,p by definition. This completes the proof. (cid:3) heorem 4.8. Theorem 1.1 holds for Π = θ ( τ ⊠ τ ) .Proof. For p ∤ cond( Π ), let f ◦ p ∈ τ ,p ⊠ τ ,p and ( f ◦ p ) ∨ ∈ τ ∨ ,p ⊠ τ ∨ ,p be the SO( V )( Z p )-invariant vectors definingthe restricted tensor products N v τ ,v ⊠ τ ,v and N v τ ∨ ,v ⊠ τ ∨ ,v , respectively. Let f τ ⊠ τ = O v f τ ⊠ τ ,v ∈ τ +1 ⊗ τ +2 , f τ ∨ ⊠ τ ∨ = O v f τ ∨ ⊠ τ ∨ ,v ∈ ( τ ∨ ) + ⊗ ( τ ∨ ) + be the normalized newforms of τ ⊠ τ and τ ∨ ⊠ τ ∨ , respectively. We assume f τ ,p ⊠ τ ,p = f ◦ p and f τ ∨ ,p ⊠ τ ∨ ,p =( f ◦ p ) ∨ for p ∤ cond( Π ). The Petersson norm k f τ ⊠ τ k of f τ ⊠ τ is defined by k f τ ⊠ τ k = h f τ ⊠ τ , ( τ ∨ , ∞ ⊠ τ ∨ , ∞ )([ a ( − , a ( − f τ ∨ ⊠ τ ∨ i SO( V ) . Let S be the set of prime divisors of cond( Π ). Fix f = f τ ⊠ τ , ∞ ⊗ f S ⊗ ( ⊗ p/ ∈ S f ◦ p ) , f ′ = ( τ ∨ , ∞ ⊠ τ ∨ , ∞ )([1 , a ( − f τ ∨ ⊠ τ ∨ , ∞ ⊗ f ′ S ⊗ ( ⊗ p/ ∈ S ( f ◦ p ) ∨ ) ,ϕ = ϕ + ∞ ⊗ ϕ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p ) , ϕ ′ = ϕ −∞ ⊗ ϕ ′ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p )for some f S ∈ τ ,S ⊠ τ ,S , f ′ S ∈ τ ∨ ,S ⊠ τ ∨ ,S , and ϕ S , ϕ ′ S ∈ S ( V ( Q S )) such that h θ ψ ( f, ϕ ) , Π ∨∞ (diag( − , − , , θ ψ − ( f ′ , ϕ ′ ) i GSp = 0 . (4.14)It is clear that f ∈ τ +1 ⊠ τ +2 and f ′ ∈ τ +1 ⊠ τ − . Note that we have the factorization of L -functions: L ( s, Π , Ad) = L ( s, τ × τ ) · L ( s, τ , Ad) · L ( s, τ , Ad) , where L ( s, τ i , Ad) is the adjoint L -function of τ i for i = 1 , 2. By Theorem 4.6 and Lemma 4.7-(1), we have ζ (2) · h θ ψ ( f, ϕ ) , Π ∨∞ (diag( − , − , , θ ψ − ( f ′ , ϕ ′ ) i GSp k f Π k = 2 · L (1 , Π , Ad) ζ (2) ζ (4) · k f Π k · ζ (2) · k f τ ⊠ τ k L (1 , τ , Ad) · L (1 , τ , Ad) · C ∞ · Z S ( f S , f ′ S , ϕ S , ϕ ′ S ) h f τ ⊠ τ ,S , f τ ∨ ⊠ τ ∨ ,S i S . (4.15)Here C ∞ = Z ∞ (cid:16) f τ ⊠ τ , ∞ , f τ ∨ ⊠ τ ∨ , ∞ , ϕ + ∞ , ω ψ − ∞ (diag( − , − , , , [1 , a ( − ϕ −∞ (cid:17) h f τ ⊠ τ , ∞ , ( τ ∨ , ∞ ⊠ τ ∨ , ∞ )([ a ( − , a ( − f τ ∨ ⊠ τ ∨ , ∞ i ∞ . Let σ ∈ Aut( C ). It is easy to see that σ (cid:18) h f , Π ∨∞ (diag( − , − , , f i GSp k f Π k (cid:19) = h σ f , σ Π ∨∞ (diag( − , − , , σ f i GSp k f σ Π k for all f ∈ Π mot and f ∈ Π ∨ mot . By the result of Sturm [Stu89], we have σ (cid:18) L (1 , τ , Ad) · L (1 , τ , Ad) ζ (2) · k f τ ⊠ τ k (cid:19) = L (1 , σ τ , Ad) · L (1 , σ τ , Ad) ζ (2) · k f σ τ ⊠ σ τ k . By Lemma 4.7, we have σ (cid:18) Z S ( f S , f ′ S , ϕ S , ϕ ′ S ) h f τ ⊠ τ ,S , f τ ∨ ⊠ τ ∨ ,S i S (cid:19) = Z S ( T σ,S f S , T ∨ σ,S f ′ S , σ ϕ S , σ ϕ ′ S ) h T σ,S f τ ⊠ τ ,S , T ∨ σ,S f τ ∨ ⊠ τ ∨ ,S i σ,S . By the Chinese remainder theorem, there exists a ∈ Q × > such that u σ,S = at for some t ∈ Q × S . Now weapply σ to both sides of (4.15). It then follows from Proposition 4.5 that ζ (2) · h θ ψ a ( σ f, ϕ σ,a ) , σ Π ∨∞ (diag( − , − , , θ ψ − a ( σ f ′ , ϕ ′ σ,a ) i GSp k f σ Π k = 2 · σ (cid:18) L (1 , Π , Ad) ζ (2) ζ (4) · k f Π k (cid:19) · ζ (2) · k f σ τ ⊠ σ τ k L (1 , σ τ , Ad) · L (1 , σ τ , Ad) · σC ∞ · Z S ( T σ,S f S , T ∨ σ,S f ′ S , σ ϕ S , σ ϕ ′ S ) h T σ,S f τ ⊠ τ ,S , T ∨ σ,S f τ ∨ ⊠ τ ∨ ,S i σ,S . (4.16)Here ϕ σ,a = ω ψ ∞ (cid:18)(cid:18) √ a √ a − (cid:19) , (cid:19) ϕ + ∞ ⊗ ω ψ S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p ) ,ϕ ′ σ,a = ω ψ − ∞ (cid:18)(cid:18) √ a √ a − (cid:19) , (cid:19) ϕ −∞ ⊗ ω ψ − S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ ′ S ⊗ ( ⊗ p/ ∈ S ϕ ◦ p ) . y the equivariance under the Weil representation, we have B ∞ (cid:18) h , ∞ · ω ψ ∞ (cid:18)(cid:18) √ a √ a − (cid:19) , (cid:19) ϕ + ∞ , ω ψ − ∞ (cid:18)(cid:18) −√ a √ a − (cid:19) , [1 , a ( − (cid:19) ϕ −∞ (cid:19) = B ∞ (cid:16) h , ∞ · ϕ + ∞ , ω ψ − ∞ (diag( − , − , , , [1 , a ( − ϕ −∞ (cid:17) , B S (cid:18) h ,S · ω ψ S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ S , ω ψ − S (cid:18)(cid:18) t − t (cid:19) , (cid:19) σ ϕ ′ S (cid:19) = B S ( h ,S · σ ϕ S , σ ϕ ′ S )(4.17)for all h , ∞ ∈ SO( V )( R ) and h ,S ∈ SO( V )( Q S ). We may assume the isomorphisms σ τ ⊠ σ τ ≃ N v σ τ ,v ⊠ σ τ ,v and σ τ ∨ ⊠ σ τ ∨ ≃ N v σ τ ∨ ,v ⊠ σ τ ∨ ,v are normalized so that σ f = f τ ⊠ τ , ∞ ⊗ T σ,S f S ⊗ ( ⊗ p/ ∈ S T σ,p f ◦ p ) , σ f ′ = ( τ ∨ , ∞ ⊠ τ ∨ , ∞ )([1 , a ( − f τ ∨ ⊠ τ ∨ , ∞ ⊗ T ∨ σ,S f ′ S ⊗ ( ⊗ p/ ∈ S T ∨ σ,p ( f ◦ p ) ∨ ) . Then, by Theorem 4.6 again together with (4.17), we see that the left-hand side of (4.16) is equal to2 · L (1 , σ Π , Ad) ζ (2) ζ (4) · k f σ Π k · ζ (2) · k f σ τ ⊠ σ τ k L (1 , σ τ , Ad) · L (1 , σ τ , Ad) · C ∞ · Z S ( T σ,S f S , T ∨ σ,S f ′ S , σ ϕ S , σ ϕ ′ S ) h T σ,S f τ ⊠ τ ,S , T ∨ σ,S f τ ∨ ⊠ τ ∨ ,S i σ,S . By our assumption (4.14), we have Z S ( f S , f ′ S , ϕ S , ϕ ′ S ) = 0. We thus conclude that σ (cid:18) L (1 , Π , Ad) ζ (2) ζ (4) · k f Π k · C ∞ (cid:19) = L (1 , σ Π , Ad) ζ (2) ζ (4) · k f σ Π k · C ∞ . Finally, it was proved in [CI19, Lemma 8.10] that C ∞ ∈ Q × . This completes the proof. (cid:3) Local Zeta Integrals In this section, we study the convergence and the Galois equivariant properties of the doubling local zetaintegrals and the Rankin-Selberg local zeta integrals defined in (2.2) and (2.6), respectively. The main resultsare Propositions 5.10 and 5.14.Let F be a non-archimedean local field of characteristic zero. Let o , ̟ , and q be the maximal compactsubring of F , a