A summation formula for triples of quadratic spaces II
aa r X i v : . [ m a t h . N T ] O c t A SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES II
JAYCE R. GETZ AND CHUN-HSIEN HSU
Abstract.
Let V , V , V be a triple of even dimensional vector spaces over a number field F equipped with nondegenerate quadratic forms Q , Q , Q , respectively. Let Y ⊂ Y i =1 V i be the closed subscheme of ( v , v , v ) such that Q ( v ) = Q ( v ) = Q ( v ). The first authorand B. Liu previously proved a Poisson summation formula for this scheme under suitableassumptions on the functions involved. In the current work we extend the formula to abroader class of test functions which necessitates the introduction of boundary terms. Thisis the first time a summation formula with boundary terms has been proven for a sphericalvariety that is not a Braverman-Kazhdan space. As an application, we prove that the Fouriertransform on Y, previously defined as a correspondence, descends to an isomorphism of theSchwartz space of Y. Contents
1. Introduction 21.1. The boundary terms 41.2. Outline 5Acknowledgements 62. Schwartz spaces 63. Braverman-Kazhdan spaces 83.1. The local Schwartz spaces 93.2. The summation formula 234. Groups and orbits 245. Local functions 265.1. A transform 275.2. The Schwartz space of Y Mathematics Subject Classification.
Primary 11F70, Secondary 11F66.The authors are thankful for partial support provided by the first author’s NSF grant DMS 1405708. Anyopinions, findings, and conclusions or recommendations expressed in this material are those of the authorsand do not necessarily reflect the views of the National Science Foundation. The first author also thanksD. Kazhdan for travel support under his ERC grant AdG 669655. Part of this paper was written while thefirst author was on sabbatical at the IBS Center for Geometry and Physics at Postech, South Korea. Hethanks the center and the Postech Math Department for their hospitality and excellent working conditions.
7. The unramified calculation 397.1. The open orbit 397.2. The identity orbit 407.3. The other orbits 418. Bounds on integrals in the nonarchimedean case 459. Bounds on integrals in the archimedean case 499.1. The open orbit 499.2. The identity orbit 529.3. The other orbits 5410. Absolute convergence 5611. The L -theory 6112. The Fourier transform 65List of symbols 70References 711. Introduction
Let d , d , d be three positive even integers, let V i = G d i a , V := ⊕ i =1 V i , and let F bea number field. Then V ( F ) is an F -vector space. For each i , let Q i be a nondegeneratequadratic form on V i . Let Y ⊂ V be the subscheme whose points in an F -algebra R aregiven by Y ( R ) : = { ( y , y , y ) ∈ V ( R ) : Q ( y ) = Q ( y ) = Q ( y ) } . In [GL19b] the first author and Liu proved a Poisson summation formula for this space. Thespace Y is a spherical variety for an appropriate reductive group, and hence the summationformula confirms a special case of conjectures of Braverman and Kazhdan (later discussed byNgˆo) [BK99, BK00, BK02, Ngˆo20], suitably extended to spherical varieties (see [Sak12]). Itis the first summation formula for a spherical variety that is not a Braverman-Kazhdan space.Here a Braverman-Kazhdan space is the affine closure of [ P, P ] \ G where G is a reductivegroup and P ≤ G is a parabolic subgroup.In this paper we broaden the class of test functions to which the formula applies. This hasthe effect of introducing “boundary terms” into the formula; we refer to Theorem 1.3 for aprecise statement. These boundary terms will be important in understanding the asymptoticbehavior of sums over Y, and ought to be related to poles of L -functions.The fact that we work with a broader class of test functions also allows us to make severalpieces of the theory intrinsic to Y in a sense we now explain. Let X P := [ P, P ] \ Sp where P ≤ Sp is the Siegel parabolic and let X = Pl( X P ) be the corresponding Braverman-Kazhdan space (see (3.0.6)). As explained in §
3, the Schwartz space S ( X ( A F )) is defined SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 3 and comes equipped with a Fourier transform F X : S ( X ( A F )) → S ( X ( A F )). We define S ( X ( A F ) × V ( A F )) using the conventions in §
2. For notational simplicity, we let F X : S ( X ( A F ) × V ( A F )) −→ S ( X ( A F ) × V ( A F ))be the automorphism given on pure tensors by F X ( f ⊗ f ) = F X ( f ) ⊗ f . In this paper weintroduce the Schwartz space S ( Y ( A F )) := Im( I : S ( X ( A F ) × V ( A F )) → C ∞ ( Y sm ( A F ))where I is defined as in (5.0.1) below. The Poisson summation formula in [GL19b] relies noton a Fourier transform from S ( Y ( A F )) to itself, but a correspondence S ( X ( A F ) × V ( A F )) S ( X ( A F ) × V ( A F )) S ( Y ( A F )) S ( Y ( A F )) I F X I Let Y sm ⊂ Y be the smooth locus. In the current paper we use our extended Poissonsummation formula to prove the following theorem: Theorem 1.1.
Assume Y sm ( A F ) is nonempty. There is a unique C -linear isomorphism F Y : S ( Y ( A F )) → S ( Y ( A F )) such that I ◦ F X = F Y ◦ I. In other words, the dotted arrow in the diagram above can be replaced by F Y and theresulting diagram is commutative. Theorem 1.1 follows from Theorem 12.1 below. Weprove in Proposition 11.3 below that S ( Y ( F v )) < L ( Y ( F v )) (with respect to an appropriatemeasure) for all places v of F . Using Theorem 1.1, in a follow-up paper with S. Leslie wewill give an explicit formula for F Y and prove that it extends to a unitary operator. Thiswill constitute a sound setup for harmonic analysis on Y ( F v ). We refer to [GK19, KM11] foranalogous work when Y is replaced by the zero locus of a single quadratic form. We wouldlike to emphasize again that the setting of the current paper is a significant leap from thesetting of these other papers because our space is not a Braverman-Kazhdan space.Before stating our summation formula in full generality, let us state it in a special case,chosen so that boundary terms do not appear for simplicity. For 1 ≤ i ≤
3, let F V i : S ( V i ( A F )) −→ S ( V i ( A F ))be the Fourier transform attached to Q i and a choice of nontrivial additive character on F \ A F . For finite places v of F let S i,v := { f ∈ S ( V i ( F v )) : f (0) = F V i ( f )(0) = 0 } and let S ,v := S ,v ⊗ S ,v ⊗ S ,v < S ( V ( F v )) . (1.0.1) JAYCE R. GETZ AND CHUN-HSIEN HSU
Thus restrictions of elements of S ,v to Y sm ( F v ) are elements of S ( Y ( F v )) by Lemma 5.7. Theorem 1.2.
Let f ∈ S ( Y ( A F )) . Assume that there are finite places v , v of F suchthat f = f v v f v v where f v and F Y ( f v ) are restrictions of elements of S ,v and S ,v ,respectively. Then X y ∈ Y ( F ) f ( y ) = X y ∈ Y ( F ) F Y ( f )( y ) . Theorem 1.2 is restated and proved as Theorem 12.7 below.1.1.
The boundary terms.
We now describe our main summation formula more precisely.Possibly confusing (but useful) notational conventions on Schwartz spaces are given in § G be the image of SL under a natural embedding SL → Sp (see (4.0.11)). Thequasi-affine scheme X P = [ P, P ] \ Sp admits a natural right action X P × G −→ X P . Over a field of characteristic zero, there are five orbits in X P under the action of G . We fixrepresentatives γ , Id , γ , γ , γ as in § G γ be the stabilizer of γ in G .We fix a nontrivial character ψ : F \ A F → C × . This gives rise to a Weil representation ρ : G ( A F ) × S ( V ( A F )) −→ S ( V ( A F )) . We will require the following assumption on f = f ⊗ f ∈ S ( X ( A F ) × V ( A F )): There arefinite places v , v of F such that f = f v f v f v v and f v ∈ C ∞ c ( X P ( F v )) , F X ( f v ) ∈ C ∞ c ( X P ( F v ))(1.1.1) ρ ( g ) f ( v ) = 0 for v V ◦ ( F ), for all g ∈ G ( A F ).(1.1.2)Here V ◦ is the open subscheme of V consisting of triples ( v , v , v ) with each v i = 0. Inpractice, condition (1.1.2) can be arranged via a local condition as explained in § ∈ S ( A F ) and let N ≤ SL be the unipotent radical of the Borel subgroup of uppertriangular matrices. To each representative γ i ∈ X P ( F ), we associate a function I ( f ) ( ξ ) = Z G γ ( A F ) \ G ( A F ) f ( γ g ) ρ ( g ) f ( ξ ) dg , ∀ ξ ∈ Y sm ( A F ) ,I ( f ) ( ξ ) = Z N ( A F ) \ G ( A F ) f ( g ) ρ ( g ) f ( ξ ) dg , ∀ ξ ∈ e Y ( A F ) ,I i ( f ⊗ Φ)( ξ, s ) = Z G γi ( A F ) \ G ( A F ) f ( γ i g ) × (cid:18)Z N ( A F ) \ SL ( A F ) Z F × ρ (∆ i ( h ) g ) f ( v )Φ( x (0 , hp i ( g )) | x | s dx × dh (cid:19) dg , ∀ ξ ∈ e Y i ( A F )(1.1.3) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 5 where ∆ i is defined as in (5.0.2), p i is defined as in (5.0.3), and the schemes Y i are definedas in §
4. Roughly, Y ( F ) is a quotient of { ( v , v , v ) ∈ V ◦ ( F ) : Q ( v ) = Q ( v ) = Q ( v ) = 0 } by an action of ( F × ) and Y i is the product of the zero locus of Q i in V ◦ i and the projectivescheme cut out of P ( V ◦ i − × V ◦ i +1 ) by Q i − − Q i +1 , where the indices are understood “modulo3” in the obvious sense.The main summation formula proved in this paper is the following theorem: Theorem 1.3.
Assume that ( f = f ⊗ f , Φ) ∈ S ( X ( A F ) × V ( A F )) × S ( A F ) where f satisfies (1.1.1) and (1.1.2) , and b Φ(0) = 2Vol( F × \ ( A × F ) ) − . One has X ξ ∈ Y ( F ) I ( f )( ξ ) + X ξ ∈ Y ( F ) I ( f )( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( f ⊗ Φ)( ξ, s )= X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( F X ( f ) ⊗ Φ)( ξ, s ) . Here b Φ denotes the Fourier transform of Φ normalized as in (6.0.4), and ( A × F ) < A × F is thesubgroup of ideles of norm 1. Theorem 1.3 is restated and proved as Theorem 6.1 below.Of course it would be desirable to remove assumptions (1.1.1) and (1.1.2). It seems likelythat (1.1.1) could be removed if one had a better understanding of the boundary terms in thesummation formula for X obtained in [GL19a]. In particular, it would be desirable to obtaina suitable geometric interpretation of these terms. It is unclear how to remove assumption(1.1.2). It has the effect of bounding the support of I ( f ) away from Y ( F ) − Y sm ( F ) andhas analogous consequences for the support of I ( f ) and I i ( f ). Let v be a finite place of F . We invite the reader to consult Proposition 7.1 to obtain a feeling for the sort of growthelements of S ( Y ( F v )) have as y ∈ Y sm ( F v ) approaches Y ( F v ) − Y sm ( F v ).1.2. Outline.
We now outline the sections of this paper. We state conventions for Schwartzspaces in §
2. In § X that was initiated in [BK02] and extended in [GL19a]. Wegive a treatment of the archimedean Schwartz space of certain Braverman-Kazhdan spacesthat avoids all K -finiteness conditions (for K a suitable compact subgroup) and includesfor the first time a description of their Fr´echet structure. This allows one to use continuityarguments. We also prove several basic results that lay the foundation for harmonic analysison S ( X ( F v )) including a Plancherel formula (see Proposition 3.12).After this, we discuss the geometric preliminaries necessary for the study of Y in §
4. Weturn in § JAYCE R. GETZ AND CHUN-HSIEN HSU action of an appropriate similitude group in § Y ( F v ) in § §
6. This summation formula is givenin terms of infinite sums of Eulerian integrals, that is, integrals that factor along the places of F (or residues of such expressions). The local integrals are computed in the unramified casein §
7. The proof of Theorem 1.3 depends on bounds on local integrals that are deferred to § §
9. We discuss the L -theory in §
11, and prove in particular that S ( Y ( F v )) < L ( Y ( F v ))with respect to a natural measure. In §
12 we construct the isomorphism F Y as describedabove and prove Theorem 1.2, restated as Theorem 12.7 below.We have appended a list of symbols for the reader’s convenience. Acknowledgements
In response to the paper [GL19b] several people including Y. Sakellaridis and Z. Yun askedthe first author about whether it was possible to define F Y ; he thanks them for this question.He also thanks D. Kazhdan for many interesting conversations on material related to thetopic of this paper, and H. Hahn for her help with editing and her constant encouragement.The authors thank D. Kazhdan for suggesting that a discussion of the boundary termsbe added to the introduction, T. Ikeda for answering a question on his paper [Ike92], andJ.-L. Colliot-Th´el`ene for pointing out his result with Sansuc in [CTS82] (see Theorem 12.3).2. Schwartz spaces
This work involves several Schwartz spaces. Assume for the moment that F is a localfield and that X and Y are quasi-affine schemes of finite type over F . Let X sm ⊂ X and Y sm ⊂ Y be the smooth loci. Any Schwartz space S ( X ( F )) we discuss will be a space offunctions on X sm ( F ). Functions in the Schwartz space need not be defined on all of X ( F ) if X is not smooth. We will not define Schwartz spaces of general quasi-affine schemes of finitetype over F . In fact obtaining a good definition for general spherical varities is an importantopen problem [Sak12]. In this subsection we explain the definition for smooth quasi-affineschemes and how to form Schwartz spaces of products X ( F ) × Y ( F ) given that the Schwartzspaces of each factor have been defined. Most of this is fairly obvious in the nonarchimedeancase but decidedly less obvious in the archimedean case. We have modeled our approach onthe treatment of the smooth case in [AG08].Assume for the moment that F is nonarchimedean. Then if X is smooth we set S ( X ( F )) = C ∞ c ( X ( F )) . More generally, if we have already defined S ( X ( F )) and S ( Y ( F )) (whether or not they aresmooth) we set S ( X ( F ) × Y ( F )) := S ( X ( F )) ⊗ S ( Y ( F ))(the algebraic tensor product). SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 7
Now assume that F is an archimedean local field and that X is smooth. In this case wedefine S ( X ( F )) as in [ES18, Remark 3.2]. Let us briefly recall the definition. Since X isquasi-affine we can choose an embeddingRes F/ R X ( F ) −→ R n in the category of real algebraic varieties such that the image is X ′ ( R ) where X ′ ⊂ G n isa closed (affine) subscheme. We refer to [BCR98, Proposition 3.2.10] for the proof. In-cidentally this illustrates that not all morphisms in the real algebraic category can comefrom morphisms of schemes, since there are many examples of quasi-affine schemes that arenot affine. For each D ∈ C h x , . . . , x n , ∂∂x , . . . , ∂∂x n i , viewed as an (algebraic) differentialoperator on C ∞ ( R n ), let | f | D := inf n sup x ∈ R n | D e f ( x ) | : e f ∈ S ( R n ) and e f | X ( F ) = f o . (2.0.1)Here S ( R n ) is the standard Schwartz space on R n . We then let S ( X ( F )) := (cid:26) f : X ( F ) → C : | f | D < ∞ for all D ∈ C (cid:20) x , . . . , x n , ∂x , . . . , ∂x n (cid:21)(cid:27) . (2.0.2)The seminorms | f | D give S ( X ( F )) the structure of a Fr´echet space in a natural manner. Thespace S ( X ( F )) and its topology do not depend on the choice of embedding [ES18, Lemma3.6(i)]. It is a quotient of a nuclear space by a closed subspace, hence nuclear.Assume that X and Y are quasi-affine schemes of finite type over F which may not besmooth. Suppose that we have defined Schwartz spaces S ( X ( F )) and S ( Y ( F )) that areadditionally Fr´echet spaces. We then define S ( X ( F ) × Y ( F )) = S ( X ( F )) b ⊗S ( Y ( F ))where the hat denotes the (complete) projective topological tensor product. Thus we obtainanother Fr´echet space. We warn the reader that we do not know whether or not the Schwartzspaces are nuclear, and hence we do not know whether the other various ways of definingtopological tensor products coincide with this one.We warn the reader that in [ES18] there is a definition of a Schwartz space for any quasi-affine scheme of finite type over the real numbers (not necessarily smooth). In the smoothcase their definition coincides with ours. In the nonsmooth case it does not, essentiallybecause functions in our Schwartz spaces need not extend to the singular set.Finally we discuss the adelic setting. Assume X and Y are quasi-affine schemes of finitetype over the number field F . Then for all finite sets S of places of F , X ( A SF ) and Y ( A SF ) aredefined as topological spaces [Con12]. If X is smooth and S contains all the infinite placeswe define S ( X ( A SF )) := C ∞ c ( X ( A SF )) (i.e. locally constant functions of compact support).Whether or not X is smooth, if S contains all infinite places and S ( X ( A SF )) is defined thenit will always be a restricted tensor product ⊗ ′ v ∤ S S ( X ( F v )) JAYCE R. GETZ AND CHUN-HSIEN HSU with respect to some basic functions b X,v ∈ S ( X ( F v )) for almost all v . If X is not smooththen b X,v will not be the characteristic function of X ( O F v ) or X sm ( O F v ) for any flat model X of X of finite type over O F .If S is a finite set of places of F including the infinite places, and if the Schwartz spaces S ( X ( A SF )) and S ( Y ( A SF )) have been defined then S ( X ( A SF ) × Y ( A SF )) := S ( X ( A SF )) ⊗ S ( Y ( A SF ))(algebraic tensor product). If S ( X ( F v )) has already been defined for all v |∞ and S is a setof infinite places of F then S ( X ( F S )) := b ⊗ v | S S ( X ( F v ))where the product is the (completed) projective topological tensor product. Similarly if S isany finite set of places of F (possibly empty) then S ( X ( A SF )) := S ( X ( F ∞− S )) ⊗ S ( X ( A ∞∪ SF ))(algebraic tensor product). 3. Braverman-Kazhdan spaces
In [GL19a], following work of Braverman and Kazhdan [BK02], the first author and Liudefined Schwartz spaces for certain Braverman-Kazhdan spaces X to be defined below, andproved a Poisson summation formula for them. In this section, we refine the definition of theSchwartz space in the archimedean case and endow it with the structure of a Fr´echet space.This technical refinement is necessary for continuity arguments.Let Sp n denote the symplectic group on a 2 n -dimensional vector space, and let P ≤ Sp n , M ≤ P denote the usual Siegel parabolic and Levi subgroup. More specifically, for Z -algebras R , set Sp n ( R ) : = (cid:8) g ∈ GL n ( R ) : g t (cid:0) I n − I n (cid:1) g := (cid:0) I n − I n (cid:1)(cid:9) ,M ( R ) : = { (cid:0) A A − t (cid:1) : A ∈ GL n ( R ) } ,N ( R ) : = { (cid:0) I n ZI n (cid:1) : Z ∈ gl n ( R ) , Z t = Z } , (3.0.1)and P = M N.
