Automorphic-twisted summation formulae for pairs of quadratic spaces
aa r X i v : . [ m a t h . N T ] F e b AUTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OFQUADRATIC SPACES
MIAO (PAM) GU
Abstract.
We prove a summation formula for pairs of quadratic spaces following theconjectures of Braverman-Kazhdan, Lafforgue, Ngˆo and Sakellaridis. We also compute thelocal factors where all the data are unramified. Introduction
Conjectures of Braverman-Kazhdan [BK00], Lafforgue [Laf14], Ngˆo [Ngˆo14] and Sakel-laridis [Sak12] suggest generalized Poisson summation formula exists for affine sphericalvarieties. In [GL19], such a summation formula is proved where the underlying scheme isbuilt out of a triple of quadratic spaces.Generalizing the result in [GL19], in the present paper we prove a summation formula forpairs of quadratic spaces twisted by a Whittaker function on GL n . We hope the summationformula provided in this paper (together with new insights) will be useful in studying theanalytic properties for higher-rank triple product L-functions.Let us make the summation formula precise. Let F be a number field and A F be its Adelering. Let d , d be two even positive integers, and let V = G d a , V = G d a be a pair of affinespaces over F equipped with non-degenerate quadratic forms Q , Q . Let V := V ⊕ V .Fix β ∈ F × and let Y ⊂ V be the closed subscheme whose points in an F -algebra R are Y ( R ) := { ( y , y ) ∈ V ( R ) : Q ( y ) = 2 β Q ( y ) } . (1.0.1)Below we will use R to denote a “test” F -algebra, sometimes without further comment.Let V ′ ⊂ V be the open subscheme of points ( v , v ) such that no v i = 0, and let Y ′ := Y ∩ V ′ . We let P Y ′ ⊂ P V be the corresponding quasi-projective scheme. This is the schemeattached to the pair of quadratic spaces mentioned above.Now let us discuss how to build the Whittaker twist mentioned above. Let τ be a cuspidalautomorphic representation of GL n ( A F ). Let H be the split orthogonal group SO n +1 . Let G ( R ) := { g = ( g , g ) ∈ GL ( R ) : det g = det g − } and let ξ s be smooth holomorphic section from the spaceInd H ( A ) Q n ( A ) ( τ ⊗ | det | s − ) . Mathematics Subject Classification.
Primary 11F70, Secondary 11F66.
Here Q n ≤ H is a parabolic subgroup with Levi GL n . In Section 3 following [Kap12] and[Sou93] we construct a family of functions H ( A F ) × C −→ C ( h, s ) W s,ψ Q ( h, . Here ψ Q is a character defined using Q , Q , and the W is an indication that this is aWhittaker function on GL n ( A F ) for τ when restricted to an appropriate Levi subgroup ofSO n +1 .We extend the Weil representation of SL ( A F ) on S ( V ( A F )) = S ( V ( A F ) × V ( A F )) to arepresentation ρ of G ( A F ) on S ( V ( A F ) × V ( A F ) × A × F ) in Section 3 via a standard procedure.define the global integral I ( f, ξ, s )( y,
1) = Z U ( A F ) \ G ( A F ) ρ ( g ) f ( y, Z N ◦ ( A F ) W s,ψ Q ( ωuι ( g ) , ψ β ( u ) dudg (1.0.2)where U ⊂ G (1.0.3)is a maximal unipotent subgroup, f ( y, ∈ S ( V ( A F ) × A × F )) is a Schwartz-Bruhat functionsuch that ρ ( g ) f ( v, u ) = 0 for all g ∈ G ( F ) and v ∈ V ′′ ( F ) , u ∈ F × where V ′′ ( R ) = { ( v , v ) ∈ V ( R ) : Q i ( v i ) = 0 } ,N ◦ is a unipotent subgroup of H (see Eq. (2.1.3)), and ι is a embedding map from G to H (see Eq. (2.2.2)).The integral I ( f, ξ, s ) mixes the arithmetic of the quadratic forms Q and Q and thecuspidal automorphic representation τ . The integral I ( f, ξ, s )( y,
1) is obviously Eulerianfor each y , and we compute the unramified local factors in Section 4. In particular, thecomputation is essentially evaluating the Bessel coefficients at points depending on y ∈ P Y ( F ).Now let us state our summation formula. Theorem 1.1.
The sum P y ∈ P Y ′ ( F ) I ( f, ξ, s )( y, , admits a meromorphic continuiation tothe whole s -plane which satisfies a functional equation X y ∈ P Y ′ ( F ) I ( f, ξ, s )( y,
1) = X y ∈ P Y ′ ( F ) I ( f, M ( τ, s ) ξ, − s )( y, . where M ( τ, s ) is the global intertwining operator. The proof of the main theorem amounts to substituting a product of two theta functions,viewed as an automorphic forms on G ( A F ), into the Rankin-Selberg integral for SO ℓ × GL n constructed in [Kap12] where we take ℓ = 2. Of course, it takes substantial work to prove thatthe resulting integral is convergent and compute its local factors (see Section 4 to Section 7). UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 3
The Rankin-Selberg integral in [Kap12] represents a Langlands-Shahidi L -function, andit is illuminating to consider our procedure from the point of view of the Langlands-Shahidimethod. It is well-known that Langlands-Shahidi L -functions may be roughly enumerated byroot systems together with a simple root. The Dynkin diagram that remains after deletingthe simple root is the Dynkin diagram of a Levi subgroup. Our construction corresponds tothe Dynkin diagram D n +4 with and the unique simpe root such that the complement of theroot is the Dynkin diagram for SL × SL × SL n .We end the introduction by describing briefly about the outline of this paper. We set upthe notation for the various algebraic groups involved in Section 2. In Section 3, we firstestablish the global integral in our case and then we prove our main theorem. In Section 4,we give the computation of the local integral when all the data are unramified. We justifyin Section 6 the absolute convergence of various integrals and the final result in Section 4.The main theorem is made rigorous by showing the absolute convergence of the sum ofthe global integrals in Section 5, Section 7 and Section 8. We give a list of symbols at theend. Acknowledgements
I would like to thank my advisor Jayce Getz for suggesting this problem and for providingrelentless support and valuable advice. I am also grateful to Eyal Kaplan for answering twoquestions related to his thesis and Spencer Leslie for helpful discussions and comments. Ialso want to thank Huajie Li, Aaron Pollack and Jiandi Zou for helpful comments.2.
Preliminaries
Groups.
For this section we let F be a field of characteristic zero. To define points of F -schemes we let R denote an F -algebra. All algebraic groups we define below are affinealgebraic groups over F .Let J k = ∈ GL k ( F )for k a positive integer. Let SO k be the special orthogonal group with respect to J k .We say that a parabolic subgroup of SO k is standard if it contains the Borel subgroup ofupper triangular matrices. Let G ( R ) := { ( g , g ) ∈ GL ( R ) : det g = det g − } and H := SO n +1 . (2.1.1)Let G ′ := SO . (2.1.2) MIAO (PAM) GU
We denote T G ′ and T H as the corresponding maximal split torus consisting of diagonalmatrices.For δ = ( δ , . . . , δ n ) ∈ Z n , we denote the element of T H as ̟ δH = diag( ̟ δ , . . . , ̟ δ n , , ̟ − δ , . . . , ̟ − δ n ) ∈ T H . Subgroups of H . Let Q H be the standard parabolic subgroup with Levi subgroupwhose points in an F -algebra R are M H ( R ) := n ( x, c ) ∧ = (cid:16) x c x ∗ (cid:17) ∈ H : ( x, c ) ∈ GL n − ( R ) × SO ( R ) , x ∗ = J n − ( t x − ) J n − o . Let N H be the unipotent subgroup whose points in an F -algebra R are N H ( R ) := z x yI x ′ z ∗ : x ∈ M ( n − × ( R ) , y ∈ M n − ( R ) , z ∈ Z H ( R ) , where z ∗ = J n − ( t z − ) J n − , x ′ = − J ( t x ) J n − z ∗ , and Z H is the unipotent radical of theBorel subgroup of upper triangular matrices of GL n − .We let Y H be the subgroup of N H whose points in an F -algebra R are Y H ( R ) = z x I x ′ I z ∗ : z ∈ Z H ( R ) , z ∗ = J n − ( t z − ) J n − , x ′ = − J ( t x ) J n − z ∗ , and we denote N ◦ the subgroup of N H whose points in an F -algebra R are N ◦ ( R ) = I n − x y zI y ′ I x ′ I n − : x ′ = − J ( t x ) J n − , y = x ′ = − J ( t y ) J n − (2.1.3)such that N ◦ is isomorphic to Y H \ N H .For x ∈ GL n ( R ) let v ( x ) := (cid:16) x J n ( t x − ) J n (cid:17) . (2.1.4)Let Q n be the standard parabolic subgroup with Levi subgroup M n whose points in an F -algebra R are M n ( R ) := { v ( x ) : x ∈ GL n ( R ) } . (2.1.5) UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 5
Let N n be the unipotent subgroup whose points in an F -algebra R are N n ( R ) := z x y x ′ z ∗ : z ∈ Z n , z ∗ = J n ( t z − ) J n , x ′ = − J ( t x ) J n z ∗ (2.1.6)where Z n is the unipotent radical of the Borel subgroup of upper triangular matrices of GL n .Accordingly, we denote Q n ⊂ H as the opposite parabolic subgroup with Levi subgroup M n , and we let N n be the corresponding unipotent radical of Q n .Let ω = β I I n − ( − n − I n − ( β ) − I (2.1.7)be a Weyl group element in H .Let Q G ′ be a subgroup of H whose points in an F -algebra R are Q G ′ ( R ) = a b c − β b d − β b ′ c ′ b ′ a − : a ∈ R × , c ∈ R, b ′ = − ba − , c ′ = − ca − , d = − β b ′ + c ′ a − . Embedding of the groups.
For the construction of the global integral, we use twoembeddings of groups. Here we give the explicit maps we use in our integral.We have a sporadic isogeny between the algebraic groups SL × SL / ± I and G ′ = SO . Itinduces a surjection G → G ′ given on points in an F -algebra by ι : G ( R ) −→ G ′ ( R ) a bc d ! , a ′ b ′ c ′ d ′ !! aa ′ − ab ′ ba ′ bb ′ − ac ′ ad ′ − bc ′ − bd ′ ca ′ − cb ′ da ′ db ′ cc ′ − cd ′ dc ′ dd ′ (2.2.1)In the construction in [Kap12, Section 2.1], the embedding of G ′ in H is given by diag( I n − , G ′′ , I n − ),where we denote G ′′ ⊂ SO as the image of the embedding of G ′ in SO ⊂ H . The map ι : G → SO is a a b b a a b b c c d d c c d d MIAO (PAM) GU a a − β b β a + βb − β a + b b a − β c a − β b + − β c + d β a − βb − β c − β d − β a − b + β − β c − β d b − β d βa + β c βa − β b + β c − βd a + β b + β c + d − β a − βb − β c + β d βb + β d − β a + c − β a − β b + β + c − β d − βa − β b + β c + βd a − β b + − β c + d − β b + d c c − β ds β c + βd − β c + d d . We define the composite map ι := ι ◦ ι : G −→ H. (2.2.2)2.3. Image of the embeddings.
Using the embedding of G in G ′ = SO and G ′ in SO (which naturally embedds in H ), we make the image of subgroups of G in SO precise.Let M be the subgroup of the maximal torus of G whose points in an F -algebra R are M ( R ) := (cid:8)(cid:0)(cid:0) m − (cid:1) , ( m
00 1 ) (cid:1) : m ∈ R × (cid:9) . (2.3.1) Lemma 2.1.
Let M ′ = ι ( M ) ⊂ SO . Then M ′ ( R ) = n(cid:16) m I m − (cid:17) : m ∈ R × o . (2.3.2) (cid:3) Let G be a subgroup of the maximal torus of SL × SL whose points in an F -algebra R are G ( R ) := (cid:8)(cid:0)(cid:0) b − (cid:1) , ( b ) (cid:1) : b ∈ R × (cid:9) . (2.3.3)We have a character λ : G ( R ) −→ R × (cid:0)(cid:0) b − (cid:1) , ( b ) (cid:1) b (2.3.4) Lemma 2.2.
Let G ′ = ι ( G ) < SO . Then G ′ ( R ) = + ( b + b − ) β ( b − b − ) β ( b − b − ) β ( b − b − ) ( b + b − ) − β ( b − b − ) β ( − ( b + b − )) − β ( b − b − ) + ( b + b − ) 1 : b ∈ R × . (cid:3) Let A be a subgroup of T G whose points in F -algebra R are A ( R ) := ( a a − ! , !! : a ∈ R × ) . (2.3.5) Lemma 2.3.
Let A ′ = ι ( A ) . Then A ′ ( R ) = a + ( a + a − ) β ( a − a − ) β ( a − a − ) β ( a − a − ) ( a + a − ) − β ( a − a − ) β ( − ( a + a − )) − β ( a − a − ) + ( a + a − ) a − : a ∈ R × . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 7 (cid:3)
Let A be the subgroup of T G whose points in F -algebra R are A ( R ) := ( ! , a a − !! : a ∈ R × ) . (2.3.6) Lemma 2.4.
Let A ′ = ι ( A ) . Then A ′ ( R ) = a + ( a + a − ) β ( − a + a − ) β ( − a + a − ) β ( − a + a − ) ( a + a − ) − β ( a − a − ) β ( − ( a + a − )) − β ( − a + a − ) + ( a + a − ) a − : a ∈ R × . (cid:3) We have T G = A A G .Let U be the maximal unipotent radical of the Borel subgroup of upper triangular matricesof G . Let N be a subgroup of the U whose points in an F -algebra R are N ( R ) := ( c β ! , − βc !! : c ∈ R ) . (2.3.7) Lemma 2.5.
Let N ′ = ι ( N ) < SO . Then N ′ ( R ) = c − c − c : c ∈ R . (cid:3) Let N be a subgroup of U whose points in an F -algebra R are N ( R ) := ( β b ! , b !! : b ∈ R ) . (2.3.8) Lemma 2.6.
Let N ′ = ι ( N ) < SO N ′ ( R ) = b − β b
01 0 0 β b − b : b ∈ R . (cid:3) Let M SL be a subgroup of SL × SL whose points in an F -algebra R are M SL ( R ) := ( m m − ! , m m − !! : m ∈ R × ) . (2.3.9) MIAO (PAM) GU
Lemma 2.7.
Let Q G = M SL N N . Then ι ( Q G )( R ) is a b c − β b d − β b ′ c ′ b ′ a − : a ∈ R × , c ∈ R, b ′ = − ba − , c ′ = − ca − , d = − β b ′ + c ′ a − . Summary.
We have given a Levi decomposition A A G N N = A G M N N of the Borel of upper triangular matrices in G ( R ) := { ( g , g ) ∈ GL ( R ) : det g = det g − } .We let G ′ := SO . Moreover, we have a commutative diagram A A G N N GG ′ := SO SO ι ιι Our main theorems are stated without the use of G ′ , but we require it in the proofs.2.5. Notations for local fields.
Let F be a global field and v a place of F . We denoteby O the ring of integers of F and O v the ring of integers of F v for nonarchimedean v . Wedenote by ̟ v a uniformizer for O v and q v := |O v /̟ v | the residual characteristic. The idelicnorm is denoted by | · | and the local norm on F v (normalized in the usual manner) is denotedby | · | v .2.6. Norms.
For γ ∈ P V ( F ), we let γ = ( γ : · · · : γ d : γ : · · · : γ d ) ∈ P V ( F ) = P ( V ⊕ V )( F ) . For F a non-Archimedean local field, we let γ be such that γ i , . . . , γ id i are integral, | γ i | = max( | γ i | , ..., | γ id | ) , i = 1 , , and | γ | = max( | γ | , | γ | ) = 1 . For the Archimedean local field we let γ be such that | γ i | , . . . , | γ id i | > | γ i | = min( | γ i | , ..., | γ id | ) , i = 1 , , and | γ | = min( | γ | , | γ | ) = 1 . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 9
Measures.
We let dx = ⊗ v dx v where dx v is normalized such that dx v ( O ) = 1. We let d × x v := ζ v (1) dx v | x | v . The Global Integral
In this section we use the Rankin-Selberg integral for SO ℓ × GL n developed in [Kap12]to deduce the expression of our global integral where we take ℓ = 2. We state and prove ourmain theorem of this paper in Theorem 3.2 assuming the absolute convergence statement.The main theorem will be made rigorous by showing the absolute convergence of the sumof the global integrals in Section 7.Let F be a number field. We first briefly recall the construction of the Rankin-Selberg in-tegral in [Kap12, Section 3]. Let τ be an irreducible automorphic representation for GL n ( A ).Let ξ s be a smooth holomorphic section from the (normalized induction) spaceInd H ( A F ) Q n ( A F ) ( τ ⊗ | det | s − ) . We have the Eisenstein series E ( h, ξ, s ) := X y ∈ Q n ( F ) \ H ( F ) ξ s ( yh, ξ is on H , and the second variable of ξ s is on Q n .Let ψ be a non-trivial additive character of F \ A . For u ∈ N H ( A ), let ψ β ( u ) = ψ ( n − X i =1 u i,i +1 + u n − ,n + 12 β u n − ,n +2 )be a character of N H ( A ), trivial on N H ( F ).Then the ψ β -coefficient of E ξ with respect to N H ( A ) is E ψ β ξ s ( h ) = Z N H ( F ) \ N H ( A ) E ( uh, ξ, s ) ψ β ( u ) du. The Rankin-Selberg integral in this case is I ( ϕ, ξ, s ) = Z G ′ ( F ) \ G ′ ( A ) ϕ ( g ) E ψ β ξ s ( g ) dg where ϕ is a cusp form on G ( A ).This global integral converges absolutely in the whole s -plane except at the poles of theEisenstein series E ψ β ξ s ( h ), and the absolute convergence follows from the rapid decay of thecusp form ϕ and the moderate growth of the Eisenstein series E ψ β ξ s ( h ). We construct the Weil representation ρ for G ( A F ) following [YZZ13], which extends theusual Weil representation of SL ( A F ) as follows: ρ ( g ) f ( γ, u ) = ρ u ( g ) f ( γ, u ) , g ∈ SL ( A ) ρ ((( a ) , (cid:0) a − (cid:1) )) f ( γ, u ) = f ( γ, a − u ) | a | − dim V , a ∈ A × F where ρ u is the usual Weil representation on SL ( A ) and f ∈ S ( V ( A F ) × A × F ).Let f ∈ S ( V ( A F ) × A × F ) be a Schwartz-Bruhat function such that ρ ( g ) f ( γ, u ) = 0 for all g ∈ G ( A F ) and ( γ, u ) ∈ V ′′ ( F ) × A × F (3.0.1)where V ′′ ( R ) = { ( γ , γ ) ∈ V ( R ) : Q ( γ ) = Q ( γ ) = 0 } .We let Θ f ( g ) = X ( γ,u ) ∈ V ( F ) × F × ρ ( g ) f ( γ, u )be the theta function on G ( A F ).Using the formula ofr I ( ϕ, ξ, s ), we define a global integral as I (Θ f , ξ, s ) = Z G ( F ) \ G ( A ) Θ f ( g ) E ψ β ξ s ( g ) dg. Since we take ρ ( g ) f (0) = 0 for all g ∈ G ( A F ), θ f ( g ) is cuspidal. Thus I (Θ f , ξ, s ) convergesabsolutely in the whole s -plane except at the poles of the Eisenstein series E ψ β ξ s ( h ) similar asthe Rankin-Selberg integral I ( ϕ, ξ, s ).By the action of Weil representation, we have I (Θ f , ξ, s ) = Z SL ( F ) \ G ( A ) X γ ∈ V ( F ) ρ ( g ) f ( γ, E ψ β ξ s ( g ) dg. We first unfold the Eisenstein series E ψ β ξ s for ℜ ( s ) large. Lemma 3.1.
For ℜ ( s ) large, we have E ψ β ξ s ( g ) = X y ∈ Q G ( F ) \ SL ( F ) Z Y H ( F ) \ N H ( A ) ξ ( ω ι ( y ) uι ( g ) , ψ β ( u ) du. (3.0.2) Proof.
By the embedding map ι from G to G ′ (see Eq. (2.2.1)), we have a long exact sequence1 → {± I } → SL ( F ) ι −→ G ′ ( F ) sn −→ H ( F, ± I ) ∼ = F × / ( F × ) → ι ( G ′ ( F )) = [ ǫ ∈ F × / ( F × ) I n − ǫ I ǫ − I n − ! ι (SL )( F ) Q G ′ ( F ) = [ ǫ ∈ F × / ( F × ) I n − ǫ I ǫ − I n − ! ι ( Q G )( F ) . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 11
Then Q G ( F ) \ SL ( F ) ∼ = ι ( Q G )( F ) \ ι (SL )( F ) ∼ = Q G ′ ( F ) \ ι ( G ′ ( F )) . Since by [Kap12, Proof of Proposition 3.1, Page 151-154], for ℜ ( s ) large one has E ψ β ξ s ( g ) = X y ∈ Q G ′ ( F ) \ ι ( G ′ ( F )) Z Y H ( F ) \ N H ( A ) ξ ( ω yuι ( g ) , ψ β ( u ) du, we obtain E ψ β ξ s ( g ) = X y ∈ Q G ( F ) \ SL ( F ) Z Y H ( F ) \ N H ( A ) ξ ( ω yuι ( g ) , ψ β ( u ) du. (cid:3) Then we have I (Θ f , ξ, s ) = Z SL ( F ) \ G ( A ) X γ ∈ V ( F ) ρ ( g ) f ( γ, × X y ∈ Q G ( F ) \ SL ( F ) Z Y η ( F ) \ N H ( A ) ξ ( ω yuι ( g ) , ψ β ( u ) dudg = Z Q G ( F ) \ G ( A ) X γ ∈ V ( F ) ρ ( g ) f ( γ, Z Y H ( F ) \ N H ( A ) ξ ( ω uι ( g ) , ψ β ( u ) dudg where ω = I n − I ( − n − I I n − . The main theorem of our paper is:
Theorem 3.2.
The sum P y ∈ P Y ′ ( F ) I ( f, ξ, s )( y, admits a meromorphic continuation to thewhole s -plane which satisfies a functional equation X y ∈ P Y ′ ( F ) I ( f, ξ, s )( y,
1) = X y ∈ P Y ′ ( F ) I ( f, M ( τ, s ) ξ, − s )( y, . where I ( f, ξ, s )( y ) = Z U ( A F ) \ G ( A F ) ρ ( g ) f ( y, Z N ◦ ( A F ) W ψ Q ξ s ( ω uι ( g ) , ψ β ( u ) dudg,W ψ Q ξ s ( ωug,
1) = Z Z n ( F ) \ Z n ( A ) ξ s ( ωuι ( g ) , z ) ψ − Q ( z ) dz, and f satisfies Eq. (3.0.1) . Proof.
We use the defining property of the action of the Weil representation on f , and mapsLemma 2.1 to Lemma 2.6 to unfold the integral.Firstly, using Lemma 2.1, Lemma 2.5, Lemma 2.6, and the action of N on f we have I (Θ f , ξ, s ) = Z Q G ( F ) \ G ( A ) X γ ∈ V ( F ) ρ ( g ) f ( γ, Z Y H ( F ) \ N H ( A ) ξ ( ω uι ( g ) , ψ β ( u ) dudg = Z N ( A ) N ( F ) M ( F ) \ G ( A ) Z N ( F ) \ N ( A ) X γ ∈ V ( F ) ρ ( r ′ g ) f ( γ, × Z Y H ( F ) \ N H ( A ) ξ s ( ω ur ′ ι ( g ) , ψ β ( u ) dudr ′ dg = Z N ( A ) N ( F ) M ( F ) \ G ( A ) Z F \ A X γ ∈ V ( F ) ρ ( g ) ψ ( c β Q ( γ ) − βcQ ( γ )) f ( γ, × Z Y H ( F ) \ N H ( A ) ξ s ( ω ucι ( g ) , ψ β ( u ) dudcdg = Z N ( A ) N ( F ) M ( F ) \ G ( A ) X γ ∈ V ( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, × (cid:18)Z Y H ( F ) \ N H ( A ) ξ s ( ω uι ( g ) , ψ β ( u ) du (cid:19) dg where the last line holds since by [Kap12, Proof of Propostion 3.1], the function g Z Y H ( F ) \ N H ( A ) ξ s ( ω uι ( g ) , ψ β ( u ) du is invariant on the left for r ′ ∈ N ( A ).Using the action of N on f we have I (Θ f , ξ, s ) = Z N ( A ) N ( A ) M ( F ) \ G ( A ) Z N ( F ) \ N ( A ) X γ ∈ V ( F ) Q ( γ )=2 β Q ( γ ) ρ ( zg ) f ( γ, × Z Y H ( F ) \ N H ( A ) ξ s ( ω uzι ( g ) , ψ β ( u ) dudzdg = Z N ( A ) N ( A ) M ( F ) \ G ( A ) Z N ( F ) \ N ( A ) X γ ∈ V ( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) × ψ ( 1 β bQ ( γ ) + 2 bQ ( γ )) f ( γ, Z Y H ( F ) \ N H ( A ) ξ s ( ω uzι ( g ) , ψ β ( u ) dudbdg = Z N ( A ) N ( A ) M ( F ) \ G ( A ) X γ ∈ V ( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 13 × Z N ( F ) \ N ( A ) Z Y H ( F ) \ N H ( A ) ξ s ( ω uzι ( g ) , ψ (4 bQ ( γ )) ψ β ( u ) dudzdg Using the action of M on f we have I (Θ f , ξ, s ) = Z N ( A ) N ( A ) \ G ( A ) X γ ∈ P ( V )( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, × Z N ( F ) \ N ( A ) Z Y H ( F ) \ N H ( A ) ξ s ( ω uzι ( g ) , ψ β ( u ) ψ (4 bQ ( γ )) dudzdg = Z N ( A ) N ( A ) \ G ( A ) X γ ∈ P ( V )( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, × Z N ( F ) \ N ( A ) Z Y H ( A ) \ N H ( A ) Z Y H ( F ) \ Y H ( A ) ξ s ( ω yuzι ( g ) , × ψ β ( yu ) ψ (4 bQ ( γ )) dydudzdg. As in [Kap13, Page 42], for fixed u and g , the function on N ( A ) z Z Y H ( F ) \ Y H ( A ) ξ s ( ω yzug, ψ β ( yu ) ψ (4 bQ ( γ )) dy is well-defined since the elements of N ( A ) and Y H ( A ) commute. Also, since z normalizes Y H ( A ), stabilizes ψ β ( y ) and ξ s ( ω z,
1) = ξ s ( ω , N ( F ).The mapping on N H ( A ) v Z N ( F ) \ N ( A ) Z Y H ( F ) \ Y H ( A ) ξ s ( ω yzug, ψ β ( yu ) ψ (4 bQ ( γ )) dydz is left-invariant by Y H ( A ). Also, z in the integral normalizes N H ( A ) and stabilizes ψ β . Thuswe can interchange uz to zu in the integral. We have I (Θ f , ξ, s ) = Z N ( A ) N ′ ( A ) \ G ( A ) X γ ∈ P ( V )( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, × Z Y H ( A ) \ N H ( A ) Z N ′ ( F ) \ N ′ ( A ) Z Y H ( F ) \ Y H ( A ) ξ s (( ω yzω − ) ω uι ( g ) , × ψ β ( yu ) ψ (4 bQ ( γ )) dydudzdg. As in [Kap13, Page 43], the double integral R N ′ ( F ) \ N ′ ( A ) R Y H ( F ) \ Y H ( A ) can be written as R ˜ Z n ( F ) \ ˜ Z n ( A ) , where ω ′ ˜ Z n ω ′− = Z n , where ω ′ = I I n − ! . We note that now the character on the group Z n is ψ ′ β ( z ) = ψ ( − Q ( γ ) z , + β z , + n − X i =3 z i,i +1 )for z ∈ Z n ( A ).We use a conjugation by d β = β I I n − ! to replace the character ψ ′ β to a character ψ Q , where ψ Q is the generic character of Z n withcoefficient m = − Q ( γ ) and m = m = · · · = m n − = 1.We then get the final form of our sum of Eulerian integrals I (Θ f , ξ, s ) = Z N ( A ) N ′ ( A ) \ G ( A ) X γ ∈ P ( V )( F ) Q ( γ )=2 β Q ( γ ) ρ ( g ) f ( γ, Z Y H ( A ) \ N H ( A ) W ψ Q ξ s ( ωug, ψ β ( u ) dudg = X γ ∈ P ( V )( F ) Q ( γ )=2 β Q ( γ ) Z U ( A ) \ G ( A ) ρ ( g ) f ( γ, Z Y H ( A ) \ N H ( A ) W ψ Q ξ s ( ωug, ψ β ( u ) dudg. The manipulations of the integral will be justified in Section 7 by showing the sum con-verges absolutely for ℜ ( s ) large. Then for ℜ ( s ) large, we have I (Θ f , ξ, s ) = X y ∈ P Y ′ ( F ) I ( f, ξ, s )( y, . (3.0.3)Thus we have that P y ∈ P Y ′ ( F ) I ( f, ξ, s )( y,
1) admits a meromorphic continuation to all s -plane.By the functional equation of the Eisenstein series E ( h, ξ, s ), we obtain the desired func-tional equation for the sum of the global integral. Also, the poles of our sum of integralscome from the poles of E ( h, ξ, s ) [GPSR87] [BG92]. (cid:3) Unramified computation