generator of the maximal ideal of o , and the cardinality of the residue field o /̟ o , respectively.Let | | be the absolute value on F normalized so that | ̟ | = q − . Fix a non-trivial additive character ψ of F .5.1. Some representation theory of GSp . In this section, we recall some results for GSp on the unita-rizability criterion of generic representations and the asymptotic behavior of Whittaker functions.Let Q and Q be the standard Siegel parabolic subgroup and the standard Klingen parabolic subgroupof GSp , respectively, defined by Q = ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∈ GSp ,Q = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∈ GSp . We denote by N Q and N Q the unipotent radical of Q and Q , respectively. The standatd Levi componentsof Q and Q are given by M Q = (cid:26) m ( A, ν ) = (cid:18) A ν t A − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ GL , ν ∈ GL (cid:27) ,M Q = m ( t, g ) = t a b νt − c d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ∈ GL g = (cid:18) a bc d (cid:19) ∈ GL ν = det( g ) ∈ GL . et M Q i = M Q i ∩ Sp for i = 1 , 2. For a character χ of F × and an irreducible admissible representation τ ofGL ( F ), let τ ⋊ χ be the normalized induced representation acting via the right translation ρ on the spaceconsisting of smooth functions f : GSp ( F ) → V τ such that f ( nm ( A, ν ) g ) = δ / Q ( m ( A, ν )) χ ( ν ) τ ( A ) f ( g )for all n ∈ N Q ( F ), m ( A, ν ) ∈ M Q ( F ), and g ∈ GSp ( F ). Here V τ is the representation space of τ and δ Q is the modulus character of Q ( F ) given by δ Q ( m ( A, ν )) = | det( A ) | | ν | − . Similarly, let χ ⋊ τ bethe normalized induced representation acting via the right translation ρ on the space consisting of smoothfunctions f : GSp ( F ) → V τ such that f ( nm ( t, g ′ ) g ) = δ / Q ( m ( t, g ′ )) χ ( t ) τ ( g ′ ) f ( g )for all n ∈ N Q ( F ), m ( t, g ′ ) ∈ M Q ( F ), and g ∈ GSp ( F ). Here δ Q is the modulus character of Q ( F ) givenby δ Q ( m ( t, g )) = | t | | det( g ) | − . Note that the central characters of τ ⋊ χ and χ ⋊ τ are equal to ω τ χ and ω τ χ , respectively. By the results of Sally and Tadi´c [ST93], any non-supercuspidal irreducible admissiblegeneric representation of GSp ( F ) is the generic subrepresentation of an induced representation in one of thefollowing types:(I) Ind GL ( F ) B ( F ) ( χ ⊠ χ ) ⋊ χ for some characters χ , χ , χ of F × such that χ = | | ± , χ = | | ± , and χ = | | ± χ ± ;(IIa) (St ⊗ µ ) ⋊ χ for some characters µ, χ of F × such that µ = | | ± / and µ = | | ± ;(IIIa) χ ⋊ (St ⊗ µ ) for some characters µ, χ of F × such that χ = and χ = | | ± ;(IVa) | | ⋊ (St ⊗ µ ) for some character µ of F × ;(Va) (St ⊗ µ ) ⋊ χ for some characters µ, χ of F × such that µ = | | and µ = | | / ;(VIa) ⋊ (St ⊗ µ ) for some character µ of F × ;(VII) χ ⋊ τ for some character χ of F × and irreducible supercuspidal representation τ of GL ( F ) such that χ = and either χ = | | ± or χ = | | ± or τ ⊗ χ | | ± = τ ;(VIIIa) ⋊ τ for some irreducible supercuspidal representation τ of GL ( F );(IXa) χ ⋊ τ for some character χ of F × and irreducible supercuspidal representation τ of GL ( F ) such that χ = | | , χ = | | , and τ ⊗ χ | | − = τ ;(X) τ ⋊ χ for some character χ of F × and irreducible supercuspidal representation τ of GL ( F ) such that ω τ = | | ± ;(XIa) τ ⋊ χ for some character χ of F × and irreducible supercuspidal representation τ of GL ( F ) such that ω τ = | | .Here St denotes the Steinberg representation of GL ( F ) and we follow [RS07] for the labelling of types. Lemma 5.1. Let Π be a non-supercuspidal irreducible admissible generic representation of GSp ( F ) . AssumeΠ is unitary. If Π is of one of the types (IIIa), (VIa), and (VIIIa), then the inducing data are unitary. Inthe remaining cases, the following conditions are satisfied: (I) | e ( χ ) | + | e ( χ ) | < and χ χ χ is unitary; (IIa) | e ( µ ) | < and µχ is unitary; (IVa) | | µ is unitary; (Va) | | / χ is unitary; (VII) | e ( χ ) | < and ω τ χ is unitary; (IXa) | | ω τ is unitary; (X) | e ( ω τ ) | < and ω τ χ is unitary; (XIa) | | / χ is unitary.Moreover, if Π is of one of the types (IVa), (Va), (IXa), and (XIa), then Π is a discrete series representation.Proof. The conditions for unitarizability were proved in [ST93, Theorem 4.4, Propositions 4.7 and 4.9]. Theassertion for discrete series representations was proved in [ST93, Theorem 4.1, Propositions 4.6 and 4.8]. (cid:3) Lemma 5.2. Let Π be an irreducible admissible generic representation of GSp ( F ) . There exist a finite set X Π of characters of T ( F ) and a positive integer N Π such that for any Whittaker function W of Π , we have W ( tk ) = δ B ( t ) / X ≤ n ≤ N Π X ≤ n ≤ N Π X η ∈ X Π η ( t )(log q | a | ) n (log q | b | ) n ϕ n ,n ,η ( a, b, k ) or some locally constant function ϕ n ,n ,η on F × F × GSp ( o ) with compact support for ≤ n , n ≤ N Π and η ∈ X Π . Here t = diag( ab, a, b − , ∈ T ( F ) . Moreover, the set X Π is given as follows: for η ∈ X Π , η (diag( ab, a, b − , η ( a ) η ( b ) with η = | | e ( ω Π ) / , η = if Π is supercuspidal and (I) ( η , η ) ∈ { ( χ, χ − ) , ( χ, χ − ) , ( χχ , χ ) , ( χχ , χ − ) , ( χχ , χ − ) , ( χχ , χ ) , ( χχ χ , χ ) , ( χχ χ , χ ) } ; (IIa) ( η , η ) ∈ { ( χ, | | / µ − ) , ( µ χ, | | / µ ) , ( | | / µχ, | | / µ ± ) } ; (IIIa) ( η , η ) ∈ { ( | | / µ, χ − ) , ( | | / µ, | | ) , ( | | / µχ, χ ) , ( | | / µχ, | | ) } ; (IVa) η = | | / µ , η = | | ; (Va) η ∈ {| | χ, | | / µχ } , η = | | / µ ; (VIa) η = | | / µ , η ∈ { , | |} ; (VII) η = | | e ( ω Π ) / , η = χ ± ; (VIIIa) η = | | e ( ω Π ) / , η = ; (IXa) η = | | e ( ω Π ) / , η = χ ; (X) η ∈ { χ, ω τ χ } , η = ; (XIa) η = ω τ χ , η = .Proof. The assertion follows from the result of Lapid and Mao [LM09, Theorem 3.1] for the special case G = GSp (see also [Jia96, p. 155, Proposition 1.1.1]). The formula for X Π is then a consequence of theexplicit formula for the semisimplification of the normalized Jacquet module of Π with respect to the parabolicsubgroup of GSp determined by the cuspidal support of Π (cf. [RS07, Tables A.3 and A.4]). For types (I)-(VIa), we consider the Borel subgroup B . For types (VII), (VIIIa), and (IXa) (resp. (X) and (XIa)), weconsider the Klingen parabolic subgroup Q (resp. Siegel parabolic subgroup Q ). (cid:3) Remark 5.3. If Π is either supercuspidal or of one of the types (VII), (VIIIa), and (IXa), then ϕ n ,n ,η = 0when | a | is sufficiently small. Thus in this case, η can be any character. We take η = | | e ( ω Π ) / so that theestimation in the proof of Proposition 5.14 is more uniform.The following lemma is on the asymptotic behavior of matrix coefficients for GL and will be used in theproof of Proposition 5.10. Lemma 5.4. Let τ be an irreducible admissible generic representation of GL ( F ) and φ a matrix coefficientof τ . There exist characters η , η of F × and locally constant functions Φ , Φ on F × GL ( o ) × GL ( o ) suchthat φ ( k a ( t ) k ) = η ( t ) | t | / Φ ( t, k , k ) + η ( t ) | t | / Φ ( t, k , k ) for ( t, k , k ) ∈ ( o r { } ) × GL ( o ) × GL ( o ) . Moreover, if τ = Ind GL ( F ) B ( F ) ( χ ⊠ χ ) for some characters χ , χ ,then η , η ∈ { χ , χ } . If τ = St ⊗ µ for some character µ , then η = η = µ | | / . If τ is supercuspidal, then η = η = | | e ( ω τ ) / .Proof. This is well-known. Indeed, the assertion follows from the explicit computation of the normalizedJacquet module of τ with respect to B (cf. [GH11, § (cid:3) Remark 5.5. If τ is supercuspidal, then Φ = Φ = 0 when | t | sufficiently small. Thus in this case, η and η can be any characters. We take η = η = | | e ( ω τ ) / so that the estimation in the proof of Proposition 5.10is more uniform.5.2. Intertwining operators. Let B be the standard Borel subgroup of Sp defined by B = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 00 0 0 0 ∗ ∗ ∗ ∗ ∈ Sp . enote by N the unipotent of B and T ⊂ B the maximal torus of Sp consisting of diagonal matrices.Let W = N Sp ( T ) /T be the Weyl group of T in Sp . For i = 1 , , , 4, let ǫ i : T → GL be the algebraiccharacter defined by ǫ i (diag( t , t , t , t , t − , t − , t − , t − )) = t i . The set of positive roots for (Sp , T ) is given by { ǫ i ± ǫ j , ǫ k | ≤ i < j ≤ , k = 1 , , , } . For each positive root ǫ , we normalize the associated embedding ι ǫ : SL → Sp as follows: ι ǫ i + ǫ j ( n ( x )) = (cid:18) x ( E i,j + E j,i )0 (cid:19) , ι ǫ i − ǫ j ( n ( x )) = (cid:18) + xE i,j − xE j,i (cid:19) ,ι ǫ i ( n ( x )) = (cid:18) xE i,i (cid:19) . (5.1)Here E i,j ∈ M , is the matrix with ( i, j )-entry equal to 1 and zero otherwise. Let N ǫ ⊂ Sp be the image ofthe unipotent radial of B under ι ǫ and identify N ǫ with G a via ι ǫ . For a character χ of T ( F ), let I ( χ ) bethe normalized induced representation acting via the right translation ρ on the space consisting of smoothfunctions f : Sp ( F ) → C such that f ( ntg ) = δ / B ( t ) χ ( t ) f ( g )for n ∈ N ( F ), t ∈ T ( F ), and g ∈ Sp ( F ). The modulus character δ B of B ( F ) is given by δ B (diag( t , · · · , t , t − , · · · , t − )) = | t | | t | | t | | t | . For w ∈ W , we define the intertwining operator M w : I ( χ ) −→ I ( χ w ) ,M w f ( g ) = Z N w ( F ) f ( wng ) dn. (5.2)Here χ w ( t ) = χ ( wtw − ) and N w = Y ǫ> wǫ< N ǫ . The Haar measure dn is normalized so that vol( N w ( o ) , dn ) = 1. The integral is absolutely convergent if χ belongs to some open subset and can be meromorphically continued to all χ . If we write w = w · · · w ℓ intoa reduced decomposition, then we have M w = M w ℓ ◦ · · · ◦ M w . (5.3)The following lemma is on the analytic and Galois equivariant properties of the intertwining integralsin the simplest case. For characters χ , χ of F × , let ind GL ( F ) B ( F ) ( χ ⊠ χ ) be the (non-normalized) inducedrepresentation on the space consisting of smooth functions f : GL ( F ) → C such that f ( n ( x ) a ( t ) d ( t ) g ) = χ ( t ) χ ( t ) f ( g )for x ∈ F , t , t ∈ F × , and g ∈ GL ( F ). Lemma 5.6. Let χ , χ be characters of F × and f ∈ ind GL ( F ) B ( F ) ( χ ⊠ χ ) . The intertwining integral Z F f ( wn ( x )) dx is absolutely convergent if e ( χ χ − ) > . We have σ (cid:18)Z F f ( wn ( x )) dx (cid:19) = Z F σ ( f ( wn ( x ))) dx for all σ ∈ Aut( C ) when both sides are absolutely convergent. roof. Indeed, we have Z F f ( wn ( x )) dx = Z | x |≤ q N f ( wn ( x )) dx + χ ( − χ ( − f (1) Z | x | >q N χ − χ ( x ) dx. Here N is sufficiently large so that f (cid:18)(cid:18) x (cid:19)(cid:19) = f (1)for all | x | < q − N . The