Apart from this section, we will only use the n = 3 case, but since it is nomore difficult to treat the general case, we include it. We define a character ω : M ( R ) −→ R × ( m m − t ) det m. (3.0.2)Let X P := [ P, P ] \ Sp n . (3.0.3)Let GSp n denote the group of similitudes and let ν : GSp n → G m SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 9 denote the similitude norm. We note that there is a left action M ab ( R ) × GSp n ( R ) × X P ( R ) −→ X P ( R )( m, g, x ) m (cid:16) I n ν ( g ) I n (cid:17) xg − . (3.0.4)One has the Pl¨ucker embedding Pl : X P −→ ∧ n G na (3.0.5)given by taking the wedge product of the last n rows of a representative g ∈ Sp n ( R ) for[ P, P ]( R ) g . We denote by X the closure of Pl( X P ): X := Pl( X P ) . (3.0.6)It is an affine variety (in fact a spherical variety, for many more details see [Li18, § X is the affine closure of X P : X aff P = X .3.1. The local Schwartz spaces.
Let F be a local field of characteristic zero. When F is archimedean, we let K ≤ Sp n ( F ) denote a maximal compact subgroup; when F isnonarchimedean, let K be an Sp n ( F )-conjugate of Sp n ( O ). For f ∈ C ∞ ( X P ( F )) and g ∈ Sp n ( F ), let f χ s ( g ) := Z M ab ( F ) δ P ( m ) / χ s ( ω ( m )) f ( m − g ) dm (3.1.1)be its Mellin-transform. Here χ s := χ | · | s and ω is defined as in (3.0.2). When this integralis well-defined, either because it converges absolutely or is meromorphically continued froma half-plane of absolute convergence it is a section of I ( χ s ) := Ind Sp n P ( χ s ◦ ω ) . Here the induction is normalized. We regard I ( χ s ) as a representation in the category ofsmooth representations (in other words we require sections to be smooth).In [GL19a] the first author and Liu defined a Schwartz space S ( X ( F ) , K ) by a refinementof the method in [BK02]. It was also denoted by S ( X ( F ) , K ) in loc cit., but we warnthe reader that in the earlier paper X denoted X P , not its affine closure. In any case theSchwartz space comes equipped with a Fourier transform F X : S ( X ( F ) , K ) −→ S ( X ( F ) , K ) . The Fourier transform depends on our choice of additive character ψ . Functions in S ( X ( F ) , K )are K -finite smooth functions on X P ( F ), and the space S ( X ( F ) , K ) contains the space C ∞ c ( X P ( F ) , K ) of compactly supported K -finite smooth test functions on X P ( F ) [GL19a,Proposition 4.7].In the nonarchimedean case, the space S ( X ( F ) , K ) does not depend on K , but it doesdepend on K in the archimedean case. This is inconvenient and aesthetically unpleasant. Thus we now explain how to define a space S ( X ( F )) independent of K in the archimedeansetting. It contains C ∞ c ( X P ( F )). In particular it contains functions that are not K -finite.We recall the meromorphic functions a w ( s, χ ) of [Ike92, § w ∈ Ω n , the setof representatives for the Weyl group of Sp n modulo the Weyl group of M given by takingthe element in each coset of smallest length. The function a w ( s, χ ) is a certain product of L -functions evaluated at arguments depending on s , χ and n . A section f ( s ) of I ( χ s ) is good if it is meromorphic and if the section g M w f ( s ) ( g ) a w ( χ, s )is holomorphic for all w ∈ Ω n , where M w is the usual intertwining operator [GL19a, (3.2)].The normalization of the measure used to define the intertwining operator depends on thechoice of an additive character ψ : F → C × . If we need to indicate this dependence, wewrite M w,ψ for M w .In the nonarchimedean setting, this is all one needs to define the Schwartz space. Thefollowing is [GL19a, Definition 4.1]: Definition 3.1.
Assume F is nonarchimedean. The Schwartz space is defined to be the spaceof right K -finite functions f ∈ C ∞ ( X P ( F )) such that for each g ∈ Sp n ( F ) and character χ of F × the integral (3.1.1) defining f χ s ( g ) is absolutely convergent for Re( s ) large enoughand defines a good section.Assume until otherwise stated that F is archimedean. In [GL19a] the first author and Liuintroduced a stronger notion of an excellent section. For real numbers A ≤ B , p ( x ) ∈ C [ x ]and meromorphic functions f : C → C let V A,B : = { s ∈ C : A ≤ Re( s ) ≤ B } , (3.1.2) | f | A,B,p : = sup s ∈ V A,B | p ( s ) f ( s ) | (3.1.3)(which may be ∞ ). Let w ∈ Ω n be the long Weyl element. A good section f ( s ) of I ( χ s ) is excellent if for all g ∈ Sp n ( F ), real numbers A < B , w ∈ { Id , w } and any polynomials p w ∈ C [ x ] such that p w ( s ) a w ( s, χ ) has no poles in V A,B one has | M w f ( s ) ( g ) | A,B,p w < ∞ . When f is K -finite this reduces to the definition in [GL19a]. The local properties of excellentsections proved in loc. cit. remain valid with identical proofs. As a warning, in [Ike92]Ikeda restricts to K -finite sections, but the local properties of intertwining operators usedin loc. cit. do not depend on K -finiteness [Wal92, Chapter 10].Consider the Lie algebra g := Lie(Sp n ( F )) . It acts on C ∞ ( X P ( F )) via the differential of the action (3.0.4) and hence we obtain an actionof U ( g ), the universal enveloping algebra of g . The action of U ( g ) commutes with taking SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 11
Mellin transforms and with M w for all w ∈ Ω n . Let b K G m be a set of representatives for the(unitary) characters of F × modulo equivalence, where χ is equivalent to χ ′ if and only if χ = | · | it χ ′ for some t ∈ R . It can be identified with the set of characters of the maximalcompact subgroup K G m of F × , which explains the notation. The following definition removes K -finiteness conditions from the analogous definition in [GL19a]: Definition 3.2.
The Schwartz space S ( X ( F )) consists of the f ∈ C ∞ ( X P ( F )) such thatfor all D ∈ U ( g ), g ∈ Sp n ( F ) and each character χ of F × , the integral (3.1.1) defining( D.f ) χ s ( g ) is absolutely convergent for Re( s ) large enough, is a good section, and satisfiesthe following condition: For all real numbers A < B , w ∈ { Id , w } , any polynomials p w ∈ C [ s ] such that p w ( s ) a w ( s, η ) has no poles for all ( s, η ) ∈ V A,B × b K G m and compact subsetsΩ ⊂ X P ( F ) one has | f | A,B,p w , Ω ,D := X η ∈ b K G m sup g ∈ Ω | M w ( D.f ) η s ( g ) | A,B,p w < ∞ . (3.1.4)To understand this definition, it is useful to point out that it is indeed possible to choose p w (independently of η ) that satisfy the given assumptions. This follows directly from thedefinition of the a w ( s, η ). In place of a K -finiteness assumption we have added a condition onthe behavior of the functions under the infinitesimal action of g (to guarantee smoothnessof sections as a function of g ) and control over the sum over η of the given sections. Inthe K -finite case, the sum over η is finite so asserting that the sum over η is bounded issuperfluous. We observe that the definition is obviously independent of K . The space of K -finite functions in S ( X ( F )) will be denoted S ( X ( F ) , K ). It coincides with the spacedenoted S ( X ( F ) , K ) in [GL19a].Let ψ : F → C × be a nontrivial character. Theorem 3.3.
There is a linear automorphism F X := F X,ψ : S ( X ( F )) −→ S ( X ( F )) . For f ∈ S ( X ( F )) , the Fourier transform F X ( f ) is the unique function in S ( X ( F )) such that F X ( f ) χ s = M ∗ w ( f χ − s ) for all (unitary) characters χ and all s ∈ C with Re( s ) ≥ . Here M ∗ w := M ∗ w ,ψ := γ ( s + − n , χ, ψ ) ⌊ n/ ⌋ Y r =1 γ (2 s − n + 2 r, χ , ψ ) M w ,ψ : I ( χ s ) −→ I ( χ − s )(3.1.5)is the normalized intertwining operator of [GL19a, (3.5)]. The γ -factors depend on a choiceof Haar measure on F which we always take to be the self-dual measure with respect to ψ . Proof.
This is [GL19a, Theorem 4.4]. We only point out that in the archimedean case anadditional argument is required to prove that the Fourier transform preserves S ( X ( F )) sincein loc. cit. only K -finite functions are treated. The argument is essentially contained in theproof of Proposition 3.7 below so we omit the details. (cid:3) By [GL19a, Proposition 4.7] S ( X P ( F )) < S ( X ( F ))(3.1.6)(see § S ( X P ( F ))). We point out that in the archimedean case it is onlystated that C ∞ c ( X P ( F ) , K ) < S ( X ( F ) , K ) in [GL19a, Proposition 4.7], but the strongerassertion given here follows from a minor modification of the same proof.To complete our discussion of S ( X ( F )) we must endow it with a topology. The expression(3.1.4) is a seminorm on S ( X ( F )) and the collection of these seminorms as A, B, p w , Ω , D vary gives S ( X ( F )) the structure of a locally convex space. Lemma 3.4.
The space S ( X ( F )) is a Fr´echet space.Proof. We first observe that we can replace the family of seminorms with a countable sub-family inducing the same topology. More specifically we can choose a countable basis of U ( g ), and restrict the ( A, B ) to lie in the set { ( − N, N ) : N ∈ Z ≥ } . Since the poles of a w ( s, η ) can only occur at points in Z (see [GL19a, (3.4)]), we can similarly restrict ourattention to a countable family of p w . Finally we can restrict attention to a countable familyof Ω by simply choosing a countable family of compact subsets of X P ( F ) whose union is X P ( F ).The countable family of seminorms described above is separating by Mellin inversion (see[GL19a, Lemma 4.3]). It follows that S ( X ( F )) is Hausdorff and metrizable. It is also clearthat it is complete. (cid:3) We observe that the inclusions C ∞ c ( X P ( F )) −→ S ( X P ( F )) −→ S ( X ( F ))are continuous in the archimedean case, where we give C ∞ c ( X P ( F )) the usual topologyfor compactly supported smooth functions on a real manifold and S ( X P ( F )) the topologyexplained in § n acts on ∧ n G na via its action on G na . For F archimedean, choose a K -invariantbilinear form ( · , · ) on ∧ n F n and set | x | = ( x, x ) [ F : R ] / . For F nonarchimedean, the standardbasis of F n induces a natural basis on ∧ n F n (given by wedge products of the standardbasis of F n in increasing order). Define the norm | x | on ∧ n F n to be the maximum normwith respect to the naturally induced basis. This norm is invariant under Sp n ( O ). In any SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 13 case, we set | g | = | Pl( g ) | (3.1.7)where Pl : X P → ∧ n G na is the Pl¨ucker embedding. Lemma 3.5.
Let ≤ β < . If F is nonarchimedean, then any f ∈ S ( X ( F )) satisfies | f ( g ) | ≪ f | g | − n +12 + β . Moreover f ( g ) = 0 for | g | sufficiently large in a sense depending on f . If F is archimedean,then for each N ∈ Z ≥ and D ∈ U ( g ) there is a continuous seminorm ν D,N,β on S ( X ( F )) such that for f ∈ S ( X ( F )) one has | D.f ( g ) | ≤ ν D,N,β ( f ) | g | − N − n +12 + β . Proof.
Assume first that F is nonarchimedean. When β = 0 the lemma is just [GL19a,Lemma 5.1] and by inspecting the proof one obtains the refined estimate stated in thecurrent lemma.For the archimedean assertion, using Mellin inversion [GL19a, Lemma 4.3], we write( ωω ( m )) − N D.f ( mk ) = δ P ( m ) / X η ∈ b K G m Z i R + σ ( D.f ) η s +2 N [ F : R ] − ( k ) η s ( ω ( m )) ds π [ F : R ] i (3.1.8)for σ sufficiently large. The factor a I n ( s, χ ) is holomorphic in the half plane Re( s ) > − and hence so is a I n ( s + N [ F : R ] , χ ). Thus using the fact that the seminorms (3.1.4) are finite,we can shift the contour to σ = − β in (3.1.8) to see that it is bounded by δ P ( m ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X η ∈ b K G m Z i R − β ( D.f ) η s +2 N [ F : R ] − ( k ) η s ( ω ( m )) ds π [ F : R ] i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ δ P ( m ) / | ω ( m ) | − β π [ F : R ] | f | A,B, ,K,D + | f | A,B,s ,K,D )where A := − β − ε + N [ F : R ] and B := − β + ε + N [ F : R ] for some ε < − β . Since | mk | = | ω ( m ) | − and δ P ( m ) = | m | − ( n +1) , we deduce the lemma in the archimedean case. (cid:3) To prove Proposition 3.7 below we require a more precise version of [GL19a, Lemma 3.3].Assume for the moment that F is archimedean and let µ ( z ) := z ( zz ) / where in the denominator we mean the positive square root. Thus if F is real µ is just thesign character. Then any character of F × can be written uniquely in the form χ = | · | it µ α where t ∈ R , α ∈ { , } when F is real, and α ∈ Z when F is complex. Lemma 3.6.
Assume F is archimedean. Let A < B be real numbers, and for w ∈ { Id , w } let p w , p ′ w ∈ C [ x ] be polynomials such that p w ( s ) a w ( s, µ α ) and p ′ w ( s ) a w ( − s, µ α ) are holomorphicand nonvanishing for all α as above and all s ∈ V A,B . Then the quotients p ′ Id ( s ) a Id ( − s, µ α ) p w ( s ) a w ( s, µ α ) , p w ( s ) a w ( s, χ ) p ′ Id ( s ) a Id ( − s, µ α ) , p Id ( s ) a Id ( s, µ α ) p ′ w ( s ) a w ( − s, µ α ) , p ′ w ( s ) a w ( − s, µ α ) p Id ( s ) a Id ( s, µ α ) are bounded by polynomials in s and α in V A,B . The key difference between this lemma and [GL19a, Lemma 3.3] is the uniformity in α ofthe bound. Proof.
In the case where F = R , there are only two options for α so the lemma followsimmediately from [GL19a, Lemma 3.3]. When F is complex, the proof of [GL19a, Lemma3.3] yields the stronger estimate stated above. (cid:3) We prove the following lemma for use in the proof of Theorem 3.15:
Proposition 3.7.
Assume F is archimedean. The space S ( X ( F )) contains S ( X ( F ) , K ) asa dense subspace. The Fourier transform F X : S ( X ( F )) −→ S ( X ( F )) is continuous.Proof. The first assertion follows from the usual argument (see [War72, § w ∈ { Id , w } . By Theorem 3.3 | M w ( D. F X ( f )) µ αs ( k ) | A,B,p w = | D.M w F X ( f ) µ αs ( k ) | A,B,p w = | D.M w M ∗ w ( f µ α − s )( k ) | A,B,p w . Using the argument of [GL19a, Lemma 3.4], but with Lemma 3.6 replacing [GL19a, Lemma3.3], this is bounded by a constant depending on
A, B timesmax(1 , | α | ) N | D.M w ′ ( f µ α − s )( k ) | A,B,p w ′ (3.1.9)for some N and an appropriate p w ′ independent of α , where w ′ = w , if w = Id , Id , if w = w . This in turn is dominated by | D ′ D.M w ′ ( f µ α − s )( k ) | A,B,p w ′ (3.1.10)for an appropriate differential operator D ′ (see [GL19a, Lemma 5.9]). The second assertionof the lemma follows. (cid:3) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 15
For the next lemma, F can be archimedean or nonarchimedean. Recall that we have aright action (3.0.4) of M ab × GSp n on X P . This yields an action on functions: for a function f on X P ( F ) and ( m, x, g ) ∈ M ab ( F ) × X P ( F ) × GSp n ( F ) L ( m ) R ( g ) f ( x ) := f (cid:16) m − (cid:16) I n ν ( g ) − I n (cid:17) xg (cid:17) . (3.1.11)This restricts to give an action on sections in I ( χ s ) for all χ and s . Lemma 3.8. If ( m, g ) ∈ M ab ( F ) × GSp n ( F ) and f ∈ S ( X ( F )) , then L ( m ) R ( g ) f ∈S ( X ( F )) . Moreover, F X ( L ( m ) R ( g ) f ) = | ν ( g ) | n ( n +1) / δ P ( m ) − L ( m − ) R ( ν ( g ) − g ) F X ( f ) . (3.1.12) Proof.