In this section we give the computation for the unramified local factor of the global integral I ( f, ξ, s ). Let F be a non-Archimedean local field.We assume all data are unramified, i.e. the local field F is absolutely unramified, thecharacter ψ is unramified, and β ∈ O × . Let τ be an irreducible unramified generic represen-tations of GL n ( F ). We denote K G = G ( O ). We assume that the matrices of Q and Q areinvertible and that the residual characteristic is not 2.Let f ∈ S ( V ( F ) × F × ) be f ( v, u ) = V ( O ) ×O × F ( v, u ) for v ∈ V ( F ) , u ∈ F × . Let ρ denotethe local Weil representation of G ( F ). Then ρ ( k ) f = f for k ∈ K G .Let W s,ψ Q ∈ ξ s = Ind H ( F ) Q n ( F ) ( W ( τ, ψ Q ) ⊗ | det | s − ) UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 15 be the unique spherical vector satisfying W s (1 ,
1) = 1. Here, as the notation indicates, weare viewing the induced representation as a space of smooth functions taking values in theWhittaker model.The Satake parameter for ξ s is defined (up to a permutation) by t ξ s = diag( χ ,s ( ̟ ) , . . . , χ n,s ( ̟ ) , χ n,s ( ̟ ) − , . . . , χ ,s ( ̟ ) − ) ∈ Sp n ( C )where each χ i : F × → C × is an unramified character and χ i,s := χ i | · | s − . We set χ s := χ ,s ⊗ · · · ⊗ χ n,s : ( F × ) n −→ C × . For an unramified character µ of split SO ( F ), the Satake parameter is t µ = diag( µ ( ̟ ) , µ ( ̟ ) − ) ∈ SO ( C ) . We have the local integral I s := I ( f, W s,ψ Q )( y ) = Z U ( F ) \ G ( F ) ρ ( g ) f ( y, Z N ◦ ( F ) W s,ψ Q ( ωuι ( g ) , ψ β ( u ) dudg which is the unramified local component of the global integral I ( f, ξ, s )( y ) by construction.We assume y is integral. We denote | y | = q − k , | y | = q − k (4.0.1)for non-negative integers k , k .In order to compute the local integral I s we will adapt a procedure in [Kap12] and makeuse of a local functional equation for the Whittaker model. This requires us to write thefunction ρ ( g ) f ( y,
1) in terms of functions lying in an appropriate Whittaker model.Using the Iwasawa decomposition with respect to the lower Borel subgroup of G ( F ) consist-ing of lower triangular matrices in G ( F ), we define functions Φ ,y , Φ ,y ∈ C ∞ ( U ( F ) \ G ( F ) /K G )(where U ( F ) is the unipotent radical of the lower Borel subgroup of G ( F )) byΦ ,y ( g ) : G ( F ) → C (cid:16) ( t ) ( a d ) κ , ( t ) (cid:16) a ′
00 ( ada ′ ) − (cid:17) κ (cid:17) H ,y (cid:16) ( a d ) , (cid:16) a ′
00 ( ada ′ ) − (cid:17)(cid:17) Φ ,y ( g ) : G ( F ) → C (cid:16) ( t ) ( a d ) κ , ( t ) (cid:16) a ′
00 ( ada ′ ) − (cid:17) κ (cid:17) H ,y (cid:16) ( a d ) , (cid:16) a ′
00 ( ada ′ ) − (cid:17)(cid:17) where a i ∈ F × , t i ∈ F and ( κ , κ ) ∈ K G and H ,y (cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17)(cid:17) = | a | d − | a | d − H ,y (cid:0) ( d ) , (cid:0) d − (cid:1)(cid:1) = c k ,k where c k ,k = 1(2 + P k k ′ =1 q k ′ (1 − ( q − P k k =1 q d − ))(2 + P k k ′ =1 q k ′ (1 − ( q − P k k =1 q d − ))(4.0.2) H ,y (cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17)(cid:17) = ( ̟ − k O F ( a ) − ∞ X k =1 q ( d − k ( q − ̟ k − k O F ( a )) × ( ̟ − k O F ( a ) − ∞ X k =1 q ( d − k ( q − ̟ k − k O F ( a )) H ,y (cid:0) ( d ) , (cid:0) d − (cid:1)(cid:1) = O × ( d )We let Φ y = Φ ,y Φ ,y , and H y = H ,y H ,y .We first show that an appropriate integral of Φ y is ρ ( g ) f ( y,
1) in the following lemma:
Lemma 4.1.
We have ρ ( g ) f ( y,
1) = Z U ( F ) Φ y ( ng ) ψ U ,Q ( n ) − dn (4.0.3) where ψ U ,Q ( n ) = ψ ( n Q ( y ) + n Q ( y )) for n = (( n )) , (( n )) ∈ U ( F ) .Proof. As a function of g , both sides of the equality are invariant under K G on the right andboth transform via the same character under U ( F ) on the left. Thus it suffices to verify theequality Eq. (4.0.3) for g = (cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17)(cid:17) g = (cid:0) ( d ) , (cid:0) d − (cid:1)(cid:1) . We have ρ (cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17)(cid:17) f ( y,
1) = | a | d | a | d V ( O ) ( a y , a y )= | a | d | a | d ̟ − k O ( a ) ̟ − k O ( a ) ρ (cid:0) ( d ) , (cid:0) d − (cid:1)(cid:1) f ( y,
1) = O × ( d ) | d | d − d = O × ( d ) . On the other hand, the right hand side of Eq. (4.0.3) in the two cases are Z F Φ y (cid:16) ( n ) (cid:16) a a − (cid:17) , ( n ) (cid:16) a a − (cid:17)(cid:17) ψ ( n Q ( y ) + n Q ( y )) dn dn Z F Φ y (cid:0) ( n ) ( d ) , ( n ) (cid:0) d − (cid:1)(cid:1) ψ ( n Q ( y ) + n Q ( y )) dn dn . We denote Φ y ( g , g ) = Φ ′ y ( g )Φ ′′ y ( g ) where ( g , g ) ∈ G ( F ), andΦ ′ y , Φ ′′ y ∈ C ∞ ( U ( F ) \ SL ( F ) / SL ( O F )) . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 17
By symmetry, it suffices to verify the equality ̟ − k O F ( a ) | a | d = Z F Φ ′ y (( n ) ( a a − )) ψ ( nQ ( y )) dt O × F ( d ) = Z F Φ ′ y (( n ) ( d )) ψ ( nQ ( y )) dt for a, d ∈ F × .Applying the Iwasawa decompostion with respect to the lower Borel subgroup of SL ( F )to ( n ) ( a a − ) and ( n ) ( d ), we have( n ) ( a a − ) = ( a a − ) (cid:0) a − n (cid:1) for | a − n | ≤ (cid:0) n − (cid:1) (cid:0) a − n an − (cid:1) (cid:0) a n − − (cid:1) for | a − n | > n ) ( d ) = ( d ) ( dn ) for | dn | ≤ (cid:0) n − (cid:1) (cid:0) dn n − (cid:1) (cid:16) ( dn ) − − (cid:17) for | dn | > ̟ − k O F ( a ) | a | d = Z | n |≤| a | H ′ y (( a a − )) ψ ( nQ ( y )) − dn (4.0.4) + Z | a | < | n | H ′ y ( (cid:0) a − n an − (cid:1) ) ψ ( nQ ( y )) − dn (4.0.5) O × F ( d ) = Z | n |≤| d | − H ′ y (( d )) ψ ( nQ ( y )) − dn (4.0.6) + Z | d | − < | n | H ′ y ( (cid:0) dn n − (cid:1) ) ψ ( nQ ( y )) − dn (4.0.7)We first verify Eq. (4.0.5).For | a | > q k , H y ( a ) = 0 and H y ( a − n ) = 0 when | a | < n , thus the right hand side of ofEq. (4.0.5) is 0, while the left hand side is also 0. Then we have the equality for | a | > q k .For | a | = q k , H y ( a − n ) = 0 when | a | < n , then we have that the left hand side ofEq. (4.0.5) is Z | n |≤| a | H y ( a ) ψ ( nQ ( y )) − dn = H y ( a ) Z | n |≤ q k ψ ( nQ ( y )) − dn = q k H y ( a ) Z O F ψ ( n̟ − k Q ( y )) − dn = q k H y ( a )= q k | a | d − ̟ − k O F ( a )= q d k so Eq. (4.0.5) is valid in this case. For | a | < q k , we denote | a | = q − k a . We have that the left hand side of Eq. (4.0.5) is H y ( a ) Z | n |≤| a | ψ ( nQ ( y )) − dn + Z | a | < | n | H y ( a − n ) ψ ( nQ ( y )) − dn = q − k a H y ( a ) + Z q − ka < | n |≤ q k H y ( n ) ψ ( mnQ ( y )) − q − k a dn = q − k a H y ( a ) + k + k a − X k =0 q − k a + k ( q − H y ( ̟ k a − k )On the other hand the right hand side is q − d k a . One can show by induction that these areequal.For Eq. (4.0.7), the right hand side is Z | n |≤| d | − H ′ y (( d )) ψ ( nQ ( y )) − dn + Z | d | − < | n | H ′ y ( (cid:0) dn n − (cid:1) ) ψ ( nQ ( y )) − dn = H ′ y (( d ))( Z | n |≤| d | − ψ ( nQ ( y )) − dn + Z | d | − < | n | H ′ y ( (cid:0) dn ( dn ) − (cid:1) ) ψ ( nQ ( y )) − dn )When d ∈ O × , the above expression is12 + P k k ′ =1 q k ′ (1 − ( q − P k k =1 q d − ) (1 + Z | dn | > H ′ y ( (cid:0) dn ( dn ) − (cid:1) ) ψ ( nQ ( y )) − dn )) = 1which is equal to the left hand side of Eq. (4.0.7).When d
6∈ O × , both the left hand side and right hand side of Eq. (4.0.7) are 0.Thus we deduce the lemma. (cid:3) Inserting the result of Lemma 4.1 to the local integral I , we obtain I s ( y ) = Z U ( F ) \ G ( F ) Z U ( F ) Φ y ( ng ) ψ U ,Q ( n ) − dn Z N ◦ ( F ) W s,ψ Q ( ωuι ( g ) , ψ β ( u ) dudg. Then by a change of variable we get I s = Z G ( F ) Φ y ( g ) Z N ◦ ( F ) W s,ψ Q ( ωug, ψ β ( u ) dudg. (4.0.8)We justify this step by showing the above integral converges absolutely for ℜ ( s ) large inLemma 6.2.Since Φ y ∈ C ∞ ( U ( F ) \ G ( F ) /K G ), applying the Iwasawa decomposition with respect tothe lower Borel subgroup B G ( F ) of G ( F ) we get I s = Z T G ( F ) Φ y ( g ) δ − B G ( g ) Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuyι ( g ) , ψ β ( u ) dudydg = Z A ( F ) Z M ( F ) Z G ( F ) Φ y ( amx ) δ − B G ( amx ) × Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuyι ( amx ) , ψ β ( u ) dudydadmdx UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 19 = Z F × Z F × Z F × Φ y ( (cid:0) a a − ( mb ) − (cid:1) , ( m b )) | am | × Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuyι ( (cid:18) am ab ( ab ) − ( am ) − (cid:19) ) , ψ β ( u ) dudyd × ad × md × b = Z F × H ′ y ( (cid:16) a a − (cid:17) ) | a | Z F × Z F × H ,y ( (cid:0) mb ) − (cid:1) , ( m b )) H ,y ( (cid:0) mb ) − (cid:1) , ( m b )) | m | × Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuyι ( a m a b ( a b ) − ( a m ) − ! ) , ψ β ( u ) dudyd × a d × md × b. By Mellin inversion, for ( (cid:0) mb ) − (cid:1) , ( m b )) ∈ M ( F ) G ( F ), we have H ,y ( (cid:0) mb ) − (cid:1) , ( m b )) = c q Z iI F + σ Z iI F + σ H y,s (1) χ s ,s ( (cid:0) m − b (cid:1) , ( m b − )) ds ds where I F = [ − πi log q , πi log q ], σ , σ ∈ R , and H y,s (1) = Z F × H ,y (cid:0) ( a ) , (cid:0) a ( a a ) − (cid:1)(cid:1) χ − s ,s (cid:0) ( a ) , (cid:0) a ( a a ) − (cid:1)(cid:1) d × a d × a , where χ s ,s (cid:0) ( a ) , (cid:0) a ( a a ) − (cid:1)(cid:1) = | a | s | a | s for a i ∈ F × and c q = ( log q πi ) . (4.0.9)We have H y,s (1) = Z F × H ,y (cid:0) ( a ) , (cid:0) a ( a a ) − (cid:1)(cid:1) | a | − s | a | − s d × a d × a = Z F × O × F ( a )( ̟ − k O F ( a ) − ∞ X k =1 q ( d − k ( q − ̟ k − k O F ( a )) | a | − s d × a = Z F × ̟ − k O F ( a ) | a | − s d × a + ( q − ∞ X k =1 q ( d − k Z F × ̟ i − k O F ( a ) | a | − s d × a = ∞ X i = − k q is + ( q − ∞ X k =1 ( ∞ X i = k − k q is )= c s ζ v ( − s ) where c s = 1 − q s + ( q − q − k s . (4.0.10)We denote χ ′ s ,s = χ s ,s ◦ H ,y ◦ δ − B G . Thus we get I s = c q Z F × H ′ y (cid:16) a a − (cid:17) | a | − Z F × Z F × Z iI F + σ Z iI F + σ H y,s (1) χ ′ s ,s ( (cid:0) mb ) − (cid:1) , ( m b )) × Z U ( F ) Z N ◦ ( F ) W s,ψ Q ωuyι a m a b ( a b ) − ( a m ) − ! , ! ψ β ( u ) dudyd × a d × md × bds ds = c q c k ,k Z iI F + σ Z iI F + σ H y,s (1) Z F × H ′ y ( (cid:16) a a − (cid:17) ) | a | − s d × a Z F × Z F × | m | − s + s + d | b | − s × Z U ( F ) Z N ◦ ( F ) W s,ψ Q (cid:18) ωuyι (cid:18) m b b − m − (cid:19) , (cid:19) ψ β ( u ) dudyd × md × bds ds . We denote I s,s ,s = Z F × Z F × | m | − s + s + d | b | − s × Z U ( F ) Z N ◦ ( F ) W s,ψ Q (cid:18) ωuyι (cid:18) m b b − m − (cid:19) , (cid:19) ψ β ( u ) dudyd × md × b. We show in Lemma 6.3 that I s,s ,s converges when ℜ ( s ) , ℜ ( − s ) large and s in somevertical strip in the right-half plane depends on ℜ ( s ) , ℜ ( − s ).We now define a new local integral I ′ s = Z T G ( F ) Φ y ( g ) δ − B G ( g ) Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q ( ωuyι ( g ) , r ( a )) ψ β ( u ) dadudydg. Apply the same process as for I s , we have I ′ s = c q c k ,k Z iI F + σ Z iI F + σ Z F × H ′ y (cid:16) a a − (cid:17) | a | − s d × a Z F × Z F × | m | − s + s + d | b | − s × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q ( ωuyι (cid:18) m b b − m − (cid:19) , r ( a )) ψ β ( u ) dadudyd × md × bds ds where r ( a ) = (cid:16) I n − a (cid:17) ∈ GL n ( F ) . We denote I ′ s,s ,s = Z F × Z F × | m | − s + s + d | b | − s × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q ( ωuyι (cid:18) m b b − m − (cid:19) , r ( a )) ψ β ( u ) dadudyd × md × bd. We show in Lemma 6.4 that I ′ s,s ,s converges when ℜ ( s ) , ℜ ( − s ) large and s in somevertical strip in the right-half plane depends on ℜ ( s ) , ℜ ( − s ).We now show I ′ s ,s is equal to our integral I s,s ,s up to an explicit constant. UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 21
Lemma 4.2.