first integral is a finite sum and the second integral converges for e ( χ χ − ) > 1. For σ ∈ Aut( C ), we have σ Z | x |≤ q N f ( wn ( x )) dx ! = Z | x |≤ q N σ ( f ( wn ( x ))) dx and σ Z | x | >q N χ − χ ( x ) dx ! = Z | x | >q N σ χ − σ χ ( x ) dx for e ( χ χ − ) > e ( σ χ σ χ − ) > 1. The assertions then follow at once. (cid:3) Let χ be a character of T ( F ). For σ ∈ Aut( C ), we have the σ -linear isomorphism I ( χ ) −→ I ( σ χ ) , f σ f with σ f ( g ) = σ ( f ( g )) for g ∈ Sp ( F ). Corollary 5.7. Let χ be a character of T ( F ) and w ∈ W . Then we have σ ( M w f ( g )) = M wσ f ( g ) for all σ ∈ Aut( C ) , f ∈ I ( χ ) , and g ∈ Sp ( F ) when both sides are absolutely convergent.Proof. The assertion is an immediate consequence of (5.3) and Lemma 5.6. (cid:3) Doubling local zeta integrals. Let H = GL and H = GL × Sp . We define embeddings ι : H −→ Sp , g (cid:18) g t g − (cid:19) ,ι : H −→ Sp , (cid:18)(cid:18) a bc d (cid:19) , (cid:18) A BC D (cid:19)(cid:19) a b a a b b c d a a b b a ′ b ′ c c d d c ′ d ′ c c d d . Here (cid:18) a ′ b ′ c ′ d ′ (cid:19) = (cid:18) a cb d (cid:19) − , A = ( a ij ), B = ( b ij ), C = ( c ij ), and D = ( d ij ). Recall we have identifySp × Sp as a subgroup of Sp vie the embedding (2.1). The image of M Q i × M Q i ⊂ Sp × Sp in Sp factorsthrough the embedding ι i , thus induces an embedding from M Q i × M Q i into H i . We identify M Q i × M Q i asa subgroup of H i in this way. Let R i be the maximal parabolic subgroup of H i defined by R = ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∈ H ,R = (cid:18) ∗ ∗ ∗ (cid:19) , ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∈ H . ut ξ i = − ! ∈ H ( F ) if i = 1 , − ! , − 12 1212 12 − ∈ H ( F ) if i = 2 . Note that we have ξ i ( m, m ) ξ − i ∈ R i ( F ) for all m ∈ M Q i ( F ).Let F ∈ I ( s ) be a holomorphic section, where I ( s ) is the degenerate principal series representation definedin § i = 1 , 2, we define the intertwining integralΨ i ( g, s ; F ) = Z N Qi ( F ) F ( δ ( n, ι i ( ξ − i g ) , s ) dn, where g ∈ H i ( F ). By [LR05, Proposition 1], the integral Ψ i ( g, s ; F ) converges absolutely for Re( s ) sufficientlylarge and admits meromorphic continuation to s ∈ C . Moreover, Ψ i ( ; F ) ∈ Ind H i ( F ) R i ( F ) ( µ i,s ), where µ i,s is thecharacter of the standard Levi component of R i ( F ) given by µ ,s (cid:18)(cid:18) A D (cid:19)(cid:19) = | det( A ) | s +3 / | det( D ) | − s +3 / ,µ ,s (cid:18)(cid:18) a d (cid:19) , (cid:18) A t A − (cid:19)(cid:19) = | a | s +2 | d | − s +2 | det( A ) | s . In the following lemma, we explicitly determine the region of convergence and prove the Galois equivariantproperty of the intertwining integrals. Lemma 5.8. Let F ∈ I ( s ) be a holomorphic section. For i = 1 , , the integral Ψ i ( g, s ; F ) converges absolutelyfor Re( s ) > − and satisfies the Galois equivariant property σ Ψ i ( g, n ; F ) = Ψ i ( g, n ; σ F ) for all σ ∈ Aut( C ) , g ∈ H i ( F ) , and odd positive integers n .Proof. Let w , w ∈ W be Weyl elements defined by w = ι ǫ ( w ) ι ǫ − ǫ ( w ) ι ǫ ( w ) , w = ι ǫ ( w ) ι ǫ − ǫ ( w ) ι ǫ − ǫ ( w ) . Here ι ǫ is the embedding defined in (5.1) for each positive root ǫ . A direct calculation shows that δ ( N Q ( F ) , δ − ∈ P ( F ) · ι − w N w ( F ) w − ι ,δ ( N Q ( F ) , δ − ∈ P ( F ) · ι − w N w ( F ) w − ι . Note that F ∈ I ( s ) ⊂ I ( χ s ), where χ s is the character of T ( F ) defined by χ s (diag( t , · · · , t , t − , · · · , t − )) = | t | s − / | t | s − / | t | s +1 / | t | s +3 / . herefore, we have Ψ ( g, s ; F ) = M w F w − ι δι ( ξ − g ) , Ψ ( g, s ; F ) = M w F w − ι δι ( ξ − g ) for g ∈ H i ( F ). Here M w i : I ( χ s ) → I ( χ w i s ) is the intertwining operator defined in (5.2) for i = 1 , 2. Theabsolute convergence for Re( s ) > − then follows immediately from Lemma 5.6. Indeed, we have χ ι ǫ ( w ) s (diag( t , · · · , t , t − , · · · , t − )) = | t | s − / | t | s − / | t | s +1 / | t | − s − / ,χ ι ǫ ( w ) ι ǫ − ǫ ( w ) s (diag( t , · · · , t , t − , · · · , t − )) = | t | s − / | t | s − / | t | − s − / | t | s +1 / . Hence M w F = M ι ǫ ( w ) ◦ M ι ǫ − ǫ ( w ) ◦ M ι ǫ ( w ) F and M w F = M ι ǫ − ǫ ( w ) ◦ M ι ǫ − ǫ ( w ) ◦ M ι ǫ ( w ) F areabsolutely convergent for Re( s ) > max {− , − , − } = − . For σ ∈ Aut( C ), we have σ χ n/ = χ n/ and σ F | s = n/ ∈ I ( n ) ⊂ I ( χ n/ ) for all odd integers n . The Galoisequivariant property for Ψ i then follows from Corollary 5.7. This completes the proof. (cid:3) Let Π be an irreducible admissible representation of GSp ( F ). Lemma 5.9. Let φ be a matrix coefficient of Π . Then σ φ is a matrix coefficient of σ Π for all σ ∈ Aut( C ) .Proof. Let σ ∈ Aut( C ). Let V Π and V Π ∨ be the representation spaces of Π and Π ∨ , respectively, and fix σ -linear isomorphisms t : V Π → V Π and t ∨ : V Π ∨ → V Π ∨ . The representations σ Π and σ Π ∨ are realized on V Π and V Π ∨ , respectively, with actions defined in (1.8). Let h , i be a non-zero equivariant bilinear pairingfor Π × Π ∨ realized on V Π × V Π ∨ . Define a bilinear pairing h , i ′ on V Π × V Π ∨ by h v, v ∨ i ′ = σ h t − v, ( t ∨ ) − v ∨ i for v ∈ V Π and v ∨ ∈ V Π ∨ . It is easy to verify that h , i ′ defines an equivariant pairing for σ Π × σ Π ∨ . Assume φ ( g ) = h