One has ( L ( m ) R ( g ) f ) χ s = δ P ( m ) − / χ s ( ω ( m ) − ) R ( g ) f χ s . Thus the assertion that L ( m ) R ( g ) f ∈ S ( X ( F )) follows immediately from the definition of S ( X ( F )). Using the identity in Theorem 3.3 several times, F X ( L ( m ) R ( g ) f ) χ s = M ∗ w (( L ( m ) R ( g ) f ) χ − s )= | ν ( g ) | n ( n +1) / δ P ( m ) − / χ − s ( ω ( m − )) R ( ν ( g ) − g ) M ∗ w ( f χ − s )= | ν ( g ) | n ( n +1) / δ P ( m ) − / χ − s ( ω ( m ) − ) R ( ν ( g ) − g ) F X ( f ) χ s = | ν ( g ) | n ( n +1) / δ P ( m ) − ( L ( m − ) R ( ν ( g ) − g ) F X ( f )) χ s . The last assertion of the lemma thus follows from the characterization of the Fourier trans-form given in Theorem 3.3. (cid:3)
Lemma 3.9.
The action of M ab ( F ) × GSp n ( F ) on S ( X ( F )) is smooth. When F isarchimedean it is continuous with respect to the Fr´echet topology on S ( X ( F )) .Proof. The statement is immediate in the archimedean case. For F nonarchimedean, since f ∈ S ( X ( F )) is right K -finite, it suffices to check that it is fixed under a compact opensubgroup of M ab ( F ) and fixed under ( I n λI n ) if λ lies in a sufficiently small compact opensubgroup of O × . Since f is right K -finite, the Mellin transforms f χ vanish if χ is sufficientlyramified. Thus by Mellin inversion [GL19a, Lemma 4.3] f is fixed under a compact opensubgroup of M ab ( F ). We can choose a compact open normal subgroup K ′ ≤ K and acompact open subgroup U of O × such that K ′ fixes f and ( I n λI n ) K ′ ( I n λ − I n ) ≤ K ′ for λ ∈ U. This induces a group homomorphism U −→ Aut(
K/K ′ ) . Since the latter group is finite, the kernel of the morphism is a compact open subgroup of O × . For λ in the kernel, (cid:0) I n λI n (cid:1) fixes f by the Iwasawa decomposition. (cid:3) Let m ( x ) := (cid:18) x − I n − x I n − (cid:19) . (3.1.13)The following lemma explains how F X,ψ changes as we change ψ : Lemma 3.10.
Let c ∈ F × and let ψ c ( x ) := ψ ( cx ) . Then F X,ψ c = | c | n − n −
24 + ⌊ n/ ⌋ ( ⌊ n/ ⌋− n −
12 ) F X,ψ ◦ L ( m ( c ⌊ n/ ⌋ +1 )) . Proof.
The character ψ enters into the definition of F X,ψ in two manners. First, it entersinto the normalization of the measure in the unnormalized intertwining operator M w as on[Ike92, p. 193]. Second, it enters into the definition of the normalized intertwining operatorvia the product of γ factors. If dx is the self-dual Haar measure on F with respect to ψ then | c | / dx is the self-dual Haar measure on F with respect to ψ c . Thus by [Tat79, (3.2.2) and(3.2.3)] γ ( s, χ, ψ c ) = | c | s − χ ( c ) γ ( s, χ, ψ ) (recall we always normalize the γ -factors using theself-dual Haar measure with respect to the given additive character) and hence we have γ ( s + − n , χ, ψ c ) ⌊ n/ ⌋ Y r =1 γ (2 s − n + 2 r, χ , ψ c )= | c | s − n χ ( c ) γ ( s + − n , χ, ψ ) ⌊ n/ ⌋ Y r =1 | c | s − n +2 r − χ ( c ) γ (2 s − n + 2 r, χ , ψ ) . Thus for f ∈ S ( X ( F )), M ∗ w ,ψ c ( f χ s ) = | c | n ( n +1)4 | c | s − n χ ( c ) ⌊ n/ ⌋ Y r =1 | c | s − n +2 r − χ ( c ) M ∗ w ,ψ ( f χ s )= | c | n ( n +1)4 | c | − n ⌊ n/ ⌋ Y r =1 | c | − n +2 r − | c | − ( n +1)(2 ⌊ n/ ⌋ +1) / M ∗ w ,ψ ( L ( m ( c ⌊ n/ ⌋ +1 ) f χ s ) . Here the | c | n ( n +1) / factor comes from renormalizing the Haar measure in the definition of M w ,ψ c . The lemma now follows from the characterization of the Fourier transform given inTheorem 3.3. (cid:3) Assume now that F is nonarchimedean. By the Iwasawa decomposition, a C -vector spacebasis for C ∞ c ( X P ( F )) Sp n ( O ) is given by the functions k := [ P,P ]( F ) m ( ̟ k )Sp n ( O ) (3.1.14)for k ∈ Z . The space S ( X ( F )) Sp n ( O ) contains C ∞ c ( X P ( F )) Sp n ( O ) but it is larger. It contains,for example, the basic function b X := X ( j ,...,j ⌊ n/ ⌋ ,k ) ∈ Z ⌊ n/ ⌋ +1 ≥ q j +4 j + ··· +2 ⌊ n/ ⌋ j ⌊ n/ ⌋ k +2 j + ··· +2 j ⌊ n/ ⌋ . (3.1.15) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 17
One has F X ( b X ) = b X [GL19a, Lemma 5.4] provided that ψ is unramified and F is absolutelyunramified.It will be convenient to isolate another family of functions in this space. Let ≥ c := X α ≥ c α . (3.1.16) Lemma 3.11.
One has ≥ c ∈ S ( X ( F )) Sp n ( O ) .Proof. One has L ( m ( ̟ ) c ) ≥ = ≥ c , so by Lemma 3.8 it suffices to show ≥ ∈ S ( X ( F )) Sp n ( O ) .Since ⌊ n/ ⌋ Y j =1 (1 − q j L ( m ( ̟ ) )) b X = ≥ , we can apply Lemma 3.8 again to deduce the result. (cid:3) The usual Schwartz space of F is dense in L ( F, dx ) and the Fourier transform extendsto an isometry of L ( F, dx ). We now prove analogues of these statements for the currentsetting. We choose a positive right Sp n ( F )-invariant Radon measure on X P ( F ) (it is uniqueup to scaling). Since X P ( F ) ⊂ X ( F ) is open and dense we extend by zero to obtain ameasure on X ( F ) and we can speak of L ( X ( F )). Let ε := m ( − . (3.1.17) Proposition 3.12.
One has S ( X ( F )) < L ( X ( F )) . The Fourier transform F X extends toyield an isometry of L ( X ( F )) . For f, f , f ∈ L ( X ( F )) one has F X ◦ F X = L ( ε n +1 ) , (3.1.18) F X ( f ) = F X ( L ( ε ) f ) , (3.1.19) Z X ( F ) f ( x ) F X ( f )( x ) dx = Z X ( F ) F X ( f )( x )( L ( ε n +1 ) f )( x ) dx . (3.1.20)Before giving the proof, we prove two lemmas. The first is an identity that was statedwith a typo in [Ike92, (1.2.3)]: Lemma 3.13.
Assume that χ : F × → C × is a character and that n = 1 . The operator M ∗ w ◦ M ∗ w : I ( χ s ) −→ I ( χ s ) is the identity. This is well-known, but for lack of a reference we give the proof.
Proof.
For s in a dense subset of C , the induced representation I ( χ s ) is irreducible and thecomposite M ∗ w ◦ M ∗ w : I ( χ s ) → I ( χ s ) is well-defined and intertwines with the action ofSL ( F ). It is therefore multiplication by a constant. We are to show that this constant is 1. For f ∈ S ( F ), let F ∧ ( f )( c, d ) : = Z F f ( x, y ) ψ ( − dx + cy ) dxdy. (3.1.21)This is an SL ( F )-equivariant Fourier transform from S ( F ) to itself. We claim M ∗ w ( f ′ χ s ) ( a bc d ) = F ∧ ( f ) ′ χ − s ( a bc d ) . (3.1.22)Here for a function f ∈ S ( F ), we let f ′ ( g ) := f ( e g );it is a function on N ( F ) \ SL ( F ). The identity is to be interpreted as an identity of mero-morphic functions in s . Since F ∧ ◦ F ∧ is the identity on S ( F ), the identity (3.1.22) implies M ∗ w ◦ M ∗ w = Id, which in turn implies the lemma.Thus we are left with proving (3.1.22). One has f ′ χ s ( ∗ ∗ c d ) = Z F × | z | χ s ( z ) f ( e (cid:0) z − z (cid:1) ( ∗ ∗ c d )) d × z = Z F × | z | χ s ( z ) f ( zc, zd ) d × z. This integral converges absolutely for Re( s ) > −
1. Hence for 0 < Re( s ) < M ∗ w ( f ′ χ s ) ( a bc d ) = γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f ( e (cid:0) z − z (cid:1) ( − ) ( x ) ( a bc d )) d × z (cid:19) dx = γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f (( − z, − xz ) ( a bc d )) d × z (cid:19) dx = γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f ( z ( − a − xc, − b − xd )) d × z (cid:19) dx. (3.1.23)If c = 0, we change variables x x − ac to arrive at γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f ( z ( − xc, − b + adc − xd )) d × z (cid:19) dx = γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f ( z ( − xc, c − − xd )) d × z (cid:19) dx. Here we have used that ad − bc = 1. We now switch the order of integration and changevariables x
7→ − z − x to see this is γ ( s, χ, ψ ) Z F × (cid:18)Z F χ s ( z ) f ( xc, c − z + xd ) d × z (cid:19) dx. (3.1.24)This is absolutely convergent for 0 < Re( s ) <
1, and this justifies switching the integrals.For any Φ ∈ S ( F ), one has an identity of meromorphic functions Z F × | z | χ − s ( z ) Z F Φ( y ) ψ ( zy ) dyd × z = γ ( s, χ, ψ ) Z F × χ s ( z )Φ( z ) d × z. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 19
The function γ ( s, χ, ψ ) is holomorphic and the integrals are absolutely convergent for 0 < Re( s ) <
1. Applying this functional equation we have that (3.1.24) is equal to Z F × χ − s ( z ) | z | (cid:18)Z F f ( cx, c − y + xd ) ψ ( zy ) dxdy (cid:19) d × z = Z F × χ − s ( z ) | z || c | − (cid:18)Z F f ( x, c − y + xc − d ) ψ ( zy ) dxdy (cid:19) d × z = Z F × χ − s ( z ) | z | (cid:18)Z F f ( x, y ) ψ ( − zdx + zcy ) dxdy (cid:19) d × z = F ∧ ( f ) ′ χ − s ( a bc d ) . The integral over z are all absolutely convergent provided Re( s ) <
1. Thus we have proven(3.1.22) for c = 0.Now assume c = 0. In this case, using the fact that ad − bc = 1 and arguing as above, wesee that (3.1.23) is γ ( s, χ, ψ ) Z F (cid:18)Z F × | z | χ s ( z ) f ( z ( − /d, − b − xd )) d × z (cid:19) dx = γ ( s, χ, ψ ) Z F × (cid:18)Z F | d | − χ s ( z ) f ( − z/d, x ) dx (cid:19) d × z = Z F × | d | − | z | χ − s ( z ) (cid:18)Z F f ( − y/d, x ) ψ ( zy ) dxdy (cid:19) d × z = Z F × | z | χ − s ( z ) (cid:18)Z F f ( y, x ) ψ ( − dzy ) dydx (cid:19) d × z = F ∧ ( f ) ′ χ − s ( a b d ) . All of these expressions are holomorphic for 0 < Re( s ) <
1. Thus we obtain (3.1.22) for c = 0. (cid:3) Lemma 3.14.
For any n and any character χ : F × → C × the operator M ∗ w ◦ M ∗ w : I ( χ s ) −→ I ( χ s ) is multiplication by χ ( − n +1 .Proof. This was stated incorrectly in [Ike92, Lemma 1.1]. The root of the error is the typoin [Ike92, (1.2.3)]. Upon correcting the typo using Lemma 3.13 the same argument provesthe current lemma. (cid:3)
Remark.
The typos in [Ike92] corrected in Lemma 3.13 and 3.14 do not affect the mainresults of [Ike92] or their proofs. Moreover they do not affect [GL19a], which makes use ofresults in [Ike92], except for the statement of [GL19a, Lemma 4.6]. The correct statement is(3.1.18) above.
Proof of Proposition 3.12.
With notation as in (3.1.13), one has an Iwasawa decomposition X P ( F ) = m ( F × ) K .
Any right Sp n ( F )-invariant measure on X P ( F ) decomposes into a product of a Haar measureon K and δ P ( m ( · )) − times a Haar measure on m ( F × ) ∼ = F × . By Lemma 3.5, for β > f ∈ S ( X ( F )) satisfies the estimate | f ( m ( x ) k ) | ≪ N δ P ( m ( x )) / N − β (3.1.25)for all N ∈ Z ≥ , x ∈ F × and k ∈ K . Moreover, in the nonarchimedian case, f ( m ( x ) k )vanishes for | δ P ( m ( x )) | small enough. Thus elements of S ( X ( F )) are square-integrable.For f ∈ S ( X ( F )), assertion (3.1.18) is a consequence of the definition of the Fouriertransform (see Theorem 3.3) and Lemma 3.14. Taking the complex conjugate of the identityof Theorem 3.3, for σ ≥ F X ( f ) χ σ + it = F X ( f ) χ σ − it = M ∗ w ( f χ − σ + it )= γ ( − σ + it − n − , χ, ψ ) ⌊ n/ ⌋ Y r =1 γ (2( − σ + it ) − n + 2 r, χ , ψ ) M w ( f χ − σ + it )= γ ( − σ − it − n − , χ, ψ ) ⌊ n/ ⌋ Y r =1 γ (2( − σ − it ) − n + 2 r, χ , ψ ) M w ( f χ − σ − it )= χ ( − M ∗ w ( f χ − σ − it ) . Using the uniqueness statement in Theorem 3.3, we deduce assertion (3.1.19).Now assume f , f ∈ S ( X ( F )). By [GL19a, (4.6)] (and using the notation of loc. cit.) onehas Z X ( F ) f ( x ) F X ( f )( x ) dx = Z X ( F ) f ( x ) X η ∈ b K G m Z iI F M ∗ w ( f η − s )( x ) c F ds πi dx = Z F × × K f ( m ( b ) k ) X η ∈ b K G m Z iI F M ∗ w ( f η − s )( m ( b ) k ) c F ds πi d × bdkδ P ( m ( b )) . (3.1.26)The first integral here converges absolutely by H¨older’s inequality (or the bound (3.1.25)).We wish to pull the sum over η and the integral over iI F outside the integral over F × × K .To justify this by the Fubini-Tonelli theorem, it suffices to show that X η ∈ b K G Z F × × K | f ( m ( b ) k ) | Z I F (cid:12)(cid:12) M ∗ w ( f η − it )( m ( b ) k ) (cid:12)(cid:12) dtd × bdkδ P ( m ( b ))(3.1.27) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 21 is finite. In the nonarchimedean case, the sum over η is finite. Using the bound from Lemma3.5, the fact F X ( f ) η s = M ∗ w ( f η − s ), and the fact that a I n ( s, η ) has no poles for Re( s ) > − (see [GL19a, (3.3)]), we deduce convergence in the nonarchimedean case. In the archimedeancase, it is still true that a I n ( s, η ) has no poles for Re( s ) > − , and hence for 0 < ǫ < thenorm |F X ( f ) | − ǫ,ǫ,p, Ω ,D is finite for all p, Ω , D . Thus we can again use Lemma 3.5 to deduce that (3.1.27) is conver-gent.Therefore, we can pull the sum over η and the integral over iI F outside the integral over F × × K in (3.1.26) to see that it is equal to X η ∈ b K G m Z iI F Z F × × K f ( m ( b ) k ) M ∗ w ( f η − s )( m ( b ) k ) c F dsd × bdk πiδ P ( m ( b )) . Consider the integral over F × × K . It is Z F × × K f ( m ( b ) k ) η s ( ω ( m ( b ))) d × bδ ( m ( b )) / M ∗ w ( f η − s )( k ) dk = Z K f η − s ( k ) M ∗ w ( f η − s )( k ) dk. Thus Z X ( F ) f ( x ) F X ( f )( x ) dx = X η ∈ b K G m Z iI F Z K f η − s ( k ) M ∗ w ( f η − s )( k ) dkds. Taking the complex conjugate of this equation and replacing f by L ( ε n +1 ) f and f by f ,we arrive at Z X ( F ) F X ( f )( x )( L ( ε n +1 ) f ( x )) dx = X η ∈ b K G m Z iI F Z K M ∗ w ( f η − s )( k )( L ( ε n +1 ) f η − s )( k ) dkds = X η ∈ b K G m Z iI F Z K M ∗ w ( f η s )( k )( L ( ε n +1 ) f η s )( k ) dkds. Thus to prove the identity (3.1.20) for f , f ∈ S ( X ( F )), it suffices to show that Z K f η − s ( k ) M ∗ w ( f η − s )( k ) dk = Z K M ∗ w ( f η s )( k )( L ( ε n +1 ) f η s )( k ) dk. (3.1.28)One has Z K f η − s ( k ) M ∗ w ( f η − s )( k ) dk = γ Z K f η − s ( k ) M w ( f η − s )( k ) dk (3.1.29)where γ := γ ( − s − n − , η, ψ ) ⌊ n/ ⌋ Y r =1 γ ( − s − n + 2 r, η , ψ ) and M w : I ( η − s ) → I ( η s ) is the usual unnormalized intertwining operator. For s ∈ i R consider the pairings I ( η s ) × I ( η s ) −→ C I ( η − s ) × I ( η − s ) −→ C given in both cases by ( ϕ , ϕ ) Z K ϕ ( k ) ϕ ( k ) dk. Since s ∈ i R , the adjoint of M w with respect to these pairings is M w − = η ( − n M w by [Wal92, § § η ( − n γ Z K M w ( f η s )( k ) f η − s ( k ) dk. Since s ∈ i R we have that γ ( s − n − , η, ψ ) ⌊ n/ ⌋ Y r =1 γ (2 s − n + 2 r, η , ψ )= γ ( − s − n − , η, ψ ) ⌊ n/ ⌋ Y r =1 γ ( − s − n + 2 r, η , ψ ) = η ( − γ. Hence the integral (3.1.29) is Z K M ∗ w ( f η s )( k )( L ( ε n +1 ) f ) η − s ( k ) dk. Thus we have proven (3.1.28), and hence (3.1.20), for f , f ∈ S ( X ( F )).The space S ( X ( F )) contains S ( X P ( F )) by (3.1.6), and S ( X P ( F )) is dense in L ( X ( F )).Hence S ( X ( F )) < L ( X ( F )) is dense. Therefore, since F X is an automorphism of S ( X ( F )),to check that F X extends to an automorphism of L ( X ( F )), it suffices to check that it is anisometry of S ( X ( F )) with respect to the L -norm. But combining (3.1.18) and (3.1.20), for f ∈ S ( X ( F )) one has that kF X ( f ) k : = Z X ( F ) F X ( f )( x ) F X ( f )( x ) dx = Z X ( F ) ( L ( ε n +1 ) f )( x ) F X ◦ F X ( f )( x ) dx = Z X ( F ) ( L ( ε n +1 ) f )( x )( L ( ε n +1 ) f )( x ) dx = Z X ( F ) f ( x ) f ( x ) dx = k f k . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 23
We deduce that F X extends to an isometry in L ( X ( F )), and hence is continuous. Since wehave already checked (3.1.18) and (3.1.19) for f ∈ S ( X ( F )), we deduce immediately thatthey remain valid for f ∈ L ( X ( F )).To see that equation (3.1.20) remains valid for f , f ∈ L ( X ( F )), consider the continuousbilinear forms h· , ·i i : L ( X ( F )) × L ( X ( F )) −→ C defined by h f , f i = Z X ( F ) f ( x ) F X ( f )( x ) dx, h f , f i = Z X ( F ) F X ( f )( x ) (cid:0) L ( ε n +1 ) f (cid:1) ( x ) dx. They agree on the dense subspace S ( X ( F )) × S ( X ( F )) and thus are identical by continuity. (cid:3) The summation formula.