In the region of convergence of I s,s ,s (see Lemma 6.3), we have I s,s ,s = γ ( s − s + s , χ ′ ⊗ τ ) − I ′ s,s ,s where γ ( s − s + s , χ ′ ⊗ τ ) is the GL × GL n gamma factor defined in [JPSS83] and χ ′ = |·| d .Proof. By [Kap12, Claim 4.1], one has I = γ ( s − ζ , χ ⊗ τ ) I where I = Z GL ( F ) \ G ′ ( F ) Z N ◦ ( F ) Z GL ( F ) ϕ ζ ( g, m, I ) × W s,ψ ( ωuι ( g ) , diag( m, I n − )) | det m | s − ζ − n − ψ β ( u ) dmdudg where ϕ ζ ( g, m, I ) ∈ V π ζ for π ζ = Ind G ′ ( F ) B G ′ ( F ) ( χ ⊗ µ ) and µ a character of SO ( F ), and I = Z GL ( F ) \ G ′ ( F ) Z N ◦ ( F ) Z GL ( F ) ϕ ζ ( g, m, I ) × Z M × ( n − ( F ) W s,ψ ( ωuyι ( g ) , r ( a )) | det m | s − ζ − n − ψ β ( u ) dadmdudg. By [Kap12, Proof of Lemma 4.1], one has I = I ′ , I = I ′ where I ′ = Z G ′ ( F ) ϕ ζ ( g, I , I ) Z N ◦ ( F ) W s,ψ ( ωuyι ( g ) , ψ β ( u ) dudydg, and I ′ = Z G ′ ( F ) ϕ ζ ( g, I , I ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ ( ωuyι ( g ) , r ( a )) ψ β ( u ) dadudydg. By the property of ϕ ζ and since we are in the unramified case, we have I ′ = Z T G ′ ( F ) χ ( g ) Z U ( F ) Z N ◦ ( F ) W s,ψ ( ωuyι ( g ) , ψ β ( u ) dudydg = Z F × Z F × χ ( m ) χ ( b ) | m | − Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuyι (cid:18) m b b − m − (cid:19) , × ψ β ( u ) dudyd × md × b and I ′ = Z T G ′ ( F ) χ ( g ) Z U ( F ) Z N ◦ Z M × ( n − ( F ) W s,ψ ( ωuyι ( g ) , r ( a )) ψ β ( u ) dadudydg = Z F × Z F × χ ( m ) χ ( b ) | m | − × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q ( ωuyι ( (cid:18) m b b − m − (cid:19) ) , r ( a )) ψ β ( u ) dadudyd × md × b where χ (cid:18) m b b − m − (cid:19) = χ ( m ) χ ( b ) | m | − where χ , χ are characters on F × .Then we have I ′ = γ ( s − ζ , χ ⊗ τ ) I ′ . Therefore in our case we deduce the result of the lemma. (cid:3)
Inserting the result to our local integral I s we get I s = c q c k ,k Z iI F + σ Z iI F + σ H y,s (1) Z F × H ′ y ( (cid:16) a a − (cid:17) ) | a | − s d × a × γ ( s − s + s , χ ′ ⊗ τ ) − I ′ s,s ,s ds ds = c q c k ,k Z iI F + σ Z iI F + σ c s ζ v ( − s − d H y,s (1) γ ( s − s + s , χ ′ ⊗ τ ) − I ′ s,s ,s ds ds . Also, since c s ζ v ( − s − d + 2) H y,s (1) converges when ℜ ( − s ) large, c s ζ v ( − s − d H y,s (1) γ ( s − s + s , χ ′ ⊗ τ ) − I ′ s,s ,s converges when ℜ ( s ) , ℜ ( − s ) large and ℜ ( s ) bounded in some vertical strip in the right halfplane depends on −ℜ ( s ) , −ℜ ( s ) as in Lemma 6.3.To compute our final result, we remain to compute I ′ s,s ,s . Since U ( F ) normalizes N ◦ ( F ) and ψ β , we can interchange uy to yu in the integral. Denoting ( r ( a ) ω ) − ( yu ) =( r ( a ) ω )( yu )( r ( a ) ω ) − , we have I ′ s,s ,s = Z F × Z F × | m | − s + s + d | b | − s × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q (cid:18) ( r ( a ) ω ) − ( yu ) r ( a ) ωι (cid:18) m b b − m − (cid:19) , (cid:19) × ψ β ( u ) dadudyd × md × b. Let ˜ ω = I n ( − n I n and let V be the subgroup V ( R ) = I n − y y s y ′ y ′ I n − ∈ H ( R ) . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 23
By [Kap12, Claim 4.2], denoting ˜ ω V = ( ˜ ω ) − V ˜ ω we have I ′ s,s ,s = Z F × Z F × | m | − s + s + d | b | − s Z M × ( n − ( F ) × Z ˜ ω V ( F ) W s,ψ Q ( vu ( a ) ωι (cid:18) m b b − m − (cid:19) , ψ β ( u ) dvdad × md × b = X k ∈ Z | ̟ k | − s + s + d Z M × ( n − ( F ) Z SO ( F ) Z ˜ ω V ( F ) W s,ψ Q ( vu ( a ) ωxι (cid:18) m m − (cid:19) , × ψ β ( u ) dvdadx where µ s = | · | − s is a character for SO ( F ).As in [Kap12, Page 161], by the structure of x ∈ ι ( G ( F )), we have ( u ( a ) ω ) − x = ω ′ ˜ ω x where ω ′ = diag( I n , − , I n ). Thus we have I ′ s,s ,s = X k ∈ Z | ̟ k | − s + s + d Z M × ( n − ( F ) Z ω ′ ˜ ω SO ( F ) Z ˜ ω V ( F ) W s,ψ Q ( vxu ( a ) ωι ( m ) , × µ s ( x ) ψ β ( u ) dvdadx. We regard dvdx integral as a function on H ( F ): h Z ω ′ ˜ ω SO ( F ) Z ˜ ω V ( F ) W s,ψ Q ( vxh, ψ β ( u ) µ s ( x ) dadudmdx We denote this function as B ψ ′ β,Q ,s Similar as in [Kap12, Page 162], our function B ψ ′ β,Q ,s is an unramified Bessel functionwhich corresponds to the Bessel functional defined for the subgroup( ˜ ω Z n ( F ) ⋉ ˜ ω V ( F )) ⋊ ω ′ ˜ ω SO ( F ) = ˜ ω R ◦ ( F )(for ω ′ ˜ ω SO ( F ) split) and representations ξ s , ψ ′ β,Q and µ , where the character ψ ′ β,Q is definedon ˜ ω R ◦ ( F ) by ψ ′ β,Q = ψ − Q ( y ) v , + n − X i =2 v i,i +1 + 12 β v n, ! ( v ∈ ˜ ω Z n ⋉ ˜ ω V ) . Substituting the function to our local integral we get I ′′ s,s ,s = X k ∈ Z | ̟ k | − s + s + d Z M × ( n − ( F ) B ψ ′ β,Q ,s ( ˜ ω u ( a ) ω̟ ˜ δ k H ) da where ˜ δ k = (0 n − , k, ∈ Z n .As in [Kap12, Claim 4.3], we have the following equality Lemma 4.3.
We have Z M × ( n − ( F ) B ψ ′ β,Q ,s ( ˜ ω u ( a ) ω̟ ˜ δ k H ) da = B ψ ′ β,Q ,s ( ̟ δ − k H ) | ̟ k | n − where δ − k = ( − k, n − ) ∈ Z n .Proof. The proof is as in [Kap12, Proof of Claim 4.3]. Let ω = v ( (cid:0) I n − (cid:1) ) ∈ M n ( F ) (seenotation in Eq. (2.1.5)). Since B W is invariant by K H , we have B ψ ′ β,Q ,s ( ˜ ω u ( a ) ω̟ ˜ δ k H ) = B ψ ′ β,Q ,s (( ˜ ω u ( a ) ˜ ω − ω )( ω − ˜ ω ω̟ ˜ δ k H ω − ˜ ω − ω ))= B ψ ′ β,Q ,s ( v ( (cid:16) a ′ I n −
00 0 1 (cid:17) )) ̟ δ − k H )where for m ∈ M ( n − × ( F ), v ( (cid:16) m I n −
00 0 1 (cid:17) ) ∈ M n ( F ). Since v ( a ′ ) ̟ δ k H = ̟ δ k H v ( a ′ ̟ δ k H ), wehave Z M × ( n − ( F ) B ψ ′ β,Q ,s ( v ( a ′ ) ̟ δ − k H ) da = Z M × ( n − ( F ) B ψ ′ β,Q ,s ( ̟ δ − k H v ( a ′ )) | ̟ k | n − da. By the structure of B ψ ′ β,Q ,s and the invariance property of W s,ψ Q , for z ∈ Z n − ( F ) (maxi-mal unipotent radical of Borel subgroup of GL n − ( F )), h ∈ H ( F ), B ψ ′ β,Q ,s (diag( z, I , z ∗ ) · h ) = ψ Q ( z ) B ψ ′ β,Q ,s ( h )where z ∗ = − J n − ( t z − ) J n − , then one can show (similar as in [Gin90] and [Sou93, Page 98])for any s ∈ F × the support of the function α B ψ ′ β,Q ,s ( v ( (cid:16) s α I n −
00 0 1 (cid:17) )is in M ( n − × ( O ). Thus we get the desired equality in the lemma. (cid:3) Then we have I ′ s,s ,s = X k ∈ Z q − ( − s + s + n − d ) k B ψ ′ β,Q ,s ( ̟ δ − k H ) . Before making explicit computation for the function B ψ ′ β,Q ,s , we first replace the corre-sponding character ψ ′ β,Q to an unramified character using the lemma below. Lemma 4.4.
We have B ψ ′ β,Q ,s ( ̟ δ − k H ) = B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H ) where ψ ′ β ( v ) = ψ ( n − X i =1 v i,i +1 + 12 β v n, ) , for ( v ∈ ˜ ω Z n ( F ) ⋉ ˜ ω V ( F )) and | − Q ( y ) | = q − k Q ( y ) .Proof. For r ′ = vt ∈ ( ˜ ω Z n ⋉ ˜ ω V ) ⋊ ˜ ω SO = ˜ ω R ◦ , we have θ Q ( r ′ ) = θ ( ar ′ a − ) . where we denote a = ̟ δ kQ ( y ) H , θ Q ( vt ) = ψ ′ β,Q ( v ) µ s ( t ), and θ ( vt ) = ψ ′ β ( v ) µ s ( t ). UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 25
Thus the function g B ψ ′ β ,s ( ag )lies in the Bessel model B ( ξ s , θ Q ) since B ψ ′ β ,s ( avg ) = B ψ ′ β ,s ( ava − ag ) = ψ ′ β ( ava − ) B ψ ′ β ,s ( ag ) = ψ ′ β,Q ( v ) B ψ ′ β ,s ( ag ) . (cid:3) Thus we get I ′ s,s ,s = X k ∈ Z q − ( − s + s + n − d ) k B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H ) . Now we proceed to compute the function.Let ∆( χ ) = ( − n det( χ i,s ( ̟ ) n − i +1 − χ i,s ( ̟ ) − ( n − i +1) ) ,D ( χ s , µ s ) = n Y i =1 χ i,s ( ̟ ) − ( n +1 − i ) n − Y i =1 (1 − χ i,s ( ̟ ) µ s ( ̟ ) q − )(1 − χ i,s ( ̟ ) µ s ( ̟ ) − q − ) . Lemma 4.5.