Π ( g ) v, v ∨ i for some v, v ∨ . Then σ φ ( g ) = h tv, tv ∨ i ′ is a matrix coefficient of σ Π . This completes theproof. (cid:3) Proposition 5.10. Let φ be a matrix coefficient of Π and F ∈ I ( s ) be a holomorphic section. The localzeta integral Z ( s, φ, F ) converges absolutely for Re( s ) sufficiently large, and satisfies the Galois equivariantproperty σZ ( n , φ, F ) = Z ( n , σ φ, σ F ) for all σ ∈ Aut( C ) and sufficiently large odd integers n . Assume that Π is essentially unitary and generic,then Z ( s, φ, F ) converges absolutely for Re( s ) ≥ .Proof. If Π is supercuspidal, then Z ( s, φ, F ) = n X i =1 F ( δ ( g i , , s ) φ ( g i )for some g , · · · , g n depending only on the support of φ and on a sufficiently small open compact subgroupof GSp ( F ) which stabilizes F . The assertions then follow at once.Suppose that Π is a subrepresentation of an induced representation of the form τ ⋊ χ or χ ⋊ τ for someirreducible admissible generic representation τ of GL ( F ) and some character χ of F × . Let η and η be thecharacters of F × depending on τ described in Lemma 5.4. Let Q = Q (resp, Q = Q ) in if Π ⊂ τ ⋊ χ (resp. Π ⊂ χ ⋊ τ ). We writeΨ = Ψ i , ι = ι i , ξ = ξ i , µ s = µ i,s , H = H i , R = R i f Q = Q i . Fix a non-zero equivariant bilinear pairing h , i on τ × τ ∨ . We may assume that φ ( g ) = Z Sp ( o ) h f ( kg ) , f ∨ ( k ) i dk for some f and f ∨ . For Re( s ) > − , we have Z ( s, φ, F ) = Z Sp ( F ) F ( δ ( g, , s ) Z Sp ( o ) h f ( kg ) , f ∨ ( k ) i dkdg = Z Sp ( o ) Z Sp ( F ) F ( δ ( g, k ) , s ) h f ( g ) , f ∨ ( k ) i dgdk = Z Sp ( o ) dk Z M Q ( F ) dm Z N Q ( F ) dn δ Q ( m ) − F ( δ ( nmk , k ) , s ) h f ( mk ) , f ∨ ( k ) i = Z Sp ( o ) dk Z M Q ( F ) dm δ Q ( m ) − Ψ( ξ ( m, , s ; ρ (( k , k )) F ) h f ( mk ) , f ∨ ( k ) i . Here we use (2.3) in the second line and Lemma 5.8 in the fourth line. Also note that we have identify M Q × M Q as a subgroup of H via ι . By the Cartan decomposition, for all ϕ ∈ L ( M Q ( F )), we have Z M Q ( F ) ϕ ( m ) dm = Z F × d × t Z | t |≤ d × t Z GL ( o ) dk ♯ (GL ( o ) a ( t ) GL ( o ) / GL ( o )) ϕ ( k t a ( t ) k )(5.4)if Q = Q ; and Z M Q ( F ) ϕ ( m ) dm = Z F × d × t Z | t |≤ d × t Z SL ( o ) dk ♯ (SL ( o ) m ( t ) SL ( o ) / SL ( o )) ϕ ( t , k m ( t ) k )(5.5)if Q = Q . Here we identify GL and GL × SL with M Q and M Q via the embeddings A (cid:18) A t A − (cid:19) , (cid:18) t, (cid:18) a bc d (cid:19)(cid:19) t a b t − c d . Note that for any Ψ ∈ Ind H ( F ) R ( F ) ( µ s ), we haveΨ( ξ ( t a ( t ) , g, s ) = | t t | s +5 / Ψ t t − t − g, s = | t t | − s +1 / Ψ − t − t − 00 1 0 − t − g, s = | t | | t | s +5 / Ψ t t − − t − g, s (5.6) or t , t ∈ F × , g ∈ H ( F ) if Q = Q ; andΨ( ξ (( t , m ( t )) , g, s ) = | t | s +5 / | t | s +3 / Ψ (cid:18) t − (cid:19) , t − t g, s = | t | − s +3 / | t | s +3 / Ψ (cid:18) − t − (cid:19) , t − t g, s (5.7)for t , t ∈ F × , g ∈ H ( F ) if Q = Q . Assume Q = Q . By Lemma 5.4, there exist locally constant functions Φ , Φ on F × GL ( o ) × GL ( o ) × Sp ( o ) × Sp ( o ) such that h f (( k a ( t ) k , k ′ ) , f ∨ ( k ′ ) i = η ( t ) | t | Φ ( t, k , k , k ′ , k ′ ) + η ( t ) | t | Φ ( t, k , k , k ′ , k ′ )for ( t, k , k , k ′ , k ′ ) ∈ ( o r { } ) × GL ( o ) × GL ( o ) × Sp ( o ) × Sp ( o ). Combining with (5.4) and (5.6), forRe( s ) > − , we see that the integral Z ( s, φ, F ) is a finite sum of integrals of the following forms: I ( τ, η, ϕ , ϕ , ϕ , s ) = Z ( F × ) | t | s +2 ω τ ( t ) | t | s +1 / η ( t ) ϕ ( t ) ϕ ( t ) ϕ ( t t ) d ( t , t ) ,I ( τ, η, ϕ , ϕ , ϕ , s ) = Z ( F × ) | t | s +2 ω τ ( t ) − | t | − s − / η ( t ) ϕ ( t ) ϕ ( t ) ϕ ( t t − ) d ( t , t ) ,I ( τ, η, ϕ , ϕ , ϕ , s ) = Z ( F × ) ω τ ( t ) − | t | s +1 / η ( t ) ϕ ( t ) ϕ ( t ) ϕ ( t − t ) d ( t , t ) , where η ∈ { η , η } and ϕ i is a locally constant function on F with compact support for i = 1 , , 3. Theintegrals I , I , I converge absolutely forRe( s ) > max {− | e ( ω τ ) | , − − e ( η ) , − + e ( ω τ ) − e ( η ) } . (5.8)For an odd positive integer n such that s = n belongs to the above region of convergence, it is easy to verifythat σI ( τ, η, ϕ , ϕ , n ) = I ( σ τ, σ η, σ ϕ , σ ϕ , n ) ,σI ( τ, η, ϕ , ϕ , ϕ , n ) = I ( σ τ, σ η, σ ϕ , σ ϕ , σ ϕ , n ) ,σI ( τ, η, ϕ , ϕ , ϕ , n ) = I ( σ τ, σ η, σ ϕ , σ ϕ , σ ϕ , n )(5.9)for all σ ∈ Aut( C ). Assume Q = Q . By Lemma 5.4, there exist locally constant functions Φ ′ , Φ ′ on F × SL ( o ) × SL ( o ) × Sp ( o ) × Sp ( o ) such that h f (((1 , k m ( t ) k ) , k ′ ) , f ∨ ( k ′ ) i = ω − τ η ( t ) | t | Φ ′ ( t, k , k , k ′ , k ′ ) + ω − τ η ( t ) | t | Φ ′ ( t, k , k , k ′ , k ′ )for ( t, k , k , k ′ , k ′ ) ∈ ( o r { } ) × SL ( o ) × SL ( o ) × Sp ( o ) × Sp ( o ). Combining with (5.5) and (5.7), forRe( s ) > − , we see that the integral Z ( s, φ, F ) is a finite sum of integrals of the following forms: I ( χ, η, ϕ , ϕ , s ) = Z ( F × ) | t | s +1 / χ ( t ) | t | s +1 / η ( t ) ϕ ( t ) ϕ ( t ) d ( t , t ) ,I ( χ, η, ϕ , ϕ , s ) = Z ( F × ) | t | s +1 / χ ( t ) − | t | s +1 / η ( t ) ϕ ( t ) ϕ ( t ) d ( t , t ) , where η ∈ { ω − τ η , ω − τ η } and ϕ i is a locally constant function on F with compact support for i = 1 , 2. Theintegrals I , I converge absolutely forRe( s ) > max {− + | e ( χ ) | , − + e ( ω τ ) − e ( η ) , − + e ( ω τ ) − e ( η ) } . (5.10)For an odd positive integer n such that s = n belongs to the above region of convergence, it is easy to verifythat σI ( χ, η, ϕ , ϕ , n ) = I ( σ χ, σ η, σ ϕ , σ ϕ , n ) ,σI ( χ, η, ϕ , ϕ , n ) = I ( σ χ, σ η, σ ϕ , σ ϕ , n )(5.11) or all σ ∈ Aut( C ). Let σ ∈ Aut( C ) and n an odd positive integer such that s = n belongs to the region ofconvergence. We conclude from (5.9) and (5.11) that σZ ( n , φ, F ) = Z Sp ( o ) dk Z M Q ( F ) dm δ Q ( m ) − σ Ψ( ξ ( m, , n ; ρ (( k , k )) F ) σ h f ( mk ) , f ∨ ( k ) i . Note that σ Ψ( ξ ( m, , n ; ρ (( k , k )) F ) = Ψ( ξ ( m, , n ; ρ (( k , k )) σ F )for m ∈ M Q , ( k , k ) ∈ Sp ( o ) by Lemma 5.8 and σ φ ( g ) = Z Sp ( o ) σ h f ( kg ) , f ∨ ( k ) i . Therefore, we have σZ ( n , φ, F ) = Z ( n , σ φ, σ F ) . Assume Π is non-supercuspidal, essentially unitary, and generic. Note that( τ ⋊ χ ) ⊗ | | t = τ ⋊ χ | | t , ( χ ⋊ τ ) ⊗ | | t = χ ⋊ ( τ ⊗ | | t )for t ∈ C . By Lemma 5.1, the inequalities (5.8) and (5.10) are satisfied when Re( s ) ≥ except when Π is either of type (IVa) or (IXa) or (XIa). Indeed, if Π is of one of the types (I), (II), (Va), and (X), then Q = Q and (5.8) is satisfied for Re( s ) ≥ . If Π is of one of the types (IIIa), (VIa), (VII), and (VIIIa),then Q = Q and (5.10) is satisfied for Re( s ) ≥ . For the remaining types (IVa), (IXa), and (XIa), Π is adiscrete series representation. Hence the integral Z ( s, φ, F ) converges absolutely for Re( s ) ≥ − by [GI14,Lemma 9.5]. This completes the proof. (cid:3) Local zeta integrals for GSp × GSp . We write n − ( w, y, x, u, v ) = ι ǫ + ǫ (cid:18) w (cid:19) ι ǫ (cid:18) y (cid:19) ι ǫ + ǫ (cid:18) x (cid:19) ι ǫ + ǫ (cid:18) u (cid:19) ι ǫ + ǫ (cid:18) v (cid:19) and n − ( w, y, x ) = n − ( w, y, x, , I ( s ) be the degenerate principal series representation defined in § α, β ∈ F and F ∈ I ( s ),define the twisted intertwining integral I ( α, β ; F , ψ ) = Z F F ( n − ( w, y, x, , ψ ( − αx + βy ) dw dy dx. Lemma 5.11. Let u, v ∈ F .(1) Suppose u ∈ F × . We have I ( α, β ; ρ ( n − (0 , , , u, F , ψ ) = | u | − s − I (cid:18) uα, β ; ρ (cid:18) ι ǫ + ǫ (cid:18) − u − (cid:19)(cid:19) F , ψ (cid:19) . (2) Suppose v ∈ F × . We have I ( α, β ; ρ ( n − (0 , , , , v )) F , ψ ) = | v | − s − I (cid:18) vα, v β ; ρ (cid:18) ι ǫ + ǫ (cid:18) − v − (cid:19)(cid:19) F , ψ (cid:19) . (3) Suppose u, v ∈ F × . We have I ( α, β ; ρ ( n − (0 , , , u, v )) F , ψ ) = | u | − s − | v | − s − I (cid:18) uvα, v β ; ρ (cid:18) ι ǫ + ǫ (cid:18) − u − (cid:19) ι ǫ + ǫ (cid:18) − v − (cid:19)(cid:19) F , ψ (cid:19) . Proof. Note that (cid:18) x (cid:19) = m ( x − ) n ( x ) (cid:18) − x − (cid:19) , (5.12) n − ( w, y, x ) ι ǫ + ǫ ( n ( u )) n − ( w, y, x ) − ∈ P ( F ) ∩ ker( δ P ) , (5.13) n − ( w, y, x ) ι ǫ + ǫ ( n ( v )) n − ( w − vxy, y, x ) − ∈ P ( F ) ∩ ker( δ P ) , . (5.14)One can easily verify that the first (resp. second) assertion follows from (5.12) and (5.13) (resp. (5.14)). Thethird assertion is a direct consequence of (1) and (2). We leave the detail to the readers. (cid:3) emma 5.12. Let F ∈ I ( s ) be a holomorphic section. The integral I ( α, β ; F , ψ ) converges absolutely for Re( s ) > − and satisfies the Galois equivariant property σ ( I ( α, β ; F , ψ ) | s = n ) = I ( α, β ; σ F , σ ψ ) | s = n for all σ ∈ Aut( C ) and integers n ≥ . Moreover, for Re( s ) > − , the integral I ( α, β ; ρ ( n − (0 , , , u, v )) F , ψ ) as a function in ( α, β, u, v ) ∈ ( F × ) is a finite sum of functions of the form ϕ , ( α ) ϕ , ( β ) ϕ , ( u ) ϕ , ( v ) + ϕ , ( uα ) ϕ , ( β ) ϕ , ( u − ) ϕ , ( v )+ ϕ , ( vα ) ϕ , ( v β ) ϕ , ( u ) ϕ , ( v − ) + ϕ , ( uvα ) ϕ , ( v β ) ϕ , ( u − ) ϕ , ( v − ) , where ϕ i,j is a locally constant function on F × so that ϕ i,j ( x ) = 0 for | x | sufficiently large and there thereexist c i,j ∈ C and character χ i,j of F × such that ϕ i,j ( x ) = c i,j · χ i,j ( x ) for | x | sufficiently small.Proof. We rewrite the integral I ( α, β ; F , ψ ) into 8 terms according to whether | x | , | y | , and | w | are sufficientlylarge or not. Note that n − ( w, y, ι ǫ + ǫ ( n ( x )) n − ( − w, − y, ∈ P ( F ) ∩ ker( δ P ) ,n − ( w, , ι ǫ ( n ( y )) n − ( − w, , ∈ P ( F ) ∩ ker( δ P ) . It follows that for all sufficiently larger integers N , N , and N depending only on F , we have I ( α, β ; F , ψ )= Z | x |≤ q N dx Z | y |≤ q N dy Z | w |≤ q N dw F ( n − ( w, y, x )) ψ ( − αx + βy )+ Z | x | >q N | x | − s − ψ ( − αx ) dx Z | y |≤ q N dy Z | w |≤ q N dw F (cid:0) n − ( w, y, ι ǫ + ǫ ( w ) (cid:1) ψ ( βy )+ Z | y | >q N | y | − s − ψ ( βy ) dy Z | x |≤ q N dx Z | w |≤ q N dw F (cid:0) n − ( w, , x ) ι ǫ ( w ) (cid:1) ψ ( − αx )+ Z | x |≤ q N dx Z | y |≤ q N dy Z | w | >q N dw | w | − s − F (cid:18) ι ǫ + ǫ (cid:18) − w − (cid:19) n − (0 , y, x ) (cid:19) ψ ( − αx + βy )+ Z | x | >q N | x | − s − ψ ( − αx ) dx × Z | y |≤ q N dy Z | w | >q N dw | w | − s − F (cid:18) ι ǫ + ǫ (cid:18) − w − (cid:19) ι ǫ (cid:18) y (cid:19) ι ǫ + ǫ ( w ) (cid:19) ψ ( βy )+ Z | y | >q N | y | − s − ψ ( βy ) dy × Z | x |≤ q N dx Z | w | >q N dw | w | − s − F (cid:18) ι ǫ + ǫ (cid:18) − w − (cid:19) ι ǫ + ǫ (cid:18) x (cid:19) ι ǫ ( w ) (cid:19) ψ ( − αx )+ Z | x | >q N | x | − s − ψ ( − αx ) dx Z | y | >q N | y | − s − ψ ( βy ) dy × Z | w |≤ q N dw F (cid:18) ι ǫ + ǫ (cid:18) w (cid:19) ι ǫ ( w ) ι ǫ + ǫ ( w ) (cid:19) + Z | x | >q N | x | − s − ψ ( − αx ) dx Z | y | >q N | y | − s − ψ ( βy ) dy Z | w | >q N | w | − s − dw × F ( ι ǫ + ǫ ( w ) ι ǫ ( w ) ι ǫ + ǫ ( w )) . We see that I ( α, β ; F , ψ ) is absolutely convergent for Re( s ) > − 1. Note that Z | x |≤ q N ψ ( ax ) dx = q N · I ̟ N + d o ( a ) , Z | x |≥ q N | x | − s − ψ ( ax ) dx = h − q − − q − s − · ( q − N ( s +1) − q ( d − s +1) | a | s +1 ) − q ( d − s +1) − | a | s +1 i · I ̟ N + d − o ( a ) , here ̟ d o is the largest fractional ideal of F on which ψ is trivial. Let Re( s ) > − 1. Combining with Lemma5.11, we deduce that each term of I ( α, β ; ρ ( n − (0 , , , u, v )) F , ψ ) as a function in ( α, β, u, v ) ∈ ( F × ) is equalto a finite sum of functions satisfying the conditions in the lemma. The Galois equivariant property then alsofollows at once. Indeed, for an integer n , we have σ ( | | n ) = | | n and σ F ( g, n ) = σ ( F ( g, n )) for all g ∈ Sp ( F )by definition. This completes the proof. (cid:3) Lemma 5.13. Let ϕ , · · · , ϕ be locally constant functions on F × so that ϕ i ( x ) = 0 for | x | sufficiently largeand there there exist c i ∈ C , character χ i of F × , and integer m i ≥ such that ϕ i ( x ) = c i · χ i ( x )(log q | x | ) m i for | x | sufficiently small. Let I ( ϕ , · · · , ϕ ) be the integral defined by I ( ϕ , · · · , ϕ ) = Z ( F × ) ϕ ( a ) ϕ ( b ) ϕ ( av − ) ϕ ( bu − v ) ϕ ( bu − ) ϕ ( u ) ϕ ( v ) d ( a, b, u, v ) . Then we have σI ( ϕ , · · · , ϕ ) = I ( σ ϕ , · · · , σ ϕ ) for all σ ∈ Aut( C ) when both sides are absolutely convergent. Similar assertion holds for the followingintegrlas: I ( ϕ , · · · , ϕ ) = Z ( F × ) ϕ ( a ) ϕ ( b ) ϕ ( av − ) ϕ ( buv ) ϕ ( u ) ϕ ( v ) d ( a, b, u, v ) ,I ( ϕ , · · · , ϕ ) = Z ( F × ) ϕ ( a ) ϕ ( b ) ϕ ( av ) ϕ ( bu − v − ) ϕ ( u ) ϕ ( v ) d ( a, b, u, v ) ,I ( ϕ , · · · , ϕ ) = Z ( F × ) ϕ ( a ) ϕ ( b ) ϕ ( av ) ϕ ( buv − ) ϕ ( bv − ) ϕ ( u ) ϕ ( v ) d ( a, b, u, v ) . Proof. We recall a type of local integral of the form Z F × χ ( x )(log q | x | ) m · ϕ ( x ) d × x, (5.15)where ϕ is a locally constant function on F with compact support, χ is a character of F × , and m ≥ e ( χ ) > 0. In this case, it is easy to verify that the integralsatisfies the Galois equivariant property σ (cid:18)Z F × χ ( x )(log q | x | ) m · ϕ ( x ) d × x (cid:19) = Z F × σ χ ( x )(log q | x | ) m · σ ϕ ( x ) d × x for all σ ∈ Aut( C ) (cf. [Gro18, Proposition A]) when both sides are absolutely convergent.We only consider the integral I ( ϕ , · · · , ϕ ). The assertion for other three integrals I , I , I can be provedin a similar way and we omit it. First we consider the case when ϕ and ϕ vanish unless | u | and | v | aresufficiently small. Then the integral I ( ϕ , · · · , ϕ ) is a finite sum of integrals of the form Z ( F × ) ϕ ′ ( a ) ϕ ′ ( av − ) ϕ ′ ( v ) d ( a, v ) · Z ( F × ) ϕ ′ ( b ) ϕ ′ ( bu − ) ϕ ′ ( u ) d ( b, u ) , where ϕ ′ i satisfies the same conditions of the lemma and ϕ ′ , ϕ ′ vanish unless | u | and | v | are sufficiently small.Firstly we consider the integral Z ( F × ) ϕ ′ ( a ) ϕ ′ ( av − ) ϕ ′ ( v ) d ( a, v ) . (5.16)When the support of ϕ ′ is contained in a bounded set away from zero, the above integral is a finite sum ofintegrals of the form (5.15). Similar assertion holds when the support of ϕ ′ is contained in a bounded setaway from zero. Therefore, we may assume ϕ ′ ( a ) = χ ( a )(log q | a | ) n I ̟ N o ( a ) , ϕ ′ ( v ) = µ ( v )(log q | v | ) m I ̟ N o ( v )for some characters χ, µ of F × and integers n, m, N , N ≥ 0. Furthermore, by the condition on ϕ ′ , we mayassume that either ϕ ′ = I c + ̟ N o (5.17) or some c ∈ F and integer N with c / ∈ ̟ N o or ϕ ′ ( x ) = ν ( x )(log q | x | ) r I ̟ N o ( x )(5.18)for some character ν of F × and integers r, N ≥ 0. Write I ̟ N o = I ̟ N o + ( I ̟ N o − I ̟ N o )for some sufficiently large N such that c ∈ ̟ − N + N o (resp. ̟ N ∈ ̟ − N + N o ) if ϕ ′ is a function of theform (5.17) (resp. (5.18)). Then the integral (5.16) over v ∈ ̟ N o r ̟ N o is a finite sum of integrals of theform (5.15). Suppose that ϕ ′ is a function of the form (5.17). Then the integral (5.16) over v ∈ ̟ N o isequal to Z | a |≤ q − N Z | v |≤ q − N χ ( a )(log q | a | ) n µ ( v )(log q | v | ) m I c + ̟ N o ( av − ) d ( a, v )= Z | a − c |≤ q − N Z ̟ N o χ ( av ) µ ( v )(log q | v | ) m (log q | cv | ) n d ( a, v ) . Now we assume that ϕ ′ is a function of the form (5.18). Then the integral (5.16) over v ∈ ̟ N o is equal to Z | a |≤ q − N Z | v |≤ q − N χ ( a )(log q | a | ) n µ ( v )(log q | v | ) m I ̟ N o ( av − ) d ( a, v )= Z | a |≤ q − N | v | Z | v |≤ q − N χ ( a )(log q | a | ) n µ ( v )(log q | v | ) m d ( a, v )= Z | a |≤ q − N Z | v |≤ q − N χ ( av )(log q | av | ) n µ ( v )(log q | v | ) m d ( a, v ) . In any case, we conclude that the integral (5.16) satisfies the Galois equivariant property, since it is a finitesum of products of integrals of the form (5.15). By a similar argument, the integral Z ( F × ) ϕ ′ ( b ) ϕ ′ ( bu − ) ϕ ′ ( u ) d ( b, u )also satisfies the Galois equivariant property.Now we consider the remaining cases. If ϕ vanishes when | u | is sufficiently small, then the integral I ( ϕ , · · · , ϕ ) is a finite sum of integrals of the form Z ( F × ) ϕ ′ ( a ) ϕ ′ ( b ) ϕ ′ ( av − ) ϕ ′ ( bv ) ϕ ′ ( v ) d ( a, b, v ) , (5.19)where ϕ ′ i satisfies the same conditions of the lemma. If ϕ vanishes when | v | is sufficiently small, then theintegral I ( ϕ , · · · , ϕ ) is a finite sum of integrals of the form Z ( F × ) ϕ ′ ( a ) ϕ ′ ( b ) ϕ ′ ( bu − ) ϕ ′ ( u ) d ( a, b, u ) , (5.20)where ϕ ′ i satisfies the same conditions of the lemma. Proceeding similarly as in the previous paragraph, onecan prove that the integrals of types (5.19) and (5.20) also satisfy the Galois equivariant property. We leavethe detail to the readers. This completes the proof. (cid:3) Let Π be an irreducible admissible generic representation of GSp ( F ). Proposition 5.14. Let W ∈ W ( Π , ψ U ) , W ∈ W ( Π ∨ , ψ − U ) , and F ∈ I ( s ) be a holomorphic section. Thelocal zeta integral Z ( s, W , W , F ) converges absolutely for Re( s ) sufficiently large, and satisfies the Galoisequivariant property σ Z ( n, W , W , F ) = Z ( n, σ W , σ W , σ F ) for all σ ∈ Aut( C ) and sufficiently large odd integers n . Assume Π is essentially unitary, then Z ( s, W , W , F ) converges absolutely for Re( s ) ≥ .Proof. By Lemma 5.2, we may assume W ( utk ) = ψ U ( u ) δ B ( t ) / η ( t )(log q | a | ) n (log q | b | ) n ϕ ( a, b, k ) ,W ( utk ) = ψ − U ( u ) δ B ( t ) / η ′ ( t )(log q | a | ) n ′ (log q | b | ) n ′ ϕ ( a, b, k ) or some η ∈ X Π and η ′ ∈ X Π ∨ , some integers 0 ≤ n , n , n ′ , n ′ ≤ N Π , and locally constant functions ϕ , ϕ on F × F × GSp ( o ) with compact support. Here u ∈ U ( F ), t = diag( ab, a, b − , ∈ T ( F ), and k ∈ GSp ( o ).Write η (diag( ab, a, b − , η ( a ) η ( b ) , η ′ (diag( ab, a, b − , η ′ ( a ) η ′ ( b ) . In the notation as in the proof of [CI19, Lemma 9.5], we have k a = e ( η ) − e ( η ′ ) − e ( ω Π ) , k b = e ( η ) , k c = 2 e ( η ′ ) + e ( ω Π ) , k d = e ( η ′ ) . Following the same argument as in the proof of that lemma, we see that the integral Z ( s, W , W , F ) convergesabsolutely forRe( s ) > max {− e ( η ) + e ( ω Π ) − , − e ( η ) − , − e ( η ′ ) − e ( ω Π ) − , − e ( η ′ ) − , − e ( η ) − e ( η ′ ) + e ( ω Π ) − , − e ( η ) − e ( η ′ ) − , − e ( η ) − e ( η ′ ) − , − e ( η ′ ) + e ( η ′ ) − e ( ω Π ) − } . When Π is essentially unitary, by Lemmas 5.1 and 5.2, one can verify case by case that the above inequalityholds for Re( s ) ≥ Z ( s, W , W , F ) satisfies the Galois equivariant property. Write W i ( tk ) = δ B ( t ) / Φ i ( a, b, k )for i = 1 , t = diag( ab, a, b − , ∈ T ( F ), and k ∈ GSp ( o ). Let(GSp ( o ) × GSp ( o )) ◦ = { ( k , k ) ∈ GSp ( o ) × GSp ( o ) | ν ( k ) = ν ( k ) } . We define ( B × B ) ◦ and ( T × T ) ◦ in a similar way. We have Z ( s, W , W , F ) = Z Z H ( F ) ˜ U ( F ) \ G ( F ) F ( ηg, s )( W ⊗ W )( g ) dg = Z (GSp ( o ) × GSp ( o )) ◦ Z Z H ( F ) \ ( T × T ) ◦ ( F ) δ ( B × B ) ◦ ( t ) − ( W ⊗ W )( tk ) × Z U ′ ( F ) \ U ( F ) F ( η ( u, tk ) ψ U ( u ) du dt dk. Note that (cid:8)(cid:0) diag( ab, a, b − , , diag( cd, c, c − d − a, c − a ) (cid:1) | a, b, c, d ∈ F × (cid:9) is a set of representatives for Z H ( F ) \ ( T × T ) ◦ ( F ). Let t = diag( ab, a, b − , , t = diag( cd, c, c − d − a, c − a ) . For u = x ∗ w ∗ y − x ∈ U ( F ) , a direct calculation gives that η ( u, t , t ) η − = p ( u, t , t ) n − ( − ac − d − w, − ac − y, ac − d − x, abc − d − − , ac − − p ( u, t , t ) ∈ P ( F ) with δ P ( p ( u, t , t )) = | a / bc − | . Therefore, Z U ′ ( F ) \ U ( F ) F ( η ( u, tk ) ψ U ( u ) du = δ ( B × B ) ◦ (( t , t )) / | a / bc − | s +1 × Z F F ( n − ( w, y, x, abc − d − − , ac − − ηk ) ψ ( − a − cdx + a − c y ) dw dy dx = δ ( B × B ) ◦ (( t , t )) / | a / bc − | s +1 · I ( a − cd, a − c ; ρ ( n − (0 , , , abc − d − − , ac − − ηk ) F , ψ ) . e conclude that Z ( s, W , W , F ) = Z (GSp ( o ) × GSp ( o )) ◦ dk Z ( F × ) d ( a, b, c, d ) | a / bc − | s +1 ω Π ( a − c ) Φ ( a, b, k ) Φ ( a − c , d, k ) × I ( a − cd, a − c ; ρ ( n − (0 , , , abc − d − − , ac − − ηk ) F , ψ )= Z (GSp ( o ) × GSp ( o )) ◦ dk Z ( F × ) d ( a, b, u, v ) | a / bv | s +1 Φ ( a, b, k ) ω Π ( v ) − Φ ( av − , bu − v, k ) × I ( bu − , av − ; ρ ( n − (0 , , , u − , v − ηk ) F , ψ ) . Here d ( a, b, u, v ) is the Haar measure on ( F × ) with vol(( o × ) , d ( a, b, u, v )) = 1. By Lemma 5.12, for k =( k , k ) ∈ (GSp ( o ) × GSp ( o )) ◦ and Re( s ) sufficiently large, the integral Z ( F × ) d ( a, b, u, v ) | a / bv | s +1 ω Π ( v ) − Φ ( a, b, k ) Φ ( av − , bu − v, k ) × I ( bu − , av − ; ρ ( n − (0 , , , u − , v − ηk ) F , ψ )is a finite sum of integrals of the forms I , I , I , I in Lemma 5.13. Let σ ∈ Aut( C ) and n an odd integer sothat the above integrals are all absolutely converge, we have σ ( | a / bv | n +1 ) = | a / bv | n +1 and σ (cid:0) I ( α, β ; ρ ( n − (0 , , , u − , v − ηk ) F , ψ ) | s = n (cid:1) = I ( α, β ; ρ ( n − (0 , , , u − , v − ηk ) σ F , σ ψ ) | s = n by Lemma 5.12. It then follows from the Galois equivariant property proved in Lemma 5.13 that σ Z ( n, W , W , F )= Z (GSp ( o ) × GSp ( o )) ◦ dk Z ( F × ) d ( a, b, u, v ) | a / bv | n +1 ω σ Π ( v ) − σ Φ ( a, b, k ) σ Φ ( av − , bu − v, k ) × I ( bu − , av − ; ρ ( n − (0 , , , u − , v − ηk ) σ F , σ ψ ) . 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