We now revert to the global setting. We define S ( X ( F ∞ )) := b ⊗ v |∞ S ( X ( F v ))where the hat denotes the projective topological tensor product. We let S ( X ( A F )) := S ( X ( F ∞ )) ⊗ O v ∤ ∞ ′ S ( X ( F v )) , (3.2.1)where the restricted direct product is with respect to the basic functions b X,v . Let ψ : F \ A F → C × be a nontrivial character. Then we have a global Fourier transform F X := F X,ψ := ⊗ v F X,ψ v : S ( X ( A F )) −→ S ( X ( A F )) . In particular, F X,ψ v ( b X,v ) = b X,v if ψ v is unramified and F v is nonarchimedean and absolutelyunramified.There is one place in [GL19a] where dropping the assumption that f ∈ S ( X ( A F )) is K ∞ -finite could potentially cause issues. This is that Eisenstein series formed from smoothsections (as opposed to K -finite sections) need not be of finite order [GL06, § X . We require an analogous formula for functions in S ( X ( A F )), sowe must proceed slightly differently.Recall that S ( X P ( F v )) < S ( X ( F v )) for all places v by (3.1.6). Theorem 3.15.
Assume that for some finite places v , v (not necessarily distinct) one has f = f v f v and F X ( f ) = F X ( f v ) F X ( f v ) with f v ∈ C ∞ c ( X P ( F v )) and F X ( f v ) ∈ C ∞ c ( X P ( F v )) . Then X γ ∈ X ( F ) f ( γ ) = X γ ∈ X ( F ) F X ( f )( γ ) . (3.2.2) Proof.
We may assume f = f ∞ f ∞ with f ∞ ∈ S ( X ( F ∞ )) and f ∞ ∈ S ( X ( A ∞ F )). Let K ∞ < Sp n ( F ∞ ) be a maximal compact subgroup and let S ( X ( F ∞ ) , K ∞ ) ≤ S ( X ( F ∞ )) be the spaceof K ∞ -finite functions. Assume first that f ∞ ∈ S ( X ( F ∞ ) , K ∞ ). Then the stated identityfollows from [GL19a, Theorem 1.1] and [GL19b, Theorem 10.1].We now argue by continuity to deduce the identity in general. Using the estimates inLemma 3.5 and the convergence argument in [GL19a, Lemma 6.4], it suffices to show the ex-istence of a sequence f n, ∞ ∈ S ( X ( F ∞ ) , K ∞ ) such that f n, ∞ → f ∞ and F X ( f n, ∞ ) → F X ( f ∞ )in the topology on S ( X ( F ∞ )). But this directly follows from the fact that S ( X ( F ∞ ) , K ∞ )is dense in S ( X ( F ∞ )) and F X is continuous by Proposition 3.7. (cid:3) We remark that Theorem 3.15 and the unitarity of F X in the nonarchimedean case (whichis part of Proposition 3.12) were already proved in [BK02], but with a different definition ofthe Schwartz space. At least at the nonarchimedean places, the two definitions should yieldthe same space of functions. At the archimedean places this is less clear. In any case, it iseasier to just prove the theorem directly than to rigorously check the compatibility of thetwo definitions. 4. Groups and orbits
For this section, F is a field of characteristic zero. For 1 ≤ i ≤
3, let V i = G d i a where d i iseven and let Q i be a nondegenerate quadratic form on V i ( F ). We put V ◦ i = V i − { } and V ◦ := V ◦ × V ◦ × V ◦ (4.0.1)and we let V ′ ⊂ V (4.0.2)be the open subscheme of ( v , v , v ) such that no two v i are zero. For an F -algebra R , recallthat Y ( R ) : = { ( y , y , y ) ∈ V ( R ) : Q ( y ) = Q ( y ) = Q ( y ) } . (4.0.3)We observe that Y sm = Y ∩ V ′ . We let Y ani ⊂ Y be the open complement of the vanishing locus of Q i (it is independent of i ). SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 25
We let GO Q i be the similitude group of ( V i , Q i ) and let ν : GO Q i → G m be the similitudenorm. We then set H ( R ) := { ( h , h , h ) ∈ GO Q ( R ) × GO Q ( R ) × GO Q ( R ) : ν ( h ) = ν ( h ) = ν ( h ) } , (4.0.4)and define λ : H ( R ) −→ R × ( h , h , h ) ν ( h ) . (4.0.5)Let e Y ( R ) := { ( y , y , y ) ∈ V ◦ ( R ) : Q ( y ) = Q ( y ) = Q ( y ) = 0 } (4.0.6)and let Y be the (quasi-affine) quotient of e Y by G m × G m , acting via the restriction of theaction G m ( R ) × G m ( R ) × V ( R ) −→ V ( R )(4.0.7) ( a , a , ( v , v , v )) ( a v , a v , ( a a ) − v ) . This quotient can be constructed by taking the affine closure Y of e Y in V and viewing Y as an open subscheme of the GIT quotient of Y by G m × G m . We observe that Y is ageometric quotient of e Y .For 1 ≤ i ≤
3, we define the scheme e Y i ( R ) : = { ( y , y , y ) ∈ V ◦ ( R ) : Q i − ( y i − ) = Q i +1 ( y i +1 ) and Q i ( y i ) = 0 } . (4.0.8)Here the indices are taken modulo 3 in the obvious sense. Let Y be the quotient of e Y by G m acting via the restriction of the action G m ( R ) × V ( R ) −→ V ( R )(4.0.9) ( a, ( v , v , v )) ( v , av , av ) . This is nothing but the product over F of the quasi-affine scheme cut out by Q in V ◦ andthe quasi-projective scheme cut out of P ( V ◦ × V ◦ ) by Q = Q . The schemes Y and Y aredefined similarly. Thus Y := e Y / G m and Y i := e Y i / G m (4.0.10)where the quotients are defined as above. Using Hilbert’s theorem 90, we deduce the followinglemma: Lemma 4.1.
The maps e Y ( F ) / ( F × ) → Y ( F ) and e Y i ( F ) /F × → Y i ( F ) are bijective. (cid:3) We often identify SL ( R ) with the subgroup G ( R ) ≤ Sp ( R ) defined as follows: G ( R ) = a b a b a b c d c d c d ∈ GL ( R ) : a i d i − b i c i = 1 for 1 ≤ i ≤ . (4.0.11)We give a set of representatives for X P ( F ) /G ( F )and the corresponding stabilizers. Let γ : = − − − γ i : = − ! i − for 1 ≤ i ≤ . (4.0.12)All four matrices are in Sp ( Z ). By [GL19b, Lemmas 2.1 and 2.2], the matrices γ i to-gether with the identity matrix, denoted by Id, form a minimal set of representatives of X P ( F ) /G ( F ) (strictly speaking, we have chosen different representatives for the γ i orbitsthan in [GL19a], but this does not affect the validity of [GL19b, Lemmas 2.1 and 2.2]). For γ ∈ X P ( F ), let G γ ≤ G be the stabilizer of γ under the right action. The following is Lemma2.3 in loc. cit. Lemma 4.2.
One has G γ ( R ) : = { (( t ) , ( t ) , ( t )) : t , t , t ∈ R, t + t + t = 0 } ,G Id ( R ) : = n(cid:16)(cid:16) b − t b (cid:17) , (cid:16) b − t b (cid:17) , (cid:16) b − t b (cid:17)(cid:17) : t , t , t ∈ R, b , b , b ∈ R × , b b b = 1 o ,G γ ( R ) : = { (( t ) , g, ( − ) g ( − )) : t ∈ R, g ∈ SL ( R ) } ,G γ ( R ) : = { (( − ) g ( − ) , ( t ) , g ) : t ∈ R, g ∈ SL ( R ) } ,G γ ( R ) : = { ( g, ( − ) g ( − ) , ( t )) : t ∈ R, g ∈ SL ( R ) } . (cid:3) Local functions
In this section, we define the local integrals required to state our summation formula andprove some of their basic properties. Let F be a local field of characteristic zero. We use theconventions on Schwartz spaces explained in §
2. For each of the 5 orbits of G ( F ) in X P ( F )given in Lemma 4.2, we will define a family of integrals. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 27
For f = f ⊗ f ∈ S ( X ( F )) ⊗ S ( V ( F )), let I ( f ) ( v ) = Z G γ ( F ) \ G ( F ) f ( γ g ) ρ ( g ) f ( y ) dg, v ∈ Y sm ( F ) I ( f ) ( v ) = Z N ( F ) \ G ( F ) f ( g ) ρ ( g ) f ( y ) dg, y ∈ Y i ( F ) . (5.0.1)Here the stabilizers G γ are computed in Lemma 4.2. These are integrals attached to the G ( F )-orbit of γ and Id, respectively.Let ∆ i : SL −→ G be defined by ∆ i ( h ) := ( I , h, ( − ) h ( − )) for i = 1 , (( − ) h ( − ) , I , h ) for i = 2 , ( h, ( − ) h ( − ) , I ) for i = 3 . (5.0.2)Moreover let p i : G ( R ) −→ SL ( R )( g , g , g ) g i +1 (5.0.3)where the indices are taken modulo 3 in the obvious sense.We need one more piece of data to define the integrals attached to the other orbits. LetΦ ∈ S ( F ). For y ∈ e Y i ( F ), 1 ≤ i ≤ s ∈ C with Re( s ) >
0, we define I i ( f ⊗ Φ)( v, s )= Z G γi ( F ) \ G ( F ) f ( γ i g ) Z N ( F ) \ SL ( F ) Z F × ρ (∆ i ( h ) g ) f ( v )Φ( x (0 , hp i ( g )) | x | s dx × dhdg. (5.0.4)In § § §
9. We observe that in the nonarchimedean case I ( f ), I ( f ), and I ( f ⊗ Φ) are defined for all f ∈ S ( X ( F ) × V ( F )) by bilinearity, and in the archimedean casethe same is true using the estimates in § A transform.
The action of H on V induces actions on Y and e Y i . For applications, itis important to understand how our integrals behave under this action.As in (4.0.5), for h ∈ H ( F ), we denote by λ ( h ) the similitude norm of h . PutΛ( h ) := (cid:16) I λ ( h ) I (cid:17) . For f ∈ S ( V ( F )), we put(5.1.1) L ( h ) f ( v ) := f ( h − v ) . For g ∈ G ( F ), we will make use of the identity [HK92, Lemma 5.1.2],(5.1.2) L ( h ) ρ ( g ) f = ρ (Λ( h ) g Λ( h ) − ) L ( h ) f . For h ∈ H ( F ) and f ∈ S ( X ( F )) let L ′ ( h ) f ( g ) := f ( γ Λ( h ) − γ − g Λ( h )) = f λ ( h ) − λ ( h ) − λ ( h ) − g Λ( h ) . (5.1.3)By the identity λ ( h ) − λ ( h ) − λ ( h ) − = λ ( h ) − λ ( h ) λ ( h ) 1 λ ( h ) − λ ( h ) − λ ( h ) Λ( h ) − ∈ [ P, P ]( F ) λ ( h ) − λ ( h ) Λ( h ) − , we have L ′ ( h ) f ( g ) = f λ ( h ) − λ ( h ) Λ( h ) − g Λ( h ) . (5.1.4)Indeed, we could have defined L ′ ( h ) f this way, but defining it as we have above makes thecomparison with [GL19b] easier. We define an action L ′ ( h ) : S ( X ( F ) × V ( F )) −→ S ( X ( F ) × V ( F ))(5.1.5)given on pure tensors by L ′ ( h )( f ⊗ f ) = L ′ ( h ) f ⊗ L ( h ) f . (5.1.6)In more detail, it sends pure tensors to elements of S ( X ( F ) × V ( F )) by Lemma 3.8 and theidentity L ′ ( h ) f = L λ ( h ) I λ ( h ) − I !! R (Λ( h )) f . (5.1.7)In the nonarchimedean case, the action defined on pure tensors extends tautologically; in thearchimedean case, the fact that it extends to the completed tensor product S ( X ( F ) × V ( F ))follows from Lemma 3.9.By abuse of notation we continue to denote by F X the map F X : S ( X ( F ) × V ( F )) −→ S ( X ( F ) × V ( F ))(5.1.8)given by F X on the first factor. Thus on pure tensors F X ( f ⊗ f ) = F X ( f ) ⊗ f . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 29
Lemma 5.1.
For f ∈ S ( X ( F )) one has F X ( L ′ ( h ) f )( g ) = | λ ( h ) | L ′ ( h ) F X ( f ) λ ( h ) − I λ ( h ) I ! g ! . Proof.
This is shown in the proof of [GL19b, Lemma 4.3]. It is also a consequence of Lemma3.8 and (5.1.7). (cid:3) As H acts on Y, e Y i , we also let L ( h ) denote the induced action on the corresponding spacesof functions. The following is [GL19b, Lemma 4.3]: Lemma 5.2.
For f ∈ S ( X ( F ) × V ( F )) and h ∈ H ( F ) one has L ( h ) I ( f ) = | λ ( h ) | − I ( L ′ ( h ) f ) ,I ( F X ( L ′ ( h ) f )) = | λ ( h ) | P i =1 d i / L (cid:18) hλ ( h ) (cid:19) I ( F X ( f )) . (cid:3) Lemma 5.3.
For v ∈ V ◦ ( F ) we have I ( L ′ ( h ) f )( v ) = | λ ( h ) | d / L ( h ) I ( f )( v , v , λ ( h ) v ) I ( F X ( L ′ ( h ) f ))( v ) = | λ ( h ) | d + d ) / L (cid:18) hλ ( h ) (cid:19) I ( F X ( f ))( v , v , λ ( h ) − v ) . Proof.