For t = ̟ δ − k + kQ ( y ) H ∈ T H ( F ) such that − k + k Q ( y ) ≥ , we have B ψ ′ β ,s ( t ) = c s ,s δ H ( t ) S ( χ s , µ s , t ) where S ( χ s , µ s , t ) = 1∆( χ s ) X ω ∈ W sgn( ω ) D ( ω χ s , µ s ) ω χ s ( t ) − where W is the Weyl group of H , and c s ,s = L ( s, µ s × τ ) L (2 s, τ, Sym ) where L ( s, µ s × τ ) = n Y i =1 (1 − χ i ( ̟ ) µ s ( ̟ ) q − s ) − (1 − χ i ( ̟ ) µ s ( ̟ ) − q − s ) − ,L (2 s, τ, Sym ) = Y ≤ i ≤ j ≤ n (1 − χ i ( ̟ ) χ i ( ̟ ) q − s ) − n Y i =1 (1 − χ i ( ̟ ) q − s ) − . Proof.
Since B ψ ′ β ,s is an unramified Bessel function, by the vanishing condition of unramifiedBessel function, B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H ) = 0 unless − k + k Q ( y ) ≥ B ψ ′ β ,s as the normalized Bessel function such that B ψ ′ β ,s = B ψ ′ β ,s /B ψ ′ β ,s (1).Since we normalize W s,ψ Q by W s,ψ Q (1 ,
1) = 1, we have B ψ ′ β ,s = W τ s (1) − B f (1) B , where W τ s (1) = X ≤ i ≤ j ≤ n (1 − χ i,s ( ̟ ) χ i,s ( ̟ ) − q − ) (see [CS80]), and B f (1) = Q ≤ i ≤ j ≤ n (1 − χ i,s ( ̟ ) χ j,s ( ̟ ) q − )(1 − χ i,s ( ̟ ) χ j,s ( ̟ ) − q − ) Q ni =1 (1 − χ i,s ( ̟ ) q − ) Q ni =1 (1 − χ i,s ( ̟ ) µ s ( ̟ ) q − )(1 − χ i,s ( ̟ ) µ s ( ̟ ) − q − ) . Using the explicit formula for the unramified Bessel functional (see [BFF97], [Kap12,Section 2.5]), we deduce the result using similar notations as in [Kap12, Section 2.5]. (cid:3)
We proceed to state the main theorem of this section. Consider c s ζ v ( − s − d ζ v ( − s ) γ ( s − s + s , χ ′ ⊗ τ ) − B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H )(4.0.11)as a product of Laurent series in q s and q s where c s is as defined in Eq. (4.0.10). Let C k ( s ) : = c q Z iI F + σ Z iI F + σ q ( s − s ) k c s ζ v ( − s − d ζ v ( − s ) × γ ( s − s + s , χ ′ ⊗ τ ) − B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H ) ds ds (4.0.12)where c q is as defined in Eq. (4.0.9). This is nothing but the product of the − k -th coefficientin q s and the k -th coefficient in q s of (4.0.11). Theorem 4.6.
For all the data unramified and ℜ ( s ) large, we have I s = c k ,k ∞ X k = − k Q ( y ) q ( n − d ) k C − k ( s ) where c k ,k is as defined in Eq. (4.0.2) .Proof. Combining all the above results we get I s = c q c k ,k Z iI F + σ Z iI F + σ c s ζ v ( − s − d H y,s (1) γ ( s − s + s , χ ′ ⊗ τ ) − I ′ s,s ,s ds ds = c q c k ,k Z iI F + σ Z iI F + σ c s ζ v ( − s − d ζ v ( − s ) γ ( s − s + s , χ ′ ⊗ τ ) − × ∞ X k = − k Q ( y ) q ( − s + s + n − d ) k B ψ ′ β ,s ( ̟ δ k + kQ ( y ) H ) ds ds = c q c k ,k ∞ X k = − k Q ( y ) q ( n − d ) k Z iI F + σ Z iI F + σ q ( − s + s ) k c s ζ v ( − s − d ζ v ( − s ) × γ ( s − s + s , χ ′ ⊗ τ ) − B ψ ′ β ,s ( ̟ δ k + kQ ( y ) H ) ds ds We will make the above manipulations rigorous in Lemma 6.5 by showing the infinite sum c s ζ v ( − s − d ζ v ( − s ) γ ( s − s + s , χ ′ ⊗ τ ) − ∞ X k = − k Q ( y ) q ( − s + s + n − d ) k B ψ ′ β ,s ( ̟ δ k + kQ ( y ) H )converges absolutely for ℜ ( s ) , ℜ ( − s ) large and ℜ ( s ) lies in some region depends on ℜ ( s ) , ℜ ( − s ). UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 27
Then we deduce the theorem. (cid:3) Bounds of the local integrals in the non-Archimedean case
Let F be a non-Archimedean local field of characteristic zero. Let K G = G ( O ), and let K H = SO n +1 ( O ). In this section we give bounds for the local factors of the global integral I ( f, ξ, s ) in the non-Archimedean case.We first bound the inner integral of our local integral using some techniques from theproof of convergence of non-Archimedean Rankin-Selberg integral for SO l +1 × GL n in [Sou93,Section 4] in Lemma 5.1 and Lemma 5.2.We give a bound for our local integral in the general case in Lemma 5.3, and then in theunramified case in Lemma 5.5.The local integral is I s := I v ( f, W s,ψ Q ) = Z U ( F ) \ G ( F ) ρ ( g ) f ( y ) Z N ◦ ( F ) W s,ψ Q ( ωug, ψ β ( u ) dudg. For t = diag( t , . . . , t n ) ∈ GL n ( F ) let t ′ := (cid:16) t w t − w (cid:17) ∈ T H ( F )(5.0.1)where w ∈ GL n ( F ) is the antidiagonal matrix.For a quasi-character η : F × → C × there is a unique real number ℜ ( η ) such that η | · | − Re( η ) is unitary. We say η is positive if ℜ ( η ) > Lemma 5.1.
Let ( n, t ′ , k ) ∈ N n ( F ) × T H ( F ) × K H , we have | W s,ψ Q ( nt ′ k, | ≤ | det t | ℜ ( s )+ n − l X j =1 c j,s η j ( t ) where c j,s ∈ C and η j are positive quasi-characters of T H ( F ) which depend on τ .Proof. We use an argument analogous to [Sou93, Lemma 4.4]. Since W s,ψ Q is K H -finite, wehave W s,ψ Q ( k, t ) = X i y i,s ( k ) W i ( t )where y i,s ( k ) are matrix coefficients of K H and W i ∈ W ( τ, ψ − Q ). Since each W i can bemajorized by a gauge, there are positive quasicharacters η j of T n such that | W i ( t ) | ≤ l X j =1 c i,j η j ( t ) , and | t i t i +1 | for i = 1 , . . . , n − k ). We have | W s,ψ Q ( na ′ k, | = | det t | ℜ ( s ) − + n | W s,ψ Q ( k, t ) | , then the assertion is clear. (cid:3) The points of the group ωN ◦ ω − in an F -algebra R are ωN ◦ ( R ) ω − = I n − v ′ v v T v I n − v ′ v ′ : v , v , v ∈ R n − , v ′ i = − t v i J n − , T ∈ M n − ( R ) . Let v ∈ H ( F ) be a unipotent element of H of the form v = c − c I n − − cI n − (5.0.2)for some c ∈ F . Lemma 5.2.
For ℜ ( s ) large, the integral Z ωN ◦ ( F ) ω − | W s,ψ Q ( uv, | du converges.Proof. For u ∈ ωN ◦ ( F ) ω − , we have uv = c − c I n − v ′ − cv v T v c + v I n − − v c − v c v ′ v ′ . (5.0.3)We denote the Iwasawa decomposition of uv as uv = nt ′ k where ( n, t ′ , k ) ∈ N n ( F ) × T H ( F ) × K H , and we denote the i -th line of uv as ( uv ) i .By Lemma 5.1, the integral is majorized by ν X j =1 c j,s Z ωN ◦ ( F ) ω − [ D ( uv )] ℜ ( s )+ n − E j ( uv ) du (5.0.4)where D ( nt ′ k ) = | det t | ,E j ( nt ′ k ) = η j ( t ) . We use the technique as in [Sou93, Lemma 1, Section 11.15]. Let { e , . . . , e n +1 } bethe standard basis of F n +1 . We take the sup-norm on ∧ p F n +1 according to the basis { e i ∧ e i ∧ · · · ∧ e i p | ≤ i < · · · < i p ≤ n } . K H preserves this norm. We have k v ∧ v ∧ · · · ∧ v p k ≤ k v k · k v k · · · k v p k , v j ∈ F n +1 . Let e n +1+ j = e − n + j − for j = 1 , . . . , n , we have UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 29 | a j +1 · · · a | = k e − ( j +1) u ′ v ∧ · · · e − u ′ v k = k ( e − ( j +1) + ( u ′ v ) n +1 − j ) ∧ · · · ( e − + ( u ′ v ) n +1 ) k≤ j Y i =0 max { , k ( uv ) n +1 − j k} = j Y i =0 [( uv ) n +1 − j ]where [( uv ) n +1 − j ] = max { , k ( uv ) n +1 − j k} and k·k denotes the sup-norm. For h = h ... h n +1 ! ∈ H ( F )we let B ( h ) = h n +2 ... h n +1 ! (5.0.5)be the bottom n rows of h .Since the coordinates of B ( uv ) appear in the coefficients of | t j +1 · · · t | = k e − ( j +1) B ( uv ) ∧ · · · e − B ( uv ) k , we have | t j +1 · · · t | − ≥ max { , k x ′ n +1 − j k , . . . , k x ′ n +1 k} . Then we have [ B ( uv )] − j ≤ | t j t j +1 | ≤ [ B ( uv )] j , j = 1 , . . . , n − B ( uv )] − n ≤ D ( u ′ v ) ≤ [ B ( uv )] − (5.0.7)where [ B ( uv )] = max { , kB ( uv ) k} and k·k is the sup-norm.Since [ B ( uv )] ≤ [ u ][ v ], we have E j ( uv ) ≤ [ B ( uv )] C ≤ [ u ] C [ v ] C (5.0.8)for some positive constant C which depends only on τ .By the structure of B ( uv ), we have[ B ( uv )] − = max { , k v k , k v k , k v k , k T k , k− v c k , k v c k} − ≤ max { , k v k , k v k , k v k , k T k} − = [ uv ] − . Thus we have D ( uv ) ≤ [ B ( uv )] − ≤ [ uv ] − (5.0.9)By Eq. (5.0.8) and Eq. (5.0.9), for ℜ ( s ) + n − − C >
0, Eq. (5.0.4) is bounded by ν X j =1 c j,s [ v ] C Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + C du (5.0.10)which converges absolutely. (cid:3) Now we proceed to bound our local integral in the general case.
Lemma 5.3.