We prove the lemma when f = f ⊗ f is a pure tensor. The general case merelyrequires more confusing notation. We have I ( L ′ ( h ) f ⊗ L ( h ) f )( v )= Z N ( F ) \ G ( F ) f λ ( h ) − λ ( h ) Λ( h ) − g Λ( h ) ρ ( g ) L ( h ) f ( v ) dg = | λ ( h ) | Z N ( F ) \ G ( F ) f ( g ) ρ λ ( h ) 1 1 λ ( h ) − Λ( h ) g Λ( h ) − L ( h ) f ( v ) dg = | λ ( h ) | d / χ ( λ ( h )) L ( h ) I ( f ⊗ f )( v , v , λ ( h ) v ) . Here we have used (5.1.2), and χ is the quadratic character attached to the quadratic form Q as in [GL19b, § χ ( λ ( h )) = 1 [GL19b, Lemma 3.2], this proves the equationfor I . The equation for I ( F X ( L ′ ( h ) f )) is computed similarly using Lemma 5.1. (cid:3) Lemma 5.4.
Given Φ ∈ S ( F ) , define Φ λ ,λ ( c, d ) = Φ( λ c, λ d ) , λ , λ ∈ F × . For ≤ i ≤ , we have I i ( L ′ ( h ) f ⊗ Φ) = | λ ( h ) | L ( h ) I i ( f ⊗ Φ λ ( h ) , ) I i ( F X ( L ′ ( h ) f ) ⊗ Φ) = | λ ( h ) | P i =1 d i / L (cid:18) hλ ( h ) (cid:19) I i ( F X ( f ) ⊗ Φ ,λ ( h ) − ) . Proof.
For any i , we have the identity λ ( h ) − λ ( h ) − λ ( h ) − γ i = λ ( h ) − λ ( h ) λ ( h ) 1 λ ( h ) − γ i Λ( h ) − ∈ [ P, P ]( F ) γ i Λ( h ) − . (5.1.9)We assume that f = f ⊗ f is a pure tensor; the general case merely requires more confusingnotation. We prove the given identities for i = 1; the i = 2 and i = 3 cases follow bysymmetry. We first prove the identity for I ( L ′ ( h ) f ⊗ L ( h ) f ⊗ Φ). For v ∈ e Y ( F ), using(5.1.9) and then (5.1.2) we have I ( L ′ ( h ) f ⊗ L ( h ) f ⊗ Φ)( v, s )= Z G γ ( F ) \ G ( F ) f (cid:0) γ Λ( h ) − g Λ( h ) (cid:1) × Z N ( F ) \ SL ( F ) Z F × ρ (( I , g ′ , ( − ) g ′ ( − )) g ) L ( h ) f ( v )Φ((0 , x ) g ′ p ( g )) | x | s d × xdg ′ dg = | λ ( h ) | Z G γ ( F ) \ G ( F ) f ( γ g ) × Z N ( F ) \ SL ( F ) Z F × ρ (cid:0) Λ( h ) (cid:0) I , (cid:0) λ ( h ) − (cid:1) g ′ (cid:0) λ ( h ) (cid:1) , ( − ) (cid:0) λ ( h ) − (cid:1) g ′ (cid:0) λ ( h ) (cid:1) ( − ) (cid:1) g Λ( h ) − (cid:1) × L ( h ) f ( v )Φ((0 , x ) g ′ p (Λ( h ) g Λ( h ) − )) | x | s d × xdg ′ dg = | λ ( h ) | Z G γ ( F ) \ G ( F ) f ( γ g ) × Z N ( F ) \ SL ( F ) Z F × L ( h ) ρ (( I , g ′ , ( − ) g ′ ( − )) g ) f ( v )Φ((0 , x ) (cid:0) λ ( h ) (cid:1) g ′ p ( g ) (cid:0) λ ( h ) − (cid:1) ) | x | s d × xdg ′ dg = | λ ( h ) | L ( h ) I ( f ⊗ f ⊗ Φ λ ( h ) , )( v, s ) . The identities involving F X follow similarly; we observe that for 1 ≤ i ≤ λ ( h ) − λ ( h ) − λ ( h ) − γ i = λ ( h ) 1 λ ( h ) − λ ( h ) − λ ( h ) γ i λ ( h ) − λ ( h ) − λ ( h ) − λ ( h ) λ ( h ) λ ( h ) Λ( h ) − SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 31 ∈ [ P, P ]( F ) γ i λ ( h ) − λ ( h ) − λ ( h ) − λ ( h ) λ ( h ) λ ( h ) Λ( h ) − and use Lemma 5.1. (cid:3) The Schwartz space of Y . Since Y ani ( F ) consists of finitely many open H ( F )-orbits,it follows from lemmas 3.9 and 5.2 that I ( f ) is smooth on Y ani ( F ) for all f ∈ S ( X ( F ) × V ( F )).In propositions 8.1 and 9.3, we will show that I ( f ) in fact extends to a smooth function on Y sm ( F ). With this in mind, we define S ( Y ( F )) := Im( I : S ( X ( F ) × V ( F )) → C ∞ ( Y sm ( F ))) . (5.2.1)This is the Schwartz space of Y ( F ). We observe that the first equality of Lemma 5.2implies in particular that the natural action of H ( F ) on C ∞ ( Y sm ( F )) preserves S ( Y ( F )). Lemma 5.5.
Let F be an archimedean local field. The kernel of the map I : S ( X ( F ) × V ( F )) −→ C ∞ ( Y sm ( F )) is closed.Proof. By the Cauchy-Schwartz inequality, | I ( f ⊗ f ) | ( y ) is bounded by Z G γ ( F ) \ G ( F ) | f | ( γ g ) max( | γ g | , N dg ! / Z G γ ( F ) \ G ( F ) max( | γ g | , − N | ρ ( g ) f | ( y ) dg ! / for any N ≥
0. Now G γ ( F ) \ G ( F ) is dense in X P ( F ) and hence the right-invariant positiveRadon measure on X P ( F ) agrees with dg up to a positive real constant. Thus the left integralis equal to Z X P ( F ) | f | ( g ) max( | g | , N dg. Using the decomposition of the measure dg as in the proof of Proposition 3.12, we see thatthis is bounded by k f k + c ν Id ,N +1 , ( f ) by Lemma 3.5, where Id is the identity in U ( g ),and c is a positive constant independent of f . By the Iwasawa decomposition the secondintegral equals Z ( F × ) × F max( m ( t, a ) , − N Z K | ρ ( k ) f | ( a − y ) dk Y i =1 | a i | − d i ! d × adt. Here m ( t, a ) = max( | ta a a | , | a | , | a | , | a | , | a − a a | , | a − a a | , | a − a a | )(5.2.2)(see the proof of [GL19b, Proposition 7.1] for more details). Taking N sufficiently large andapplying Lemma 9.2 in the special case D = Id, r = e i = 0 we deduce that the linear form f I ( f )( y ) is continuous for every y ∈ Y sm ( F ). The kernel in the statement of the lemmais the intersection of the kernels of these continuous linear forms. (cid:3) We endow S ( Y ( F )) = S ( X ( F ) × V ( F )) / ker I with the quotient topology (which is Fr´echet).The integrals I ( f ) depend on the choice of additive character ψ used to define the Weilrepresentation ρ ψ . We write I ψ ( f )(5.2.3)for I ( f ) defined using the Weil representation ρ ψ . Lemma 5.6.
Let c ∈ F × and ψ c ( x ) := ψ ( cx ) . Then I ψ c ( f ⊗ f )( y ) = γ ( Q , ψ c ) γ ( Q , ψ ) | c | − P i =1 d i / I ψ (cid:18) L (cid:18) c I c − I (cid:19) R ( cI I ) f ⊗ f (cid:19) ( cy ) . In particular, the Schwartz space S ( Y ( F )) is independent of the choice of ψ .Proof. Let B ≤ SL be the Borel subgroup of upper triangular matrices and let w = (( − ) , ( − ) , ( − )) ∈ SL ( F ) . Since N ( F ) w B ( F ) is dense in SL ( F ), we have I ψ c ( f ⊗ f )( y )= Z F × F × ( F × ) f ( γ ( t ) w ( x ) ( a a − )) ρ ψ c (( t ) w ( x ) ( a a − )) f ( y ) dtdxd × a. Observe that ρ ψ c (( t ) w ( x ) ( a a − )) f ( y )= ψ ( ct Q ( y )) γ ( Q , ψ c ) Z V ( F ) ρ ψ c (( x ) ( a a − )) f ( u ) Y i =1 ψ ( cu ti J i y i ) du i = ψ ( c − t Q ( cy )) γ ( Q , ψ c ) Z V ( F ) ρ ψ ( a a − ) f ( u ) Y i =1 ψ ( cx i Q i ( u i )) ψ ( u ti J i cy i ) du i = | c | P i =1 d i / γ ( Q , ψ c ) γ ( Q , ψ ) − ρ ψ ( (cid:0) c − t (cid:1) w ( cx ) ( a a − )) f ( cy ) . Here γ ( Q , ψ ) := Q i =1 γ ( Q i , ψ ) is the product of the Weil indices and J i is the matrix of Q i .The factor of | c | P i =1 d i / appears because we have to renormalize the self-dual Haar measureswith respect to ψ c so that they are self-dual with respect to ψ . Taking a change of variables t ct , x i c − x i , we see that I ψ c ( f ⊗ f )( y ) is | c | P i =1 d i / γ ( Q , ψ c ) γ ( Q , ψ ) − times | c | − Z F × F × ( F × ) f ( γ ( ct ) w (cid:0) c − x (cid:1) ( a a − )) ρ ψ (( t ) w ( x ) ( a a − )) f ( cy ) dtdxd × a = | c | − Z F × F × ( F × ) f ( γ (cid:0) c − (cid:1) ( t ) w ( x ) ( a a − ) ( c )) ρ ψ (( t ) w ( x ) ( a a − )) f ( cy ) dtdxd × a SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 33 = | c | − I ψ (cid:18) L (cid:18) c I c − I (cid:19) R ( cI I ) f ⊗ f (cid:19) ( cy ) . The fact that the Schwartz space is preserved now follows from Lemma 3.8 and Lemma5.2. (cid:3)
For F archimedean or nonarchimedean, let S := Im( S ( V ( F )) −→ C ∞ ( Y sm ( F )))where the implicit map is restriction of functions. We observe that C ∞ c ( Y sm ( F )) < S . Moreover, we have the following result:
Lemma 5.7.
One has S < S ( Y ( F )) .Proof. For f ∈ S ( V ( F )), choose Φ i ⊗ f i ∈ C ∞ c ( G ( F )) ⊗ S ( V ( F )) for 1 ≤ i ≤ n such that n X i =1 Z G ( F ) Φ i ( g ) ρ ( g ) f i dg = f. In the nonarchimedean case it is clear that this is possible, and in the archimedean case itfollows from a well-known theorem of Dixmier-Malliavin. Then for y ∈ Y ( F ) one has n X i =1 Z G γ ( F ) \ G ( F ) Z G γ ( F ) Φ i ( ng ) dn ! ρ ( g ) f i ( y ) dg = f ( y ) . (5.2.4)Let γ G ( F ) ∼ = G γ ( F ) \ G ( F ) be the orbit of γ in X ( F ). Since C ∞ c ( X P ( F )) < S ( X ( F ))(see (3.1.6)) we have C ∞ c ( γ G ( F )) < S ( X ( F )) and hence C ∞ c ( γ G ( F )) ⊗ S ( V ( F )) < S ( X ( F ) × V ( F )) . (5.2.5)Thus the lemma follows from (5.2.4). (cid:3) We now revert to the adelic setting, bearing in mind the conventions on Schwartz spacesexplained in §
2. Let F be a number field. The obvious global analogue of (5.0.1) yields amap S ( Y ( A F )) = Im( I : S ( X ( A F ) × V ( A F )) → C ∞ ( Y sm ( A F ))) . To check that this is well-defined, one uses the computation of the basic function for Y F v b Y,v := I ( b X,v ⊗ V ( O v ) )(5.2.6)in Proposition 7.1 below. We define S ( Y ( F ∞ )) := Im( I : S ( X ( F ∞ ) × V ( F ∞ )) → C ∞ ( Y sm ( F ∞ )))The map I has closed kernel by a trivial modification of the proof of Lemma 5.5 and we give S ( Y ( F ∞ )) the quotient topology. We then have S ( Y ( A F )) := S ( Y ( F ∞ )) ⊗ O v ∤ ∞ ′ S ( Y ( F v ))where the restricted direct product is taken with respect to the b Y,v . The summation formula
Our goal in this section is to prove our main summation formula, Theorem 6.1 (stated inthe introduction as Theorem 1.3), modulo some convergence statements that we prove laterin the paper. We require the following assumptions on f = f ⊗ f ∈ S ( X ( A F ) × V ( A F )):There are finite places v , v of F (not necessarily distinct) such that f = f v f v f v v and f v ∈ C ∞ c ( X P ( F v )) , F X ( f v ) ∈ C ∞ c ( X P ( F v ))(6.0.1) ρ ( g ) f ( v ) = 0 for v V ◦ ( F ), for all g ∈ G ( A F ).(6.0.2)Assume momentarily that f = f v f v for some place v of F (not necessarily distinct from v and v ) and that ρ ( w ) f v ( v ) = 0 for v V ◦ ( F v ), for all w ∈ Γ , (6.0.3)where Γ is the subgroup of SL ( F ) generated by ( − ). Then by definition of the Weilrepresentation f satisfies (6.0.2).We will also require that Φ ∈ S ( A F ) satisfies b Φ(0) = 2Vol( F × \ ( A × F ) ) − , where b Φ( x, y ) = Z A F Φ( t , t ) ψ ( xt + yt ) dt dt (6.0.4)and ( A × F ) < A × F is the subgroup of ideles of norm 1. For simplicity we let F X : S ( X ( A F ) × V ( A F )) −→ S ( X ( A F ) × V ( A F ))be the automorphism given on pure tensors by F X ( f ⊗ f ) = F X ( f ) ⊗ f .The main summation formula proved in this paper is the following: Theorem 6.1.
Assume that ( f = f ⊗ f , Φ) ∈ S ( X ( A F ) × V ( A F )) × S ( A F ) where f satisfies (6.0.1) and (6.0.2) , and b Φ(0) = 2Vol( F × \ ( A × F ) ) − . One has X ξ ∈ Y ( F ) I ( f )( ξ ) + X ξ ∈ Y ( F ) I ( f )( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( f ⊗ Φ)( ξ, s )= X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( F X ( f ) ⊗ Φ)( ξ, s ) . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 35
Here I ( f ) ( ξ ) = Z G γ ( A F ) \ G ( A F ) f ( γ g ) ρ ( g ) f ( ξ ) dg , ∀ ξ ∈ Y sm ( A F ) ,I ( f ) ( ξ ) = Z N ( A F ) \ G ( A F ) f ( g ) ρ ( g ) f ( ξ ) dg , ∀ ξ ∈ Y ( A F ) ,I i ( f ⊗ Φ)( ξ, s ) = Z G γi ( A F ) \ G ( A F ) f ( γ i g ) × Z N ( A F ) \ SL ( A F ) Z F × ρ (∆ i ( h ) g ) f ( v )Φ( x (0 , hp i ( g )) | x | s dx × dhdg , ∀ ξ ∈ e Y i ( A F )(6.0.5)where ∆ i is defined as in (5.0.2) and G γ is the stabilizer of γ . In the remainder of the paper,when we speak of boundary terms , we mean the summands in Theorem 6.1 involving I and I i .We will prove this theorem in this section assuming the absolute convergence statementsgiven in propositions 10.2 and 10.5. We will indicate precisely when these propositions areused below. After this section, much of the remainder of the paper is devoted to provingthese convergence statements.Computing formally one has Z G ( F ) \ G ( A F ) X γ ∈ X ( F ) f ( γg )Θ f ( g ) dg = X γ ∈ X ( F ) /G ( F ) Z G γ ( F ) \ G ( A F ) f ( γg )Θ f ( g ) dg = X a Z G γa ( A F ) \ G ( A F ) f ( γ a g ) Z [ G γa ] Θ f ( g g ) dg dg (6.0.6)where the sum is over a set of representatives for X P ( F ) /G ( F ). This set has 5 elementsrepresented by γ i , 0 ≤ i ≤ G γ a are givenexplicitly by Lemma 4.2, and we will use this lemma without further comment below.We start with the γ contribution. It is computed as in the proof of [GL19b, Theorem5.3]: Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [ G γ ] Θ f ( g g ) dg dg = X ξ ∈ Y ( F ) I ( f )( ξ ) . Strictly speaking, the proof of [GL19b, Theorem 5.3] assumed f was finite under a maximalcompact subgroup of Sp ( F ∞ ), but the same proof is valid given our work in § We now turn to the Id term. Using the definition of the Weil representation, we have thatthis term is Z G Id ( A F ) \ G ( A F ) f ( g ) Z [ G m × G m ] X ξ ∈ V ( F ) Q ( ξ )= Q ( ξ )= Q ( ξ )=0 ρ (cid:16)(cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17) , (cid:0) ( a a ) − a a (cid:1)(cid:17) g (cid:17) f ( ξ ) da × da × dg = Z G Id ( A F ) \ G ( A F ) f ( g ) Z A × F × A × F X ξ ∈ e Y ( F ) / ( F × ) ρ (cid:16)(cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17) , (cid:0) ( a a ) − a a (cid:1)(cid:17) g (cid:17) f ( ξ ) da × da × dg. Here ( F × ) acts as in (4.0.7).Thus using Lemma 4.1 we conclude that the above is equal to Z G Id ( A F ) \ G ( A F ) f ( g ) Z A × F × A × F X ξ ∈ Y ( F ) ρ (cid:16)(cid:16)(cid:16) a a − (cid:17) , (cid:16) a a − (cid:17) , (cid:0) ( a a ) − a a (cid:1)(cid:17) g (cid:17) f ( ξ ) da × da × dg = Z N ( A F ) \ G ( A F ) f ( g ) X ξ ∈ Y ( F ) ρ ( g ) f ( ξ ) dg = X ξ ∈ Y ( F ) I ( f )( ξ ) , where N ≤ GL is the unipotent radical of the standard Borel subgroup of upper triangularmatrices. This formal computation is justified by Proposition 10.2.We finally turn to the γ i , 1 ≤ i ≤
3, terms. Let Φ ∈ S ( A F ) be a function satisfying b Φ(0) = 2Vol( F × \ ( A × F ) ) − . We prove in Proposition 10.5 below that the sum X ξ ∈ Y i ( F ) I i ( f ⊗ Φ)( ξ, s )(6.0.7)converges absolutely and defines a holomorphic function of s for Re( s ) sufficiently large.Moreover, we show that it admits a meromorphic continuation to the s plane and its residueat s = 1 is Z G γi ( A F ) \ G ( A F ) f ( γ i g ) Z [ G γi ] X ξ ∈ V ( F ) ρ ( hg ) f ( ξ ) dhdg. Thus altogether we have shown that Z G ( F ) \ G ( A F ) X γ ∈ X ( F ) f ( γg )Θ f ( g ) dg = X ξ ∈ Y ( F ) I ( f )( ξ ) + X ξ ∈ Y ( F ) I ( f )( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( f ⊗ Φ)( ξ, s ) . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 37
On the other hand by Theorem 3.15 Z G ( F ) \ G ( A F ) X γ ∈ X ( F ) f ( γg )Θ f ( g ) dg = Z G ( F ) \ G ( A F ) X γ ∈ X ( F ) F X ( f )( γg )Θ f ( g ) dg. Replacing f by F X ( f ) in the argument above we see that this is X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + X ξ ∈ Y ( F ) I ( F X ( f ))( ξ ) + Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( F X ( f ) ⊗ Φ)( ξ, s ) . Thus assuming the absolute convergence statements in propositions 10.2 and 10.5 we haveproven Theorem 6.1.
Corollary 6.2.
Under the assumptions of Theorem 6.1, for h ∈ H ( A F ) we have X ξ ∈ Y ( F ) L ( h ) I ( f )( ξ ) + | λ ( h ) | − d / X ξ ∈ Y ( F ) L ( h ) I ( f )( ξ , ξ , λ ( h ) ξ )+ | λ ( h ) | − Res s =1 3 X i =1 X ξ ∈ Y i ( F ) L ( h ) I i ( f ⊗ Φ)( ξ, s )= | λ ( h ) | − P i =1 d i / X ξ ∈ Y ( F ) L (cid:18) hλ ( h ) (cid:19) I ( F X ( f ))( ξ )+ | λ ( h ) | − d / d / X ξ ∈ Y ( F ) L (cid:18) hλ ( h ) (cid:19) I ( F X ( f ))( ξ , ξ , λ ( h ) − ξ )+ | λ ( h ) | − P i =1 d i / Res s =1 3 X i =1 X ξ ∈ Y i ( F ) L (cid:18) hλ ( h ) (cid:19) I i ( F X ( f ) ⊗ Φ)( ξ, s ) . Proof.
If (6.0.1) is valid for f , then it is valid for L ′ ( h ) f by Lemma 5.1. If (6.0.2) is validfor f , then it is valid for L ( h ) f by (5.1.2). Thus by Lemma 5.2 and Theorem 6.1, we have | λ ( h ) | X ξ ∈ Y ( F ) L ( h ) I ( f ⊗ f )( ξ ) = X ξ ∈ Y ( F ) I ( L ′ ( h ) f ⊗ L ( h ) f )( ξ )= X ξ ∈ Y ( F ) I ( F X ( L ′ ( h ) f ) ⊗ L ( h ) f )( ξ )+ X ξ ∈ Y ( F ) ( I ( F X ( L ′ ( h ) f ) ⊗ L ( h ) f )( ξ ) − I ( L ′ ( h ) f ⊗ L ( h ) f )( ξ ))+ Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( F X ( L ′ ( h ) f ) ⊗ L ( h ) f ⊗ Φ)( ξ, s ) − Res s =1 3 X i =1 X ξ ∈ Y i ( F ) I i ( L ′ ( h ) f ⊗ L ( h ) f ⊗ Φ)( ξ, s ) . Therefore by lemmas 5.2, 5.3, and 5.4, we have X ξ ∈ Y ( F ) L ( h ) I ( f )( ξ ) + | λ ( h ) | − d / X ξ ∈ Y ( F ) L ( h ) I ( f )( ξ , ξ , λ ( h ) ξ )+ Res s =1 3 X i =1 X ξ ∈ Y i ( F ) L ( h ) I i ( f ⊗ Φ λ ( h ) , )( ξ, s )= X ξ ∈ Y ( F ) | λ ( h ) | − P i =1 d i / L ( λ ( h ) − h ) I ( F X ( f ))( ξ )+ | λ ( h ) | − d / d / L ( λ ( h ) − h ) X ξ ∈ Y ( F ) I ( F X ( f ))( ξ , ξ , λ ( h ) − ξ )+ Res s =1 3 X i =1 X ξ ∈ Y i ( F ) | λ ( h ) | − P i =1 d i / L ( λ ( h ) − h ) I i ( F X ( f ) ⊗ Φ ,λ ( h ) − )( ξ, s ) . (6.0.8)We now observe that Theorem 6.1 is still valid (with the same proof) if the Φ on theFourier transform side of the equality is replaced by any Φ ′ ∈ S ( A F ) satisfying b Φ ′ (0) =2Vol( F × \ ( A × F ) ) − . Thus in (6.0.8) we now replace Φ with | λ ( h ) | Φ ,λ ( h ) and | λ ( h ) | − Φ λ ( h ) − , in I i ( f ⊗ Φ λ ( h ) , ) and I i ( F X ( f ) ⊗ Φ ,λ ( h ) − ) respectively. Observing that | λ ( h ) | L ( h ) I i ( f ⊗ Φ λ ( h ) ,λ ( h ) )( ξ, s ) = | λ ( h ) | − s +1 L ( h ) I i ( f ⊗ Φ)( ξ, s )and | λ ( h ) | − L (cid:18) hλ ( h ) (cid:19) I i ( F X ( f ) ⊗ Φ λ ( h ) − ,λ ( h ) − )( ξ, s ) = | λ ( h ) | s − L (cid:18) hλ ( h ) (cid:19) I i ( F X ( f ) ⊗ Φ)( ξ, s )we deduce the corollary. (cid:3)
Remark.
There is an ostensible asymmetry in the argument of the I function in Corollary6.2, but it is illusory. To be more precise, | λ ( h ) | d / I ( f )( h − ( ξ , ξ , λ ( h ) ξ ))= | λ ( h ) | d / Z N ( A F ) \ G ( A F ) f ( g ) ρ ( g ) f ( h − ( ξ , ξ , λ ( h ) ξ )) dg = Z N ( A F ) \ G ( A F ) f ( g ) ρ (cid:16)(cid:16) I , I , (cid:16) λ ( h ) λ ( h ) − (cid:17)(cid:17) g (cid:17) f ( h − ( ξ , ξ , ξ )) dg = Z N ( A F ) \ G ( A F ) f ( g ) ρ (cid:16)(cid:16)(cid:16) λ ( h ) λ ( h ) − (cid:17) , I , I (cid:17) g (cid:17) f ( h − ( ξ , ξ , ξ )) dg = | λ ( h ) | d / I ( f )( h − ( λ ( h ) ξ , ξ , ξ )) . Here we have taken a change of variables g (cid:16)(cid:16) λ ( h ) λ ( h ) − (cid:17) , I , (cid:16) λ ( h ) − λ ( h ) (cid:17)(cid:17) g. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 39
Similarly | λ ( h ) | d / I ( f )( h − ( ξ , ξ , λ ( h ) ξ )) = | λ ( h ) | d / I ( f )( h − ( ξ , λ ( h ) ξ , ξ )) . The unramified calculation
For this section, F is a local field of residual characteristic p with ring of integers O thatis unramified over Q p . We let ψ : F → C × be an unramified nontrivial character, and weassume that χ Q is unramified and V ( O ) is fixed under ρ ( K ) where K = SL ( O ). Let c := [ P,P ]( F ) ̟ − c I ̟ c I Sp ( O ) . We also let ≥ α = ∞ X c = α c . We recall that the basic function for X , by definition, is b X := ∞ X j,k =0 q j k +2 j . In this section we compute the unramified functions I ( b X ⊗ V ( O ) )( v ) , I i ( b X ⊗ V ( O ) ⊗ O )( v, s )where 1 ≤ i ≤
3. In each case we will compute these integrals and then give bounds on them.Technically speaking, the bounds should be proved first to ensure the absolute convergenceof the integrals with which we are working, but we feel that giving the formal computationfirst and then proving absolute convergence makes the argument easier to follow.7.1.
The open orbit.
Let χ Q ( a , a , a ) := χ Q ( a ) χ Q ( a ) χ Q ( a )(7.1.1)where χ Q i is the (quadratic) character attached to Q i as in [GL19b, § b Y := I ( b X ⊗ V ( O ) )(7.1.2)given by [GL19b, Proposition 6.3]: Proposition 7.1.
For v ∈ Y sm ( F ) , one has b Y ( v ) = ∞ X j =0 Z O (cid:18) Q ( v ) a a a ̟ j (cid:19) V ( O ) (cid:16) va̟ j (cid:17) χ Q ( ̟ j a ) Y i =1 (cid:18) | a i | q j (cid:19) − d i / d × a where the integral is over a , a , a ∈ O satisfying max( | a − a a | , | a − a a | , | a − a a | ) ≤ . (cid:3) The identity orbit.Proposition 7.2.
Suppose v = ( v , v , v ) ∈ e Y ( F ) . Assume moreover that | v | = | v | = 1 .One has I ( b X ⊗ V ( O ) )( v ) = ∞ X k =0 ∞ X j =0 q j Z V ( O ) ( a − v ) χ Q ( a ) Y i =1 | a i | − d i / d × a i where the integral is over those a − , a − , a ∈ O such that | a a a | = q − k − j . As a function of v the integral is supported in V ( O ) .Proof. Let T ≤ G be the maximal torus of diagonal matrices. We use the Iwasawa decom-position to write dg = dndtdkδ P ∩ G ( t )where dn, dt and dk are Haar measures on N ( F ), T ( F ) and K . We assume that K andits intersections with the other subgroups here have measure 1. Then we obtain I ( b X ⊗ V ( O ) ) ( v ) = Z N ( F ) \ G ( F ) b X ( g ) ρ ( g ) V ( O ) ( v ) dg = Z ( F × ) b X (cid:0) a − a (cid:1) ρ (cid:0) a − a (cid:1) V ( O ) ( v ) Y i =1 | a i | d × a i = Z ( F × ) b X (cid:0) a − a (cid:1) V ( O ) ( a − v ) χ Q ( a ) Y i =1 | a i | − d i / d × a i = Z ( F × ) ∞ X k =0 ∞ X j =0 q j k +2 j (cid:0) a − a (cid:1) V ( O ) ( a − v ) χ Q ( a ) Y i =1 | a i | − d i / d × a i = ∞ X k =0 ∞ X j =0 q j Z V ( O ) ( a − v ) χ Q ( a ) Y i =1 | a i | − d i / d × a i , where the integral is over those a ∈ ( F × ) such that | a a a | = q − k − j . We now employ our assumption that v and v have at least one entry that is a unit in O × .Then the integral is supported in the set of a such that a − , a − ∈ O , a ∈ O . It follows in particular that as a function of v the integral is supported in V ( O ). (cid:3) Here is the corresponding bound:
SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 41
Lemma 7.3.
Let ǫ > . For v ∈ V ◦ ( F ) with | v | = | v | = 1 , we have that Z N ( F ) \ G ( F ) | b X ( g ) ρ ( g ) V ( O ) ( v ) | dg ≤ C | v | − d / − ǫ V ( O ) ( v ) for some constant C > depending on ǫ , which equals for q sufficiently large.Proof. Arguing as in the proof of Proposition 7.2 we have Z N ( F ) \ G ( F ) | b X ( g ) ρ ( g ) V ( O ) ( v ) | dg = ∞ X k =0 ∞ X j =0 q j Z V ( O ) ( a − v ) Y i =1 | a i | − d i / d × a i (7.2.1)where the integral is over those a ∈ ( F × ) such that | a a a | = q − k − j , and a − , a − ∈ O , a ∈ O . The above is ∞ X k =0 ∞ X j =0 ∞ X k =0 ∞ X k =0 V ( O ) ( ̟ − k − j − k − k v ) q j + k (2 − d / k (2 − d / − ( k +2 j + k + k )(2 − d / . This expression vanishes unless v ∈ V ( O ). The largest that k + 2 j + k + k can beis the minimal valuation of an entry in v . We therefore obtain a bound of (ord( v ) +1) | v | − d / . (cid:3) The other orbits.Proposition 7.4.
Suppose v = ( v , v , v ) ∈ e Y ( F ) . For Re( s ) > one has I ( b X ⊗ V ( O ) ⊗ O )( v, s )= ζ (2 s ) ∞ X j =0 Z O (cid:18) Q ( v ) a a a (cid:19) V ( O ) (cid:18) v ̟ j a , v a , v a (cid:19) χ Q ( a ) | a | s q j (1 − d /
2) 3 Y i =1 | a i | − d i / d × a i where the integral is over a ∈ F × ∩ O , a , a ∈ F × such that | a | − ≥ max (cid:0) | a a − | , | a − a | (cid:1) . (7.3.1)An expression for the integrals I and I can be obtained by symmetry. Proof.
To evaluate this we begin by decomposing the Haar measure on SL ( F ) using theIwasawa decomposition: dh = dn dtdkδ B ( t ) . Here dn , dt and dk is a Haar measure on N ( F ) , T ( F ) and K . Moreover B ≤ SL is theBorel subgroup of upper triangular matrices and the measures are normalized so that theintersection of each subgroup with K has measure 1. Then I ( b X ⊗ V ( O ) ⊗ O )( v, s ) = Z F × × (∆ (SL )( F ) \ SL ( F ) × SL ( F )) b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , g (cid:17)(cid:17) × Z N ( F ) \ SL ( F ) Z F × ρ (cid:16)(cid:16) a − a (cid:17) , ( h, ( − ) h ( − )) g ) (cid:17) V ( O ) ( v ) × O (cid:16) (0 , x ) hp (cid:16)(cid:16) a − a (cid:17) , g (cid:17)(cid:17) | x | s d × xdh | a | da × dg = Z F × × SL ( F ) b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , g, I (cid:17)(cid:17) × Z N ( F ) \ SL ( F ) Z F × ρ (cid:16)(cid:16) a − a (cid:17) , hg, ( − ) h ( − ) (cid:17) V ( O ) ( v ) O ((0 , x ) hg ) | x | s d × xdh | a | da × dg. We bring the integral over N ( F ) \ SL ( F ) outside the outer integral and take a change ofvariables g h − g to arrive at Z N ( F ) \ SL ( F ) Z F × × SL ( F ) b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , h − g, I (cid:17)(cid:17) × Z F × ρ (cid:16)(cid:16) a − a (cid:17) , g, ( − ) h ( − ) (cid:17) V ( O ) ( v ) O ((0 , x ) g ) | x | s d × x | a | da × dg ! dh. Since ( I , ∆ (SL )( F )) is contained in the stabilizer of γ this is equal to Z N ( F ) \ SL ( F ) Z F × × SL ( F ) b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , g, ( − ) h ( − ) (cid:17)(cid:17) × Z F × ρ (cid:16)(cid:16) a − a (cid:17) , g, ( − ) h ( − ) (cid:17) V ( O ) ( v ) O ((0 , x ) g ) | x | s d × x | a | da × dg ! dh = Z ( F × ) × F b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17)(cid:17) × Z F × ρ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17) V ( O ) ( v ) O (0 , xa ) | x | s d × xdt Y i =1 | a i | d × a i = Z ( F × ) × F × F × b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17)(cid:17) × ψ ( t Q ( v )) V ( O ) ( a − v ) O ( xa ) χ Q ( a ) | x | s d × xdt Y i =1 | a i | − d i / d × a i = ζ (2 s ) Z ( F × ) × F b X (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17)(cid:17) × ψ ( t Q ( v )) V ( O ) ( a − v ) χ Q ( a ) | a | s dt Y i =1 | a i | − d i / d × a i . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 43
Here we have used the fact that χ Q = 1. Applying the definition of the basic function wearrive at ζ (2 s ) Z ( F × ) × F ∞ X j,k =0 q j k +2 j (cid:16) γ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17)(cid:17) (7.3.2) × ψ ( t Q ( v )) V ( O ) ( a − v ) χ Q ( a ) | a | s dt Y i =1 | a i | − d i / d × a i . To proceed further we recall that γ := ∗ − and hence (cid:12)(cid:12)(cid:12) γ (cid:16)(cid:16) a − a (cid:17) , ( t ) (cid:16) a − a (cid:17) , (cid:16) a − a (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ a a a a − − a − a t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max (cid:0) | a | , | a a a − | , | a a − a | , | a a a t | (cid:1) . Note that Z | a a a t |≤ q − k − j ψ ( t Q ( v )) dt = 1 | a a a | q k +2 j O (cid:18) Q ( v ) ̟ k +2 j a a a (cid:19) . Thus (7.3.2) is equal to ζ (2 s ) ∞ X j,k =0 Z q k O (cid:18) Q ( v ) ̟ k +2 j a a a (cid:19) V ( O ) ( a − v ) χ Q ( a ) | a | s Y i =1 | a i | − d i / d × a i − ζ (2 s ) ∞ X j,k =0 Z q k +1 O (cid:18) Q ( v ) ̟ k +2 j +1 a a a (cid:19) V ( O ) ( a − v ) χ Q ( a ) | a | s Y i =1 | a i | − d i / d × a i , where the first integral is over a , a , a ∈ F × such that q − k − j ≥ max (cid:0) | a | , | a a a − | , | a a − a | (cid:1) , and the second unmarked integral is over a , a , a ∈ F × such that q − k − j − ≥ max (cid:0) | a | , | a a a − | , | a a − a | (cid:1) . Exploiting cancellation in the sum over k , we see that this is ζ (2 s ) ∞ X j Z O (cid:18) Q ( v ) ̟ j a a a (cid:19) V ( O ) ( a − v ) χ Q ( a ) | a | s Y i =1 | a i | − d i / d × a i where the integral is over q − j ≥ max (cid:0) | a | , | a a a − | , | a a − a | (cid:1) . We change variables a ̟ j a to see that this is ζ (2 s ) ∞ X j =0 Z O (cid:18) Q ( v ) a a a (cid:19) V ( O ) (cid:18) v ̟ j a , v a , v a (cid:19) χ Q ( a ) | a | s q j (1 − d /
2) 3 Y i =1 | a i | − d i / d × a. Here the integrals are over 1 ≥ max( | a | , | a a a − | , | a a − a | ). (cid:3) Recall that the V i ( F ) are just F d i and hence come equipped with the standard basis. Wedefine ord( v ⊗ v ) , (resp . | v ⊗ v | )(7.3.3)to be the minimum of the orders (resp. maximum of the norms) of the entries of v ⊗ v withrespect to the natural induced basis on V ( F ) ⊗ V ( F ), etc. We will use obvious variants ofthis notation below. Lemma 7.5.