For ℜ ( s ) large enough, we have I s ≪ Z F × Z F × | ˜ f ( a y , a y ) || a | ℜ ( s )+ d − n − | a | ℜ ( s )+ d − n − d × a d × a where ˜ f ∈ S ( V ( F )) .Proof. Applying the Iwasawa decomposition of U ( F ) \ G ( F ) with respect to the usual Borelsubgroup of G ( F ), since W s,ψ Q is smooth, it suffices to bound Z T G ( F ) Z K G | ρ ( ak ) f ( y, | δ − B G ( a ) Z N ◦ ( F ) | W s,ψ Q | ( ωuι ( ak ) , dudadk. By the defining property of the Weil representation, the above integral is Z G Z A ( F ) Z A ( F ) Z K G | ρ ( xa a k ) f ( y, | δ − B G ( a a ) × Z N ◦ ( F ) | W s,ψ Q | ( ωuι ( xa a k ) , duda da dk = Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | − × Z N ◦ ( F ) | W s,φ Q | ( ωuι (cid:16)(cid:16) a a − b − (cid:17) , (cid:16) a a − b (cid:17)(cid:17) ι ( k ) , dud × bd × a d × a dk where ι (cid:16)(cid:16) a a − b − (cid:17) , (cid:16) a a − b (cid:17)(cid:17) is I n − a a + ( b − a a − + ba − a ) β ( b − a a − − ba − a ) β ( b − a a − − ba − a ) β ( b − a a − − ba − a ) ( b − a a − + ba − a ) − β ( b − a a − − ba − a ) β ( − ( b − a a − + ba − a )) − β ( b − a a − − ba − a ) + ( b − a a − + ba − a ) ( a a ) − I n − . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 31
Apply the Iwasawa decomposition to SO ( F ) we have for b ∈ F × + ( b + b − ) β ( b − b − ) β ( b − b − ) β ( b − b − ) ( b + b − ) − β ( b − b − ) β ( − ( b + b − )) − β ( b − b − ) + ( b + b − ) ! = (cid:18) c − c − c (cid:19) (cid:18) ⌊ b ⌋ ⌊ b ⌋ − (cid:19) k ′ where k ′ ∈ SO ( O ) ⊂ K H , ⌊ b ⌋ = b if | b | ≤ ⌊ b ⌋ = b − otherwise, and c = − β − if ⌊ b ⌋ = b β − if ⌊ b ⌋ = b − . Also we have (cid:18) c − c − c (cid:19) (cid:18) ⌊ b ⌋ ⌊ b ⌋ − (cid:19) = (cid:18) ⌊ b ⌋ ⌊ b ⌋ − (cid:19) (cid:18) c ⌊ b ⌋ − − c ⌊ b ⌋ − − c ⌊ b ⌋ − (cid:19) . Thus our local integral is majorized by Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | − × Z N ◦ ( F ) | W s,ψ Q | ( ωu diag( I n − , a a , I , ( a a ) − , I n − ) tn ′ k ′′ ι ( k ) , dud × bd × a d × a dk where we denote t = I n − ⌊ b − a a − ⌋ ⌊ ba − a ⌋ I n − , (5.0.11) n ′ = I n − c ⌊ ba − a ⌋ − c ⌊ ba − a ⌋ − c ⌊ ba − a ⌋ I n − , (5.0.12) k ′′ = (cid:18) I n − k ′ I n − (cid:19) . (5.0.13)Since ω I n − a a ⌊ b − a a − ⌋ ⌊ ba − a ⌋ ( a a ) − I n − ω − = a a ⌊ b − a a − ⌋ I n − I n − ⌊ ba − a ⌋ ( a a ) − and by the property of we have Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z N ◦ ( F ) | W s,ψ Q | ( ωun ′ k ′′ ι ( k ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a dk which is Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z N ◦ ( F ) | W s,ψ Q | (( ωuω − )( ωn ′ ω − ) ωk ′′ ι ( k ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a dk. Since ωn ′ ω − = − β c ⌊ ba − a ⌋ − − β c ⌊ ba − a ⌋ I n − β c ⌊ ba − a ⌋ I n − , we have [ ωn ′ ω − ] ≪ β |⌊ b − a a − ⌋| − . Then the above integral is majoraized by Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z ωN ◦ ( F ) ω − | W s,ψ Q | ( u ( ωn ′ ω − ) ωk ′′ ι ( k ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a dk. Since ωk ′′ k ∈ K H , we apply Lemma 5.2. Then the local integral is majorized by ν X j =1 c j,s Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 (5.0.14) ×|⌊ b − a a − ⌋| ℜ ( s ) − n − − C η j (diag( a a , ⌊ b − a a − ⌋ , I n − ))(5.0.15) × ( Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + C du ) d × bd × a d × a dk (5.0.16)where η j are positive quasi-characters which depend only on τ .Also, for u ∈ ωN ◦ ( F ) ω − , if we denote the Iwasawa decomposition of u ( ωn ′ ω − ) = nt ′ k (using notations as in Lemma 5.1), we have diag( a a , ⌊ b − a a − ⌋ , I n − ) t lies in the supportof a gauge on GL n ( F ), there are constants c such that (cid:12)(cid:12)(cid:12)(cid:12) a a |⌊ b − a a − ⌋| t t (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Thus by Lemma 5.2 we have | a a | ≤ c [ u ] |⌊ b − a a − ⌋| − (5.0.17) UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 33
Thus we have | η j | (diag( a a , ⌊ b − a a − ⌋ , I n − )) ≤ c [ u ] c |⌊ b − a a − ⌋| − c − c for some positive integers c , c depend only on τ for j = 1 , . . . , ν .Thus the integral is majorized by ν X j =1 c j,s Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 × |⌊ b − a a − ⌋| ℜ ( s ) − n − − C − c − c ( Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + C +2 c du ) d × bd × a d × a dk. Then for ℜ ( s ) large, the above sum of integrals is majorized by a constant times Z F × Z F × Z K G | ρ ( k ) ˜ f ( a y , a y ) || a | d | a | d | a a | ℜ ( s ) − n +12 d × a d × a dk. where ˜ f ∈ S ( V ( F )).Let ˜˜ f ( v ) = R K G ρ ( k ) f ( v ) where v ∈ V ( F ), we have ˜ f ∈ S ( V ( F )). The integral is equal to Z F × Z F × | ˜˜ f ( a y , a y ) || a | d | a | d | a a | ℜ ( s ) − n +12 d × a d × a which converges for ℜ ( s ) large enough. Thus we get the result in the lemma. (cid:3) Now we give a bound for the local integral in the unramified case. For all the dataunramified as in Section 4. Suppose F is such that q ≥ n . Let f = V ( O F ) ×O × F .Since in the unramified case, ρ ( g ) f ( y,
1) and W s,ψ Q is right invariant by K G and K H , afterapplying the Iwasawa decomposition, we have I s = Z M ( F ) Z G ( F ) ρ ( mx ) f ( y ) δ − B G ( mx ) Z N ◦ ( F ) W s,ψ Q ( ωuι ( mx ) , ψ ( u ) dudmdx. Lemma 5.4.
For a = ( a , . . . , a n ) ∈ GL n ( F ) where | a | = q − k ≥ | a | = q − k ≥ · · · ≥ | a n | = q − k n where k n ≤ , there exists a positive integer c which depends on τ such that | W τ ( a ) | ≤ δ B GL n ( a ) q − k n nc | det a | − c where W τ is the unramified Whittaker function for τ .Proof. The result follows from arguments in [JPSS79, Section 2.4] for k n = 0 by twisting thecorresponding rational representation of GL n ( C ) in the explicit formula of W τ ( a ). (cid:3) Lemma 5.5.
For ℜ ( s ) large enough, we have I s ≤ ζ v ( ℜ ( s ) + c τ ) Z F × Z F × | V ( O F ) ( a y , a y ) || a | ℜ ( s )+ d − n − − c | a | ℜ ( s )+ d − n − − c d × a d × a . where c > , c τ are integers depend only on τ . Proof.
As in Lemma 5.3, and since in the unramified case | β | = 1 , | | = 1, ρ ( k ) f ( y ) = f ( y )for k ∈ K G , we have I s ≤ Z F × Z F × Z F × | f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z ωN ◦ ( F ) ω − | W s,ψ Q | ( u ( ωn ′ ω − ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a . Using the notation as in Lemma 5.1 and Lemma 5.2, we write u ( ωn ′ ω − ) = nt ′ k , byEq. (5.0.6) and Eq. (5.0.7) we have | t n | ≤ [ B ( u ( ωn ′ ω − ))] n − ≤ ([ u ] |⌊ b − a a − ⌋| − ) n − . Also, by the property of W s,ψ Q , we have | W s,ψ Q | ( u ( ωn ′ ω − ) , diag( a a , ⌊ b − a a − ⌋ , I n − ))= | D ( u ( ωn ′ ω − )) | ℜ ( s )+ n − W τ (diag( a a , ⌊ b − a a − ⌋ , I n − ) t )Thus by Lemma 5.4, for ℜ ( s ) large we have | W τ | (diag( a a , ⌊ b − a a − ⌋ , I n − ) t ) ≤ [ u ] −ℜ ( s ) − n − ([ u ] |⌊ λ ( x ) ⌋| − ) ( n − nc | det(diag( a a , ⌊ b − a a − ⌋ , I n − ) t ) | − c . By Eq. (5.0.7), we have | det a | − c ≤ ([ u ] |⌊ b − a a − ⌋| − ) c n . Thus we obtain | W τ | (diag( a a , ⌊ b − a a − ⌋ , I n − ) t ) ≤ | a a | − c [ u ] −ℜ ( s ) − n − + c n − c n |⌊ b − a a − ⌋| − c n ( n − Thus we have I s ≤ Z F × Z F × Z F × | f ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n − − c × |⌊ b − a a − ⌋| ℜ ( s ) − n − − c n ( n − Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + c n − c n dud × a d × a . Then for ℜ ( s ) large, we have I s ≤ Z F × Z F × Z F × | V ( O F ) ×O × F ( a y , a y , b ) || a | d | a | d | a a | ℜ ( s ) − n − − c × Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + c n − c n dud × bd × a d × a = Z F × Z F × | V ( O F ) ( a y , a y ) || a | d | a | d | a a | ℜ ( s ) − n − − c × Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − + c n − c n dud × a d × a UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 35
We have Z ωN ◦ ( F ) ω − [ u ] −ℜ ( s ) − n − +2 c n ( n − − c du ≤ ∞ X k =1 Z [ u ]= q k q − ( ℜ ( s )+ n + − c n ( n − c ) k du ≤ q −ℜ ( s )+( c +1) n − ( c +2) n − c − − q −ℜ ( s )+( c +1) n − ( c +2) n − c − = ζ v ( ℜ ( s ) − ( c + 1) n + ( c + 2) n + c + ) . Thus we obtain I s ≤ ζ v ( ℜ ( s ) − ( c + 1) n + ( c + 2) n + c + ) × Z F × Z F × | V ( O F ) ( a y , a y ) || a | d | a | d | a a | ℜ ( s ) − n − − c d × a d × a = ζ v ( ℜ ( s ) + c τ ) Z F × Z F × | V ( O F ) ( a y , a y ) || a | d | a | d | a a | ℜ ( s ) − n − − c d × a d × a where c τ is some integer depends only on τ .Thus we deduce the lemma. (cid:3) Convergence of local integrals in unramified computation
Let F be a non-Archimedean local field. In this section we prove absolute convergence forvarious local integrals that appear in the unramified computation in Section 4.The points of the group ωN ◦ U ′ ω − in an F -algebra R are ωN ◦ U ′ ( R ) ω − = c I n − − αc v ′ v v T v I n − − αc v ′ αc − c αc v ′ αc − c : c , c ∈ R, v , v , v ∈ R n − , T ∈ M ( n − ( R ) . where U ′ = ι ( U ) and α = β . Lemma 6.1.
For ℜ ( s ) large, the integral Z ωN ◦ U ′ ( F ) ω − W s,ψ Q ( uv, du converges where v is a unipotent element of H ( F ) defined as in Eq. (5.0.2) . Proof.
For u ∈ ωN ◦ U ′ ( R ) ω − , we have uv = c c − c I n − − αc v ′ − cv v T v c + v I n − − v c − v c − αc v ′ αc − c αc v ′ αc c + αc − c where c ∈ F .Considering the Iwasawa decomposition of u ∈ ωN ◦ ( F ) U ′ ( F ) ω − in H (using the notationas in Lemma 5.1), we have uv = na ′ k where ( n, a ′ , k ) ∈ N n ( F ) × T H ( F ) × K H and a ′ = diag( a, , ω a − ω ) for a ∈ GL n ( F ). Wedenote the i -th line of uv as ( uv ) i . By a similar argument as in Lemma 5.2, we have[ B ( uv )] − j ≤ | a j a j +1 | ≤ [ B ( uv )] j where j = 1 , . . . , n and [ B ( uv )] = max { , kB ( uv ) k} and k·k is the sup-norm, and[ B ( uv )] − n ≤ D ( uv ) = | det( a ) | ≤ [ B ( uv )] − . Also, arguing as in Lemma 5.1 and Lemma 5.2, the integral in the lemma is majorized by ν X j =1 c j,s [ v ] C Z ωN ◦ U ′ ( F ) ω − [ u ] −ℜ ( s ) − n − + C du (6.0.1)where C is a positive constant which depends only on τ .Thus we deduce the desired inequalities in the lemma. (cid:3) Lemma 6.2.
The integral I s = Z G ( F ) Φ y ( g ) Z N ◦ ( F ) W s,ψ Q ( ωuι ( g ) , ψ β ( u ) dudg. (6.0.2) converges absolutely for ℜ ( s ) large enough.Proof. By the Iwasawa decomposition with respect to the lower Borel subgroup of G ( F ), wehave I s = Z U ( F ) Z T G ( F ) Φ y ( vg ) δ − B G ( g ) Z N ◦ ( F ) W s,ψ Q ( ωuvι ( g ) , ψ β ( u ) dudvdg. Since Φ y is invariant under U ( F ), we have I s = Z T G ( F ) Φ y ( g ) δ − B G ( g ) Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuvι ( g ) , ψ β ( u ) dudvdg = Z G ( F ) Z A ( F ) Z A ( F ) Φ y ( xa a ) δ − B G ( xa a ) × Z U ( F ) Z N ◦ ( F ) W s,ψ Q ( ωuvι ( xa a ) , ψ β ( u ) dudvdxda da . UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 37
Using the Iwasawa decomposition of SO ( F ) as in Lemma 5.3, we have | I s | ≤ Z F × Z F × Z F × | Φ y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) | a a | ℜ ( s ) − n +5 |⌊ ba a − ⌋| ℜ ( s ) − n +3 × Z U ( F ) Z N ◦ ( F ) | W s,ψ Q | (( ωuvn ′ , diag( a a , ⌊ b − a a − ⌋ , I n − )) dudvd × bd × a d × a = Z F × Z F × Z F × | Φ y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) | a a | ℜ ( s ) − n +5 |⌊ ba a − ⌋| ℜ ( s ) − n +3 × Z U ( F ) Z N ◦ ( F ) | W s,ψ Q | (( ωuvω − )( ωn ′ ω − ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) × dudvd × bd × a d × a = Z F × Z F × Z F × | Φ y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) | a a | ℜ ( s ) − n +5 |⌊ ba a − ⌋| ℜ ( s ) − n +3 × Z ωN ◦ U ′ ( F ) ω − | W s,ψ Q | ( u ( ωn ′ ω − ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a . where n ′ is as defined in Eq. (5.0.12).By Eq. (6.0.1) in Lemma 6.1 and a similar argument as in Lemma 5.3, the above integralis majorized by a finite sum of integrals of the form Z F × Z F × Z F × | Φ y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) | a a | ℜ ( s ) − n +5 |⌊ ba a − ⌋| ℜ ( s ) − n +3 − c × ( Z ωN ◦ U ′ ( F ) ω − [ u ] −ℜ ( s ) − n − + c du ) d × bd × a d × a where c , c > τ .Substituting the above result into our local integral while using the exiplicit formula forΦ y , we have for ℜ ( s ) large the above integral is majorized by Z F × Z F × Z F × | H ,y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) | H ,y | ( (cid:16) a ba − (cid:17) , (cid:16) a b − a − (cid:17) ) × | a a | ℜ ( s ) − n +5 d × bd × a d × a = Z F × Z F × | a | ℜ ( s )+ d − n +32 | a | ℜ ( s )+ d − n +32 ( ̟ − k O F ( a ) − ∞ X k =1 q ( d − k ( q − ̟ k − k O F ( a )) × ( ̟ − k O F ( a ) − ∞ X k =1 q ( d − k ( q − ̟ k − k O F ( a )) da × da × which converges absolutely for ℜ ( s ) large by a similar computation as for H y,s ,s (1) inSection 4. (cid:3) Lemma 6.3.
There exists positive integers C , C , C , C which depends on ( τ, d , d , n ) suchthat I s,s ,s = Z F × Z F × χ ′ s ,s ( (cid:0) mb ) − (cid:1) , ( m b )) Z U ( F ) Z N ◦ ( F ) × W s,ψ Q ( ωuyι ( (cid:18) m b b − m − (cid:19) ) , ψ β ( u ) dudyd × md × b converges absolutely for ℜ ( s ) + ℜ ( − s ) + C < ℜ ( s ) < ℜ ( s ) + 2 ℜ ( − s ) + C ,C < ℜ ( s ) , C < ℜ ( − s ) Proof.
As in Lemma 6.2, I s,s ,s is majorized by a finite sum of integrals of the form Z F × Z F × | m | − s + s + d | b | − s | m | ℜ ( s ) − n +5 |⌊ b ⌋| ℜ ( s ) − n +3 − c Z ωN ◦ U ′ ( F ) ω − [ u ] −ℜ ( s ) − n − + c dud × md × b where c , c > τ .Also, as in Lemma 5.3, we have | m | ≤ c ′ [ u ] |⌊ b ⌋| − . where c ′ is a conatant depends only on τ .Thus for ℜ ( s ) large enough, the integral is majorized by a finite sum of integrals of theform Z F × | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 − c d × b Z ωN ◦ U ′ ( F ) ω − Z | m |≤ c ′ [ u ] |⌊ b ⌋| − | m | − s + s + d + ℜ ( s ) − n +5 × [ u ] −ℜ ( s ) − n − + c d × mdu = Z F × | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 − c d × b Z ωN ◦ U ′ ( F ) ω − Z | m |≤ c ′ [ u ] |⌊ b ⌋| − | m | − s + s + d + ℜ ( s ) − n +5 × [ u ] −ℜ ( s ) − n − + c d × mdu The integral Z | m |≤ c ′ [ u ] |⌊ b ⌋| − | m | − s + s + d + ℜ ( s ) − n +5 d × m converges absolutely when − s + s + d ℜ ( s ) − n + 5 > . Thus when ℜ ( s ) > s − s − d + n −
5, the integral I ′ s,s ,s is majorized by a finite sum ofintegrals of the form Z F × | b | − s ⌊ b ⌋ −ℜ ( s )+2 s − s + d + n − − c d × b Z ωN ◦ U ′ ( F ) ω − [ u ] − s + s + d − n +112 + c du The above integral converges absolutely when − s + s + d − n + 112 + c < − C, − ℜ ( s ) + 2 s − s + d + n − − c − s > , UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 39 ℜ ( s ) − s + 2 s − d − n + 7 + c − s < C is a positive integers which depends on τ . Thus we deduce the statement in thelemma. (cid:3) Let X ⊂ H be the unipotent subgroup of H whose points in an F -algebra R are X ( R ) = I n − v ′ − αc − αc v ′ c αc v T v v I n − v ′ − αc : c , c ∈ F, v , v , v ∈ F n − , T ∈ M n − ( F ) where α = β . Lemma 6.4.
There are constants C > , C , C , C , C which depend on ( τ, d , d , n ) suchthat I ′ s,s ,s = Z F × Z F × χ ′ s ,s ( (cid:0) mb ) − (cid:1) , ( m b )) × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) W s,ψ Q ( ωuyι ( g ) , r ( e )) ψ β ( u ) dedudyd × md × b converges absolutely for ℜ ( s ) + 2 C C + 1 ℜ ( − s ) + C < ℜ ( s ) < (2 C − ℜ ( s ) + 2 C ℜ ( − s )2 C + 1 + C C < ℜ ( s ) , C < ℜ ( − s ) . Proof.
We use the approach in the proof of [Sou93, Proposition 11.16].As in Lemma 6.3, we have I ′ s,s ,s ≤ Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( ωuyn ′ , r ( e )diag( m, ⌊ b ⌋ , I n − )) dedudyd × md × b where r ( e ) = I n − e . Since I n − e m ⌊ b ⌋ I n − = ⌊ b ⌋ I n − m I n − m − e , we have I ′ s,s ,s ≤ Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( v ( r ( m − e )) ωuyn ′ , diag( ⌊ b ⌋ , I n − , m )) dedudyd × md × b where the symbol v ( r ( m − e )) ∈ M n ( F ) is as defined in Eq. (2.1.4).By a change of variable we have I ′ s,s ,s ≤ Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 | m | n × Z U ( F ) Z N ◦ ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( v ( r ( e )) ωuyn ′ , diag( ⌊ b ⌋ , I n − , m )) dedudyd × md × b = Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z N ◦ ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( v ( r ( e ))( ωuω − )( ωyω − )( ωn ′ ω − ) , diag( ⌊ b ⌋ , I n − , m )) × dedudyd × md × b Since for u ∈ N ◦ ( F ), ωuω − ∈ I n − v ′ v v T v I n − v ′
10 0 v ′ , and v ( r ( e )) I n − v ′ v v T v I n − v ′
10 0 v ′ = I n − v ′ v ′ v T + e ′ v ′ − ev v v I n − v ′ v ( r ( e ))we have | W s,ψ Q | ( v ( r ( e ))( ωuω − )( ωyω − )( ωn ′ ω − ) , diag( ⌊ b ⌋ , I n − , m ))= | W s,ψ Q | ( u ′ v ( r ( e ))( ωyω − )( ωn ′ ω − ) , diag( ⌊ b ⌋ , I n − , m ))where u ′ = I n − v ′ v ′ v T + e ′ v ′ − ev v v I n − v ′ . Since I n − e c I n − = − c e c I n −
00 0 1 I n − e UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 41 and for y ∈ U ( F ), ωyω − = c I n − − αc − αc αc − c αc αc − c I n − = v ( (cid:16) c I n − (cid:17) ) I n − − αc − αc − c αc αc I n − where c , c ∈ F , α = β , we have | W s,ψ Q | ( u ′ v ( r ( e ))( ωyω − )( ωn ′ ω − ) , diag( ⌊ b ⌋ , I n − , m ))= | W s,ψ Q | ( u ′ v ( (cid:16) − c e c I n −
00 0 1 (cid:17) ) y ′ ( ωn ′ ω − ) v ( r ( e )) , diag( ⌊ b ⌋ , I n − , m ))where y ′ = − c e c I n − − αc e αc αc c e − c − αc − c − αc e ′ e ′ c e c e ′ − αc e ′ I n − c e ′ αc e αc where e ′ = − J n − t e .Since I n − v ′ v ′ v T + e ′ v ′ − ev v v I n − v ′ v ( (cid:16) − c e c I n −
00 0 1 (cid:17) )= v ( (cid:16) − c e c I n −
00 0 1 (cid:17) ) I n − v ′ v ′ − c v ′ v T + e ′ v ′ − ev − c ev v + c v v I n − v ′ , we have | W s,ψ Q | ( u ′ v ( (cid:16) − c e c I n −
00 0 1 (cid:17) ) y ′ ( ωn ′ ω − ) v ( r ( e )) , diag( ⌊ b ⌋ , I n − , m ))= | W s,ψ Q | ( v ( (cid:16) − c e c I n −
00 0 1 (cid:17) ) u ′′ y ′ ( ωn ′ ω − ) v ( r ( e )) , diag( ⌊ b ⌋ , I n − , m ))= | W s,ψ Q | ( u ′′ y ′ ( ωn ′ ω − ) v ( r ( e )) , diag( ⌊ b ⌋ , I n − , m ))where u ′′ = I n − v ′ v ′ − c v ′ v T + e ′ v ′ − ev − c ev v + c v v I n − v ′ Thus by change of variables v v − c v − c e ′ v v − αc ev v − αc e ′ T T − e ′ v ′ + ev − c e ′ e, the integral I s,s ,s is bounded by Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z X ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( x ( ωnω − ) v ( r ( e )) , diag( ⌊ b ⌋ , I n − , m )) dedx d × md × b Now we proceed to decompose v ( r ( e )).We have I n − e = I n − e I n − , and we apply the Iwasawa decomposition with respect to the standard Borel subgroup ofGL n ( F ) I n − e = n e t e k e where n e = diag( I , n ′ e ) where n ′ e lies in the unipotent radical of the standard Borel subgroupof GL n − ( F ), t e = ( t , . . . , t n ) where t = t = 1, k e ∈ GL n ( O ).By the structure of this decomposition, we have[ e ] ≤ | t n | = | t · · · t n − | − . The integral I s ,s is majorized by Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z X ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( x ( ωnω − ) v ( n e ) , diag( ⌊ b ⌋ , I n − , m ) t e ) dedx d × md × b. By a change of variable x v ( n e )( x ( ωnω − )) v ( n e ) − the above integral is Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 × Z X ( F ) Z M × ( n − ( F ) | W s,ψ Q | ( x ( ωnω − ) , diag( ⌊ b ⌋ , I n − , m ) t e ) dedx d × md × b. UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 43
Thus by similar arguments as in Lemma 6.1 the integral is majorized by a finite sum ofintegrals of the form Z F × Z F × | m | − s + s + d + ℜ ( s )+5 | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 − C × Z X ( F ) Z M × ( n − ( F ) [ x ] −ℜ ( s ) − n − + C η j (diag( ⌊ b ⌋ , I n − , m ) t e ) dedudyd × md × b. where η j is some positive quasi-character depends on τ and C is some positive integer whichdepends on τ .Using the notation as in Lemma 6.1, denoting the Iwasawa decomposition of x = na ′ k ,diag( ⌊ b ⌋ , I n − , m ) at e lies in the support of a gauge on GL n ( F ), we have | a a t | ≤ , | a i t i a i +1 t i +1 | ≤ , | t n − mt n | ≤ i = 3 , . . . , n − e ] ≤ | t · · · t n − | − ≤ [ x ] C ′ |⌊ b ⌋| − C ′ where C ′ is some positive integer. By [Sou93, Proposition 11.15, Lemma 2] we havemax {| t i t i +1 | , | t i +1 t |} ≤ [ e ] n ≤ [ x ] nC ′ |⌊ b ⌋| − nC ′ Thus we have | m | ≥ [ x ] − C |⌊ b ⌋| C where C is a positive integer depends on τ .Then the integral I ′ s ,s is majorized by a finite sum of integrals of the form Z F × | b | − s |⌊ b ⌋| ℜ ( s ) − n +3 − c × Z X ( F ) Z | m |≥ [ x ] − C |⌊ b ⌋| C | m | − s + s + d + ℜ ( s )+5+ c [ x ] −ℜ ( s ) − n − + c dx d × md × b. where c , c > c are constants depend on τ .Similar as in Lemma 6.3, the above integral converges absolutely when − s + s + d ℜ ( s ) + 5 + c < , − C ( − s + s + d ℜ ( s ) + 5 + c ) − ℜ ( s ) − n −
12 + c < − C ′′ , C ( − s + s + d ℜ ( s ) + 5 + c ) + ℜ ( s ) − n + 3 − c − s > , − C ( − s + s + d ℜ ( s ) + 5 + c ) + ℜ ( s ) − n + 3 − c − s < C ′′ is a positive integer which depends on τ . Thus we deduce the result in the lemma. (cid:3) Lemma 6.5.
The infinite sum ∞ X k = − k Q ( y ) c s ζ v ( − s − d ζ v ( − s ) q ( − s + s + n − d ) k γ ( s − s + s , χ ′ ⊗ τ ) − B ψ ′ β ,s ( ̟ δ − k + kQ ( y ) H ) converges absolutely when ℜ ( s ) − ℜ ( s ) − − d ≤ ℜ ( s ) ≤ ℜ ( s ) − ℜ ( s ) + n + 1 − d , ℜ ( s ) > C , ℜ ( − s ) > C where C , C are constants depends on ( n, d , d ) .Proof. By the formula of c s and B ψ ′ β ,s ,s , it suffices to show ∞ X k = − k Q ( y ) q ( − s + s − n − d ) k ζ v ( − s − d ζ v ( − s ) γ ( s − s + s , χ ′ ⊗ τ ) − X ω ∈ W ω χ s ( ̟ δ k ) − converges absolutely in the region in the lemma.Since ζ v ( − s − d + 2) ζ v ( − s ) when ℜ ( − s ) > , ℜ ( − s − d > γ ( s − s + s , χ ′ ⊗ τ ) − converges when ℜ ( s − s + s + d + 52 ) ≥ ∞ X k =0 q − ( s − s + n +1 − d ) k X ω ∈ W ω χ s ( ̟ δ k ) − converges absolutely in the region.We have | χ s ( ̟ δ k ) − | = q ( s − ) k | χ ( ̟ − k ) | . By [JS81, Corollary 2.5] we have | ω χ ( ̟ − k ) | < q k for any ω ∈ W and k >
0. Then it suffices to observe ∞ X k =0 q − ( s − s + n +1 − d ) k q sk and ∞ X k =0 q − ( s − s + n +1 − d ) k q − ( s − k converges in the region. Then we deduce the result. (cid:3) UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 45 Convergence of the local integral in the Archimedean case
In this section we we give bounds for the local integrals in the Archimedean case. Let F be an Archimedean local field. Let K G be the maximal compact subgroup of G ( F ), and let K H be the maximal compact subgroup of H ( F ).As in the non-Archimedean case, the local integral in the Archimedean case is I s := I ( f, W s,ψ Q ) = Z U ( F ) \ G ( F ) ρ ( g ) f ( y, Z N ◦ ( F ) W s,ψ Q ( ωuι ( g ) , ψ β ( u ) dudg As in the non-Archimedean case, we first use some techniques as in the proof of absoluteconvergence of the Archimedean local integral for the Rankin-Selberg integral for SO l × GL n in [Sou93, Section 5] to bound our inner integral in Lemma 6.1 and Lemma 6.2.For t ∈ GL n ( F ) let t ′ := (cid:16) t w t − w (cid:17) (7.0.1)where w ∈ GL n ( F ) is the antidiagonal matrix. Lemma 7.1.
Let ( n, t ′ , k ) ∈ N n ( F ) × T H ( F ) × K H , there is a positive integer N such thatthere is a constant c s such that | W s,ψ Q ( nt ′ k ) | ≤ c s | det t | ℜ ( s )+ n − k t k N for t = diag( t , t , . . . , t n − , , where k t k = 1 + n − X i =1 | t i | + n − X i =1 | t i | − . Proof.