For v = ( v , v , v ) ∈ V ◦ ( F ) and r > , the integral Z ( N ( F ) × ∆ (SL )( F )) \ G ( F ) | b X ( γ g ) |× Z N ( F ) \ SL ( F ) | ρ (( I , h, ( − ) h ( − )) g ) V ( O ) ( v ) | O ((0 , x ) hp ( g )) | x | r d × xdhdg vanishes unless v ∈ V ( O ) . It is bounded by Cζ (2 r ) ζ (2 r + d / − | v | − d / | v | − d / max ( | v ⊗ v | , | v | ) − d / − r × ((max(0 , ord( v ⊗ v ) − ord( v )) + ζ (2 r + d / − for some constant C > which equals for q sufficiently large. Thus if Re( s ) = r thefunction I ( b X ⊗ V ( O ) ⊗ O )( v, s ) admits the same bounds on its magnitude and support.Proof. Arguing as in the proof of Proposition 7.4 we have that the integral in the lemma isequal to ζ (2 r ) ∞ X j,k =0 Z q k V ( O ) ( a − v ) | a | − r Y i =1 | a i | − d i / d × a i where the integral is over a , a , a ∈ F × such that q − k − j ≥ max (cid:0) | a | , | a a a − | , | a a − a | (cid:1) . Taking a change of variables a ̟ k +2 j a we see that this is equal to ζ (2 r ) ∞ X j,k =0 Z q ( k +2 j )( d / − q k V ( O ) (cid:18) v a ̟ k +2 j , v a , v a (cid:19) | a | − r Y i =1 | a i | − d i / d × a i (7.3.4) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 45 where the integral is over a ∈ O ∩ F × , a , a ∈ F × such that1 ≥ max( | a a a − | , | a a − a | ) . Thus as a function of v the integral is supported in v ∈ V ( O ). We also deduce that if a isin the support of the integral for a given v then | v | ≤ | a | ≤ | a | − | a | ≤ | v | − | a | . Thus (7.3.4) is bounded by ζ (2 r ) times(1 + ord( v )) | v | − d / Z | v |≤| a |≤| v | − | a | , | v |≤| a | | a | − d / | a | − d / − r d × a d × a ≤ (1 + ord( v )) | v | − d / | v | − d / Z max( | v || v | , | v | ) ≤| a | (1 + ord( v ⊗ v ) − ord( a )) | a | − d / − r d × a = (1 + ord( v )) | v | − d / | v | − d / max( | v || v | , | v | ) − d / − r × Z ≤| a | (1 + ord( v ⊗ v ) − min(ord( v ⊗ v ) , ord( v )) − ord( a )) | a | − d / − r d × a = (1 + ord( v )) | v | − d / | v | − d / max( | v || v | , | v | ) − d / − r × (cid:0) max (0 , ord( v ⊗ v ) − ord( v )) ζ (2 r + d / −
1) + ζ (2 r + d / − (cid:1) . (cid:3) Bounds on integrals in the nonarchimedean case
In this section F is a characteristic zero nonarchimedean local field, and we fix( f = f ⊗ f , Φ) ∈ S ( X ( F ) × V ( F )) × S ( F ) . We bound the integrals attached to these functions that appeared in the proof of Theorem6.1. These bounds will be used to deduce the absolute convergence statements of propositions10.2 and 10.5 below. All implicit constants in this section are allowed to depend on f ⊗ Φ.First we pause to justify an assertion made in § Proposition 8.1.
We have S ( Y ( F )) ⊂ C ∞ ( Y sm ( F )) . Proof.
Fix v = ( v , v , v ) ∈ Y sm ( F ) and let v ′ ∈ Y sm ( F ). By symmetry we can and doassume that | v || v | 6 = 0. It suffices to show Z G γ ( F ) \ G ( F ) | f ( γ g ) || ρ ( g ) f ( v ) − ρ ( g ) f ( v ′ ) | dg = 0for | v − v ′ | sufficiently small. We can and do choose κ v ∈ R > such that if | v − v ′ | < κ v then | v ′ i | = | v i | for 2 ≤ i ≤
3, and | v | = | v ′ i | if v = 0. For the remainder of the proof we assume | v − v ′ | < κ v . Using the Schwartz inequality and Lemma 3.5, the integral is bounded by k f k Z G γ ( F ) \ G ( F ) ≥ c ( γ g ) | ρ ( g ) f ( v ) − ρ ( g ) f ( v ′ ) | dg ! / . Using the Iwasawa decomposition, it suffices to show for all c ∈ Z that Z m ( t,a ) ≤ q − c (cid:18)Z K | ρ ( k ) f ( a − v ) − ψ ( − t Q ( v ) + t Q ( v ′ )) ρ ( k ) f ( a − v ′ ) | dk (cid:19) Y i =1 | a i | − d i ! d × adt is zero for | v − v ′ | sufficiently small, where m ( t, a ) is defined as in (5.2.2). When v = 0the integral is supported in the set of a i such that | v i | q − N ≤ | a i | ≤ q − c for each i for some N depending on f . When v = 0 the integral is supported in the set of a i such that | v i | q − N ≤ | a i | ≤ q − c for i = 2 ,
3, and q − N + c | v || v | ≤ q c | a || a | ≤ | a | ≤ q − c . In either case,the support of the integral, as a function of a , lies in a compact set independent of v ′ (since | v − v ′ | < κ v ). Thus the integral over t has support in a set that is independent of v ′ . Inparticular, if | v − v ′ | is sufficiently small, then ψ ( − t Q ( v ) + t Q ( v ′ )) = 1, so the integral aboveis Z m ( t,a ) ≤ q − c (cid:18)Z K | ρ ( k ) f ( a − v ) − ρ ( k ) f ( a − v ′ ) | dk (cid:19) Y i =1 | a i | − d i ! d × adt Since the vector space h ρ ( k ) f i k ∈ K is finite dimensional, and the integral over a is supportedin a compact set independent of v ′ (since | v − v ′ | < κ v ), for v ′ close enough to v the integralabove vanishes. (cid:3) Proposition 8.2.
For v ∈ V ◦ ( F ) , one has Z N ( F ) \ G ( F ) | f ( g ) ρ ( g ) f ( v ) | dg ≪ Y i =1 | v i | − d i / . The integral is supported in the set of v satisfying | v ⊗ v ⊗ v | ≪ . The function I ( f )( v ) satisfies the same bounds on its magnitude and support.Proof. We decompose the Haar measure as in the proof of Proposition 7.2 to see that theintegral in the proposition is equal to Z N ( F ) \ G ( F ) | f ( g ) ρ ( g ) f ( v ) | dg = Z ( F × ) × K (cid:12)(cid:12) f (cid:0)(cid:0) a − a (cid:1) k (cid:1) ρ (cid:0)(cid:0) a − a (cid:1) k (cid:1) f ( v ) (cid:12)(cid:12) Y i =1 | a i | d × a i ! dk = Z ( F × ) × K (cid:12)(cid:12) f (cid:0)(cid:0) a − a (cid:1) k (cid:1) ρ ( k ) f ( a − v ) (cid:12)(cid:12) Y i =1 | a i | − d i / ! d × a i dk. (8.0.1)Now (cid:12)(cid:12)(cid:0) a − a (cid:1) k (cid:12)(cid:12) = | a a a | . (8.0.2) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 47 and f itself satisfies the bound | f ( g ) | ≪ f | g | − by Lemma 3.5. Moreover by the same lemma the support of f (cid:0)(cid:0) a − a (cid:1) k (cid:1) as a function of a is contained in the set of a satisfying | a a a | ≪ . Thus (8.0.1) is bounded by a constant times
Z e f ( a − v ) Y i =1 | a i | − d i / d × a i , (8.0.3)where e f ( v ) := Z K | ρ ( k ) f ( v ) | dk, (8.0.4)and the integral is over a ∈ ( F × ) such that | a a a | ≪
1. Therefore (8.0.3) is bounded bya constant times Y i =1 Z | a i |≫| v i | | a i | − d i / d × a i ≪ Y i =1 | v i | − d i / . We bound the support of (8.0.3) as a function of v as follows. Since e f is compactly supportedwe have that | v i | ≪ f | a i | for 1 ≤ i ≤
3. Since | a a a | ≪ | v || v || v | ≪ (cid:3) Proposition 8.3.
For r > , as a function of v ∈ V ◦ ( F ) , the integral Z ( N ( F ) × ∆ (SL )( F )) \ G ( F ) | f ( γ g ) |× Z N ( F ) \ SL ( F ) Z F × | ρ (( I , h, ( − ) h ( − )) g ) f ( v )Φ( x (0 , hp ( g )) || x | r d × xdhdg has support in | v | ≪ . It is bounded by a constant times ζ (2 r ) ζ (2 r + d / − | v | − d / | v | − d / max ( | v ⊗ v | , | v | ) − d / − r × ( C + max(0 , ord( v ⊗ v ) − ord( v )) + ζ (2 r + d / − for some constant C ≥ . If r = Re( s ) , the function I ( f ⊗ Φ)( v, s ) satisfies the same boundson its magnitude and support.Proof. Decomposing the Haar measure and arguing as in the proof of Proposition 7.4, wesee that the integral equals Z ( F × ) × F × K × F × (cid:12)(cid:12)(cid:12) f (cid:16) γ (cid:16)(cid:16) a − a (cid:17) k , ( t ) (cid:16) a − a (cid:17) k , (cid:16) a − a (cid:17) k (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) ρ (cid:16)(cid:16) a − a (cid:17) k , ( t ) (cid:16) a − a (cid:17) k , (cid:16) a − a (cid:17) k (cid:17) f ( v )Φ((0 , xa ) k ) (cid:12)(cid:12)(cid:12) × | x | r | a a a | d × xdtda × da × da × dk dk dk . By Lemma 3.5 one has | f ( g ) | ≪ f | g | − and that the support of f ( g ) is contained in theset of g satisfying | g | ≪ f
1. Now, as in the proof of Proposition 7.4 (cid:12)(cid:12)(cid:12) γ (cid:16)(cid:16) a − a (cid:17) k , ( t ) (cid:16) a − a (cid:17) k , (cid:16) a − a (cid:17) k (cid:17)(cid:12)(cid:12)(cid:12) = max( | a | , | a a a − | , | a a − a | , | a a a t | )=: m ′ ( t, a ) . (8.0.5)Taking a change of variable x xa − , for some n ∈ Z the integral above is bounded by aconstant times q nr ζ (2 r ) Z ( F × ) × Fm ′ ( t,a ) ≪ m ′ ( t, a ) − e f ( a − v ) dt | a | − r Y i =1 | a i | − d i / d × a i where e f is defined as in (8.0.4). For some N ∈ Z ≥ sufficiently large we can write theintegral here as ∞ X k = − N q k Z e f ( a − v ) dt | a | − r Y i =1 | a i | − d i / d × a i where the integral is over t, a such that m ′ ( t, a ) = q − k . This is bounded by ∞ X k = − N Z q k e f ( a − v ) | a | − r Y i =1 | a i | − d i / d × a i . Here the integral is now over a such that m ′ ( a ) ≤ q − k where m ′ ( a ) := max( | a | , | a a a − | , | a a − a | ) . Taking a change of variables a ̟ k a one arrives at ∞ X k = − N Z q kd / e f (cid:18) v ̟ k a , v a , v a (cid:19) | a | − r Y i =1 | a i | − d i / d × a i where the integral is now over a , a , a such that1 ≥ max( | a | , | a a a − | , | a a − a | ) . From this we deduce the bound on the support and the bound on its magnitude just as inthe proof of Lemma 7.5. (cid:3)
SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 49 Bounds on integrals in the archimedean case
In this section F is an archimedean local field. We estimate the local integrals defined in §
5. The bounds obtained in this section will be used to prove propositions 10.2 and 10.5, theabsolute convergence statements used in the proof of Theorem 6.1. As usual, the bounds inthe archimedian case are slightly harder to prove than in the nonarchimedian case, but thebasic outline of the proofs is the same. We fix( f = f ⊗ f , Φ) ∈ S ( X ( F ) × V ( F )) × S ( F )The following is a rephrasing of [GL19a, Lemma 8.1]: Lemma 9.1.
Let
A, B ∈ R > , C ∈ R ≥ and let x ∈ F × . Assume A > B and A = B + C .One has Z F × max( | a − x | , − A | a | − B max( | a | , − C da × ≪ A,B,C max( | x | , − min( A − B,C ) | x | − B . (cid:3) This will be used several times below.9.1.
The open orbit.Lemma 9.2.
Given r, e i , N ∈ R ≥ , D ∈ U (Lie( V ( F ))) , let M : ( F × ) × F × K × V ( F ) → R be the function M ( a, t, k, v ) := max( m ( t, a ) , − N | t | r | Dρ ( k ) f | ( a − v ) Y i =1 | a i | − d i − e i ! , (9.1.1) where m ( t, a ) is defined as (5.2.2) . For v ∈ V ′ ( F ) , there is a compact neighborhood U of v ,and a continuous integrable function M ′ on ( F × ) × F × K such that for all v ′ ∈ UM ( a, t, k, v ′ ) ≤ M ′ ( a, t, k ) . Moreover, given N i ∈ Z ≥ there exists a continuous seminorm ν ′ on S ( V ( F )) such that Z ( F × ) × F × K M ( a, t, k, v ) dkd × adt ≤ ν ′ ( f ) ( Q i =1 max( | v i | , − N i | v i | − d i − e i − r , if v ∈ V ◦ ( F ) , Q i = j max( | v i | , − N i | v i | − d i − e i − r − d j − e j , if v j = 0 , provided N ≥ i { N i , d i + e i + r } .Proof. By the continuity of Weil representation and compactness of K , for any C , C , C ∈ Z ≥ , there exists a continuous seminorm ν D,C ,C ,C on S ( V ( F )) such that for all ( k, v ) ∈ ( F × ) × K × V ( F ) we have | Dρ ( k ) f | ( v ) ≤ ν D,C ,C ,C ( f ) Y i =1 max( | v i | , − C i ! . Let U be a compact neighborhood of v such that for v ′ ∈ U, if v i = 0 then v ′ i = 0. Choose v ′ ∈ U with minimum norm. Put M ′ ( a, t, k ) := ν D,C ,C ,C ( f ) max( m ( t, a ) , − N | t | r Y i =1 max( | a − i v ′ i | , − C i | a i | − d i − e i . Then M ( a, t, k, v ) ≤ M ′ ( a, t, k ) for all v ∈ U . Thus to prove the lemma it suffices to showthat for all v ∈ V ′ ( F ) one has Z ( F × ) × F max( m ( t, a ) , − N | t | r Y i =1 max( | a − i v i | , − C i | a i | − d i − e i d × a i ! dt ≤ ( Q i =1 max( | v i | , − N i | v i | − d i − e i − r , if v ∈ V ◦ ( F ); Q i = j max( | v i | , − N i | v i | − d i − e i − r − d j − e j , if v j = 0 , (9.1.2)provided that N ≥ i { N i , d i + e i + r } . We break the integral into m ( t, a ) ≤ m ( t, a ) >
1. Suppose v ∈ V ◦ ( F ). In the range m ( t, a ) ≤
1, we have | t | ≤ | a a a | − and | a | ≤
1. Therefore the integral is bounded by Z | a |≤ Y i =1 max( | a − i v i | , − C i | a i | − d i − e i − r d × a ≤ Y i =1 Z F × max( | a i | , − N i max( | a − i v i | , − C i | a i | − d i − e i − r d × a i . (9.1.3)In the range m ( t, a ) >
1, applying the inequality(9.1.4) m ( t, a ) ≥ max( | ta a a | ,
1) max( | a | ,
1) max( | a | ,
1) max( | a | , , the contribution of this part of the integral is bounded by Z ( F × ) × F max( | ta a a | , − N/ | ta a a | r Y i =1 max( | a i | , − N/ max( | a − i v i | , − C i | a i | − d i − e i − r ! d × adt ≪ N,r Y i =1 Z F × max( | a i | , − N/ max( | a − i v i | , − C i | a i | − d i − e i − r d × a i , provided N > r + 2. Since N ≥ i N i by assumption the integral is bounded by (9.1.3).The assertion then follows from Lemma 9.1 by setting A = 2 C i , B = r + d i + e i − , C = 2 N i and choosing C i so that A − B > C for each i .Now assume v ∈ V ′ ( F ) − V ◦ ( F ). By symmetry we may assume v = 0. If m ( t, a ) ≤ | t | ≤ | a a a | − , | a | ≤ , | a a | ≤ | a | . Therefore the contribution of | m ( t, a ) | ≤ Z | a |≤ , | a a |≤| a | | a | − d − e − r Y i =2 max( | a − i v i | , − C i | a i | − d i − e i − r d × a d × a d × a SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 51 ≤ Y i =2 Z F × max( | a i | , − N i max( | a − i v i | , − C i | a i | − d i − e i − r − d − e d × a i . (9.1.5)Applying the inequality m ( t, a ) ≥ max( | ta a a | ,
1) max( | a | ,
1) max( | a | ,
1) max( | a | ,
1) max( | a − a a | , m ( t, a ) > N times Z ( F × ) max( | a − a a | , − N/ Y i =1 max( | a i | , − N/ max( | a − i v i | , − C i | a i | − d i − e i − r ! d × a (9.1.6)since N > r + 5. In the range | a a | ≤ | a | , the integral is dominated by (9.1.5) since N ≥ i N i . In the range | a a | ≥ | a | , one has that Z | a |≤| a a | max( | a | , − N/ | a | − d − e − r +2 N/ d × a ≪ N,e ,r min( | a a | , − d − e − r +2 N/ . Since 2 N/ ≥ d + e + r , the integral (9.1.6) is also dominated by (9.1.5). The assertionnow follows from Lemma 9.1 by setting A = 2 C i , B = 2 r + d + e + d i + e i − , C = 2 N i and choosing C i so that A − B > C for i = 2 , (cid:3) Proposition 9.3.