We argue analogously as in [Sou93, Lemma 5.2]. One has W s,ψ Q ( nt ′ k ) = | det t | s + n − W s,ψ Q ( k, t ) . By the property of W s,ψ Q , there is a continuous seminorm p on the space τ and a positiveinteger N such that | W s,ψ Q ( k, t ) | ≤ k t k N p ( W s,ψ Q ( k, . Since p ( W s,ψ Q ( k, K H , we deduce the lemma. (cid:3) Lemma 7.2.
For v as defined in Eq. (5.0.2) as an element in H ( F ) , we have for ℜ ( s ) large Z ωN ◦ ( F ) ω − | W s,ψ Q ( uv, | du converges.Proof. As in Lemma 5.2, for u ∈ ωN ◦ ( F ) ω − and v a unipotent element of the formEq. (5.0.2), we denote the Iwasawa decomposition of uv as uv = nt ′ k where ( n, t ′ , k ) ∈ N n ( F ) × T H ( F ) × K H , and we denote the i -th line of uv as ( uv ) i . By Lemma 7.1, there is a positive integer N such that for ξ τ,s there is a constant c s suchthat | W s,ψ Q ( nt ′ k, | ≤ c s | det t | ℜ ( s )+ n − | ω τ ( t n ) |k t k N (7.0.2)where t = diag( t t · . . . · t n , t · . . . · t n , . . . , t n − t n , t n ), k t k = k t − n t k , and ω τ is the centralcharacter of τ . We assume N is even, then k a k N is a sum of positive quasicharacters.As in the non-Archimedean case, we denote B ( uv ) = ( uv ) n +2 ... ( uv ) n +1 ! .Using the technique as in [Sou93, Section 7.3, Lemma 3] we get(1 + kB ( uv ) k ) − n ≤ det( t ) ≤ (1 + kB ( uv ) k ) − . where kB ( uv ) k denotes the standard norm on M n × (2 n +1) ( F ), andmax { t j t j +1 , t j +1 t j } ≤ (1 + kB ( uv ) k ) n , j = 1 , . . . , n − . (7.0.3)Similar to the non-Archimedean case, we have(1 + kB ( uv ) k ) − = (1 + sup {k v k , k v k , k v k , k T k , k− v c k , k v c k} ) − ≤ (1 + sup {k v k , k v k , k v k , k T k} ) − = (1 + k u k ) − By Eq. (7.0.2) we have | W s,ψ Q | ( uv, ≤ X j c s (1 + k u k ) − ℜ ( s )2 − n − | χ j ( t ) | . (7.0.4)where χ j are positive quasi-characters which depend on τ .By Eq. (7.0.3), we have | χ j ( t ) | ≤ (1 + kB ( uv ) k ) C ≤ (1 + k u k ) C (1 + k v k ) C for some positive constant C which depends on τ .Thus we have Z ωN ◦ ( F ) ω − | W s,ψ Q ( uv, | du ≤ X j c s (1 + k v k ) C Z ωN ◦ ( F ) ω − (1 + k u k ) − ℜ ( s )2 − n − + C du (7.0.5)which converges for ℜ ( s ) large. (cid:3) We now proceed to bound our local integral I s . Lemma 7.3.
For y = ( y , y ) ∈ P Y ′ ( F ) , we have I ≪ Z F × Z F × | f ′ ( a y , a y ) || a | ℜ ( s )+ d − n − − c | a | ℜ ( s )+ d − n − − c d × a d × a for ℜ ( s ) large enough where f ′ ∈ S ( V ( F )) is nonnegative and c ∈ R ≥ depends only on τ as in Lemma 5.5. UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 47
Proof.
Applying the Iwasawa decomposition of G ( F ) with respect to the standard Borelsubgroup, we have I s = Z T G ( F ) Z K G ρ ( ak ) f ( y, δ − B G ( a ) Z N ◦ ( F ) W s,ψ Q ( ωuι ( ak ) , ψ β ( u ) dud × a d × a dk. By the action of Weil representation it suffices to bound Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d | a | d | a a | − × Z N ◦ ( F ) | W s,ψ Q | ( ωuι (cid:16)(cid:16) a a − b − (cid:17) , (cid:16) a a − b (cid:17)(cid:17) ι ( k ) , dud × bd × a d × a dk where ι (cid:16)(cid:16) a a − b − (cid:17) , (cid:16) a a − b (cid:17)(cid:17) is I n − a a + ( b − a a − + ba − a ) β ( b − a a − − ba − a ) β ( b − a a − − ba − a ) β ( b − a a − − ba − a ) ( b − a a − + ba − a ) − β ( b − a a − − ba − a ) β ( − ( b − a a − + ba − a )) − β ( b − a a − − ba − a ) + ( b − a a − + ba − a ) ( a a ) − I n − . As in the non-Archimedean case, applying the Iwasawa decomposition of SO ( F ) we havefor b ∈ F × + ( b + b − ) β ( b − b − ) β ( b − b − ) β ( b − b − ) ( b + b − ) − β ( b − b − ) β ( − ( b + b − )) − β ( b − b − ) + ( b + b − ) ! = (cid:18) c − c − c (cid:19) (cid:16) a a − (cid:17) k ′ where k ′ ∈ K SO ⊂ K H .For F = R , K SO = t SK ′ S where K ′ = diag(SO(2 , R ) , t SJ ′ S = J and J ′ =diag( − I , k·k denote the Euclidean vector norm. Using the above decomposition,we have k ( β ( − ( b + b − )) , − β ( b − b − ) , + ( b + b − )) k = k (0 , , a − ) k k , k ( β ( b − b − ) , ( b + b − ) , − β ( b − b − )) k = k (0 , , − ca − ) k ′ k Since the action of K SO preserve the Euclidean norm, we get | a − | = k (0 , , a − ) k = k ( β ( − ( b + b − )) , − β ( b − b − ) , + ( b + b − )) k , | ca − | ≤ k (0 , , − ca − ) k = k ( β ( b − b − ) , ( b + b − ) , − β ( b − b − )) k For F = C , K SO ∼ = SO(3 , R ). Similar as in the real case, the action of K SO preserve k·k .We have | a − | = k (0 , , a − ) k = k ( β ( − ( b + b − )) , − β ( b − b − ) , + ( b + b − )) k | ca − | ≤ k ( β ( b − b − ) , ( b + b − ) , − β ( b − b − )) k Thus in both cases, we have | a | ≪ max( | b | , | b − | ) − = min( | b | , | b − | ) | ca − | ≪ max( | b | , | b − | ) . We denote ⌊ b ⌋ = b if | b | ≤ ⌊ b ⌋ = b − otherwise. Since (cid:18) c − c − c (cid:19) (cid:16) a a − (cid:17) = (cid:16) a a − (cid:17) (cid:18) ca − − c a − − ca − (cid:19) , the local integral is majorized by Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d − | a | d − | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z N ◦ ( F ) | W s,ψ Q | ( ωunk ′′ ι ( k ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a dk which is Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d − | a | d − | a a | ℜ ( s ) − n +12 |⌊ b − a a − ⌋| ℜ ( s ) − n − × Z ωN ◦ ( F ) ω − | W s,ψ Q | ( u ( ωnω − ) ωk ′′ ι ( k ) , diag( a a , ⌊ b − a a − ⌋ , I n − )) dud × bd × a d × a dk where n = I n − ca − − c a − − ca − I n − , k ′′ = (cid:18) I n − k ′ I n − (cid:19) . Since ωnω − = − β ca − − β c a − I n − β ca − I n − , by Eq. (7.0.4) and Eq. (7.0.5) in Lemma 7.2, the local integral is majorized by X j c s Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d − | a | d − | a a | ℜ ( s ) − n +12 × |⌊ b − a a − ⌋| ℜ ( s ) − n − | χ j | (diag( a a , ⌊ b − a a − ⌋ , I n − ))(1 + k− β ca − k ) C × ( Z ωN ◦ ( F ) ω − (1 + k u k ) − ℜ ( s )2 − n − + C du ) d × bd × a d × a dk. Since | ca − | ≪ |⌊ b ⌋| − , the above sum is majorized by X j c s Z F × Z F × Z F × Z K G | ρ ( k ) f ( a y , a y , b ) || a | d − | a | d − | a a | ℜ ( s ) − n +12 × |⌊ b − a a − ⌋| ℜ ( s ) − n − − C ′ | χ j ( a a ) || χ j ( ⌊ b − a a − ⌋ ) |× ( Z ωN ◦ ( F ) ω − (1 + k u k ) − ℜ ( s )2 − n − + C du ) d × bd × a d × a dk. UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 49 where χ j , χ j are positive quasi-characters and C ′ is some positive integer depends only on τ . Then for ℜ ( s ) large, the local integral is majorized by a constant times a finite sum ofintegrals of the form Z F × Z F × Z K G | ρ ( k ) ˜ f ( a y , a y ) || a | d − | a | d − | a a | ℜ ( s ) − n +12 | a a | c j d × a d × a dk = Z F × Z F × | ˜˜ f ( a y , a y ) || a | ℜ ( s )+ d − n − cj | a | ℜ ( s )+ d − n − cj d × a d × a . where c j is some integer depends on τ , ˜ f ∈ S ( V ( F )) and ˜˜ f ( v ) = R K G ρ ( k ) f ( v ) for v ∈ V ( F ).Then we deduce the lemma. (cid:3) Absolute Convergence
In this section we handle the absolute convergence of the sum of the global integral inthe main theorem, which will make the proof of the main theorem rigorous. The prooffollows from the absolute convergence of the local integrals in the Archimedean case and thenon-Archimedean case.
Lemma 8.1.
The sum of the global integral X y ∈ P Y ′ ( F ) I ( f, ξ, s )( y,
1) = X y ∈ P Y ′ ( F ) Z U ( A F ) \ G ( A F ) ρ ( g ) f ( y, Z N ◦ ( A F ) W s,ψ Q ( ωuι ( g ) , ψ ( u ) dudg converges absolutely for ℜ ( s ) large enough.Proof. Let y = ( y , y ) ∈ P Y ′ ( F ). Let S be a finite set of places of F which includes theinfinite places and all the finite places such that q v < n , f S = V ( b O S ) × Q v S O × v , τ v unramified, | β | v = 1, and ρ v ( k ) f v = f v for k ∈ G ( O v ) for v S .By Lemma 5.3, Lemma 5.5 and Lemma 7.3, for ℜ ( s ) large we have I ( f, ξ, s )( y,
1) = Y v |∞ I v ( f v , W s,ψ Q ) Y v ∈ S −∞ I v ( W s,ψ Q ) Y v / ∈ S I v ( W s,ψ Q ) ≪ Y v |∞ Z F × v Z F × v | f ′ v ( a y , a y ) || a | ℜ ( s )+ d − n − − c v | a | ℜ ( s )+ d − n − − c v d × a d × a × Y v ∈ S −∞ Z F × Z F × | f ′ v ( a y , a y ) || a | ℜ ( s )+ d − n − − c v | a | ℜ ( s )+ d − n − − c v d × a d × a × Y v / ∈ S ζ v ( ℜ ( s ) + c τ ) Z F × Z F × | V ( O F ) ( a y , a y ) || a | ℜ ( s )+ d − n − − c v × | a | ℜ ( s )+ d − n − − c v d × a d × a ≪ ζ F ( ℜ ( s ) + c τ ) Z A × F Z A × F | f ′ ( a y , a y ) || a | ℜ ( s )+ d − n − − c × | a | ℜ ( s )+ d − n − − c d × a d × a where c > τ and f ′ ∈ S ( V ( A F )).Thus the sum of the global integral is majorized by a finite sum of a sum of integrals ofthe form X y ∈ P Y ′ ζ F ( ℜ ( s ) + c τ ) Z A × F Z A × F | f ′ ( a y , a y ) || a | ℜ ( s )+ d − n − − c | a | ℜ ( s )+ d − n − − c d × a d × a which, when ℜ ( s ) large, is majorized by X y ∈ P V ζ F ( ℜ ( s ) + c τ ) Z A × F Z A × F | f ′ ( a y , a y ) || a | ℜ ( s )+ d − n − − c | a | ℜ ( s )+ d − n − − c d × a d × a . This is a degenerate Eisenstein series which converges absolutely for ℜ ( s ) large enough. (cid:3) This completes the proof of the main theorem.
UTOMORPHIC-TWISTED SUMMATION FORMULAE FOR PAIRS OF QUADRATIC SPACES 51
List of symbols A subgroup of T G (2.3.5) A subgroup of T G (2.3.6) G { g = ( g , g ) ∈ GL ( R ) : det g = det g − } (2.1.1) G ′ SO (2.1.2) G subgroup of T G (2.3.3) H SO n +1 (2.1.1) I ( f, ξ, s ) integral 1.0.2 M subgroup of T G M SL subgroup of maximal torus of SL × SL (2.3.9) M n Levi subgroup of Q n (2.1.5) µ irreducible unramified character of SO N subgroup of U N subgroup of U N ◦ unipotent subgroup of H (2.1.3) N n unipotent radical of Q n (2.1.6) N n opposite unipotent radical of Q n ψ Q character involving Q , Q P Y projective subscheme of Y P Y ′ projective subscheme of Y ′ Q quadratic form on V Q i quadratic form on V i Q n standard parabolic subgroup of H Q n opposite parabolic subgroup of H ω Weyl group element of H (2.1.7) S ( V ( F v )), S ( V ( A F ))) Schwartz spaces 1 ρ Weil representation 3 τ irreducible cuspidal representation of GL n T G maximal torus of G T H maximal torus of H f Theta function 3 U maximal unipotent subgroup of G (1.0.3) U opposite of U V V × V V i quadratic space of even dimension 1 W s,ψ Q function on H ξ s smooth holomorphic section in the space Ind HQ n ( τ ⊗ | det | s − ) 3 Y { v ∈ V ( R ) : Q ( v ) = 2 β Q ( V ) } (1.0.1) Y ′ subscheme of Y such that no y i = 0 1 References [BFF97] Daniel Bump, Solomon Friedberg, and Masaaki Furusawa. Explicit formulas for the Waldspurgerand Bessel models.
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