We have S ( Y ( F )) < C ∞ ( Y sm ( F )) . Moreover, for f ∈ S ( Y ( F )) and D ∈ U (Lie( V ( F ))) , | Df ( v ) | Y i =1 max( | v i | , N i | v i | ( d i +deg D − / ! is bounded on Y sm ( F ) for all N i ∈ Z .Proof. Let v ∈ Y sm ( F ) and D ∈ U (Lie( V ( F ))). Let ∆ : F → F be the diagonal embedding.Using the notation of Lemma 9.2 there is a neighborhood U of v such that the expression | f ( γ (cid:0) t )1 (cid:1) (cid:0) a − a (cid:1) k ) Dρ ( (cid:0) t )1 (cid:1) (cid:0) a − a (cid:1) k ) f ( v ) | (9.1.7)is dominated by a finite sum of functions of the form | f ( γ (cid:0) t )1 (cid:1) (cid:0) a − a (cid:1) k ) | max( m ( t, a ) , N M ( a, t, k, v ) / | a | − where M ( a, t, k ) is defined using various parameters f , e i , r depending on D . We recall that m ( t, a ) = | γ (cid:0) t )1 (cid:1) (cid:0) a − a (cid:1) k | by (5.2.2). Thus by the Leibniz integral rule, to prove S ( Y ( F )) < C ∞ ( Y sm ( F )) it sufficesto check that Z F × ( F × ) × K | f ( γ (cid:0) t )1 (cid:1) (cid:0) a − a (cid:1) k ) max( m ( t, a ) , N | M ( a, t, k, v ) / | a | − | a | d × adtdk ≤ Z G γ ( F ) \ G ( F ) | f ( γ g ) max( | γ g | , N | dg ! / (cid:18)Z F × ( F × ) × K M ( a, t, k, v ) d × adtdk (cid:19) / converges for N sufficiently large (here we have used the Cauchy-Schwartz inequality). Theleft integral converges by the argument in the proof of Lemma 5.5, and the right convergesby Lemma 9.2. To obtain the bound one simply keeps track of which parameters r and e i are required in the argument above in terms of deg D . (cid:3) Remark.
By mimicking the proof above one can also bound Df ( v ) when v i = 0 for some i .9.2. The identity orbit.Proposition 9.4.
Let v ∈ V ◦ ( F ) . Given a positive integer N ′ and ǫ > , there are contin-uous seminorms ν on S ( X ( F )) and ν ′ (depending on N ′ , ǫ ) on S ( V ( F )) such that one hasthe bound Z N ( F ) \ G ( F ) | f ( g ) ρ ( g ) f ( v ) | dg ≤ ν ( f ) ν ′ ( f ) max {| v || v || v | , } − N ′ Y i =1 | v i | − d i / − ǫ . The function I ( f )( v ) admits the same bound.Proof. By symmetry, we may assume d ≥ d ≥ d . Recall the seminorms ν D,N,β mentionedin Lemma 3.5. Arguing as in the proof of Proposition 8.2, we see that the integral in theproposition is bounded by max( ν Id ,N, ( f ) , ν Id , , ( f )) times Z ( F × ) | a a a | − max( | a a a | , − N e f ( a − v ) Y i =1 | a i | − d i / d × a i , where e f is defined as in (8.0.4). By the continuity of Weil representation and compactness of K , for any N , N , N ∈ Z ≥ , there exists a continuous seminorm ν ′ depending on N , N , N such that the integral is bounded by ν ′ ( f ) times Z ( F × ) max( | a a a | , − N Y i =1 max( | a − i v i | , − N i | a i | − d i / da × i = Z ( F × ) max( | a | , − N max( | a − ( a a ) v | , − N | a | − d / | a a | d / Y i =2 max( | a − i v i | , − N i | a i | − d i / da × i Here we have taken a change of variables a ( a a ) − a . For the remainder of the proofall implicit constants are allowed to depend on N , N , N , N, and we assume N i − d i / > N i − − d i − / i where N = d = 0. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 53
Taking
N > N + d / A = N , B = d / C = N tothe a integral we see that the above is bounded by Z ( F × ) max( | a a v | , − N + d / | a a v | − d / Y i =2 max( | a − i v i | , − N i | a i | ( d − d i ) / d × a i = | v | − d / Z ( F × ) max( | a v | , − N + d / | a | − d / × max( | a − a v | , − N max( | a − v | , − N | a | ( d − d ) / da × da × . (9.2.2)Here we have taken a change of variables a a − a . The integral Z F × max( | a − a v | , − N max( | a − v | , − N | a | ( d − d ) / da × (9.2.3)breaks into the sum of four integrals Z | a |≤ min (cid:16) | a || v | , | v | (cid:17) + Z min (cid:16) | a || v | , | v | (cid:17) < | a |≤ | a || v | + Z | a || v | < | a |≤ max (cid:16) | a || v | , | v | (cid:17) + Z max (cid:16) | a || v | , | v | (cid:17) ≤| a | and this is bounded by a constant (depending only on N , N ) times | v | − N min (cid:16) | a || v | , | v | (cid:17) N +( d − d ) / + G d ,d ( a , v , v ) + (cid:18) | a || v | (cid:19) N × (cid:18) | v | − N (cid:18) max (cid:16) | a || v | , | v | (cid:17) N − N +( d − d ) / − (cid:16) | a || v | (cid:17) N − N +( d − d ) / (cid:19) + max (cid:16) | a || v | , | v | (cid:17) − N +( d − d ) / (cid:19) where G d ,d ( a , v , v ) = (cid:16) | a || v | (cid:17) ( d − d ) / − min (cid:16) | a || v | , | v | (cid:17) ( d − d ) / , if d = d ;log (cid:16) | a || v | (cid:17) − log min (cid:16) | a || v | , | v | (cid:17) , if d = d . Thus (9.2.3) is bounded by a constant times F ( a , v , v ) := | v | ( d − d ) / (cid:16) | a || v || v | (cid:17) N , if | a | < | v || v | ; | v | ( d − d ) / (cid:16) | a || v || v | (cid:17) ( d − d ) / , if | a | ≥ | v || v | and d = d (cid:16) | a || v || v | (cid:17) , if | a | ≥ | v || v | and d = d . (9.2.4)Here we have used our assumption (9.2.1).Thus the original integral (9.2.2) is bounded by a constant times | v | − d / Z F × max( | a v | , − N + d / | a | − d / F ( a , v , v ) da × = Y i =1 | v i | − d i / ! Z F × max( | a | c, − N + d / | a | − d / min( | a | , N F ′ ( a ) da × , where c = | v || v || v | , and F ′ ( a ) = ( max( | a | , ( d − d ) / , if d = d ;log(max( | a | , , if d = d . Here we have changed variables a a ( | v || v | ). The assertion of the proposition nowfollows from taking a change of variable a a − and applying Lemma 9.1 with A = N − d / , B = ǫ < / , C = N − d / − ǫ . (cid:3) The other orbits.Proposition 9.5.
Let r = Re( s ) > and N ∈ Z > , and assume N > max(2 r + d / − , d / − . For v ∈ V ◦ ( F ) , there are continuous seminorms ν on S ( X ( F )) and ν ′ dependingon N on S ( V ( F )) such that Z ( N ( F ) × ∆ (SL )( F )) \ G ( F ) | f ( γ g ) |× Z N ( F ) \ SL ( F ) Z F × | ρ (( I , h, ( − ) h ( − )) g ) f ( v )Φ((0 , x ) hp ( g )) || x | r d × xdhdg ≤ Ψ(2 r ) ν ( f ) ν ′ ( f ) | v | − N − d / max ( | v | , | v | ) − r − d / − d / where Ψ : R > → R is an analytic function. The function I ( f ⊗ Φ)( v, s ) admits the samebound.Proof. By Lemma 3.5 one has | f ( g ) | ≤ ν Id , N , ( f ) | g | − − N for any N ≥
0. Thus arguing asin Proposition 8.3, we see that the integral is bounded by ν Id ,N ( f ) Z ( F × ) × F m ′ ( t, a ) − − N (cid:18)Z K × F × (cid:12)(cid:12) ρ ( k , k , k ) f ( a − v )Φ((0 , x ) k ) (cid:12)(cid:12) | x | r d × xdk dk dk (cid:19) × dt | a | − r Y i =1 | a i | − d i / d × a i , (9.3.1)where m ′ ( t, a ) = max( | a | , | a a a − | , | a a − a | , | a a a t | ). For simplicity we assume f = ⊗ i =1 f i . The general case merely requires more annoying notation. Using Schwartz inequalityon the second copy of K , the inner integral is bounded by Y i =1 , Z K | ρ i ( k i ) f i ( a − i v i ) | dk i ! (cid:18)Z K | ρ ( k ) f ( a − v ) | dk (cid:19) / Z F × (cid:18)Z K | Φ((0 , x ) k ) | dk (cid:19) / | x | r d × x. The last term is Ψ(2 r ) for an appropriate analytic function Ψ : R > → R . By the continuityof the Weil representation and compactness of K , for any N , N , N ∈ Z > , there exists acontinuous seminorm ν ′ N ,N ,N on S ( V ( F )) such that the integral (9.3.1) is bounded by ν ′ N ,N ,N ( f ) Z ( F × ) × F m ′ ( t, a ) − N − | a | − r Y i =1 max( | a − i v i | , − N i | a i | − d i / da × dt. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 55
From now on all implicit constants are allowed to depend on N , N , N , N . Let m ′′ ( t, a ) := max(1 , | a a − | , | a − a | , | a a t | ) = max( | a a − | , | a − a | , | a a t | ) . We assume without loss of generality that N > N + d /
2. We then write the above integralas the product of Z F × | a | − N − d / max( | a − v | , − N da × ≪ | v | − N − d / (9.3.2)and Z ( F × ) × F m ′′ ( t, a ) − N − | a | − r Y i =2 max( | a − i v i | , − N i | a i | − d i / da × da × dt. (9.3.3)Now consider (9.3.3). We write it as the sum of Z | a |≥| a | | a | − N − d / − r max( | a − v | , − N (9.3.4) × Z F max( | a − | , | a t | ) − N − max( | a − v | , − N | a | − d / da × da × dt and Z | a | > | a | | a | − N − d / max( | a − v | , − N (9.3.5) × Z F max( | a − | , | a t | ) − N − max( | a − v | , − N | a | − d / − r da × da × dt. Executing the t integral in (9.3.4), we see that it is bounded by a constant times Z | a |≥| a | | a | − N − d / − r max( | a − v | , − N max( | a − v | , − N | a | N − d / da × da × = Z ≥| a | | a | − d / − d / − r max( | a − v | , − N max( | a − a − v | , − N | a | N − d / da × da × , (9.3.6)where the latter equation is obtained by taking a change of variables a a a . Similarly(9.3.5) is bounded by a constant times Z | a | > | a | − N − d / max( | a − a − v | , − N max( | a − v | , − N | a | − d / − d / − r da × da × (9.3.7)Carrying out the integral over a directly in (9.3.6) we see that it is bounded by a constanttimes Z F × | a | − d / − d / − r max( | a − v | , − N max( | a − v | , − N da × . (9.3.8) provided N − d / >
0. This bound is also valid for (9.3.7) provided N + d / > N .The integral above is bounded by a constant timesmax ( | v | , | v | ) − r − d / − d / provided N i > r + d / d / − i = 2 , (cid:3) Absolute convergence
In this section, we prove the absolute convergence statements that make the proof of thesummation formula in § F. For the remainder of the section, we fix( f = f ⊗ f , Φ) ∈ S ( X ( A F ) × V ( A F )) × S ( A F ) . All implicit constants are allowed to depend on f ⊗ Φ. For y i ∈ V ◦ i ( A F ), we let | y i | := Y v | y i | v . Lemma 10.1.
Let / > ǫ > and a finite set of places S containing the infinite placesbe given. For y ∈ V ◦ ( A F ) such that | y | v = | y | v = 1 for all v S , there exists a Schwartzfunction Ψ ∈ S (( V ⊗ V ⊗ V )( A F )) (depending on S, ǫ ) such that Z ( N ) ( A F ) \ G ( A F ) | f ( g ) ρ ( g ) f ( y ) | dg ≤ Ψ( y ⊗ y ⊗ y ) Y i =1 | y i | − d i / − ǫ . (10.0.1) The function I ( f ) satisfies the same bound.Proof. Let S ′ ⊃ S be a finite set of places including the infinite places such that f S ′ = b S ′ X , f S ′ = V ( b O S ′ ) , ψ S is unramified, F v is absolutely unramified for v S , and V ( b O S ′ ) is fixed by ρ (SL ( b O S ′ )). Then by Lemma 7.3, Proposition 8.2, and Proposition 9.4, given a sufficientlylarge positive integer N the integral is bounded by a constant depending on N, ǫ times Y v |∞ max( | y ⊗ y ⊗ y | v , − N Y i =1 | y i | − d i / − ǫv × Y v | S ′ −∞ | y ⊗ y ⊗ y | − ǫv Y i =1 | y i | − d i / v Y v S ′ | y | − ǫ | y | − d / v ! Ψ ∞ ( y ⊗ y ⊗ y )for some Ψ ∞ ∈ S (( V ⊗ V ⊗ V )( A ∞ F )). (cid:3) Proposition 10.2.
The sum X ξ ∈ Y ( F ) Z N ( A F ) \ G ( A F ) | f ( g ) ρ ( g ) f ( ξ ) | dg, is finite. SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 57
Proof.
Let { a j ⊂ O : 1 ≤ j ≤ k } be a set of representatives for the ideal classes of O . For every ξ i ∈ V i ( F ) we can choosean α ∈ F × such that αξ i ∈ V i ( O ) and the greatest common denominator gcd( αξ i ) of thecoefficients of αξ i is a j for some 1 ≤ j ≤ k . Using this observation, we see that the sum inthe proposition is bounded by a constant times k X j ,j =1 X ξ ∈ V ( O ) × V ( O ) × V ◦ ( F ) / ( O × ) gcd( v )= a j , gcd( v )= a j Z N ( A F ) \ G ( A F ) | f ( g ) ρ ( g ) f ( ξ ) | dg. Here ( O × ) < ( F × ) acts via the action (4.0.7). Thus it suffices to fix a pair of ideals b and b and prove convergence of the sum X ξ ∈ V ( O ) × V ( O ) × V ◦ ( F ) / ( O × ) gcd( ξ )= b , gcd( ξ )= b Z N ( A F ) \ G ( A F ) | f ( g ) ρ ( g ) f ( ξ ) | dg. Let S be a finite set of places including the infinite places such that b i b O S = b O S for each i .Then by Lemma 10.1, there exists β ∈ F × and Ψ ∈ S (( V ⊗ V ⊗ V )( A F )) such that thesum above is bounded by X ξ ∈ V ◦ ( O ) × V ◦ ( O ) × β − V ◦ ( O ) / ( O × ) gcd( ξ )= b , gcd( ξ )= b Ψ( ξ ⊗ ξ ⊗ ξ ) Y i =1 | ξ i | − d i / − ǫ ≤ X ξ ∈ β − ( V ⊗ V ⊗ V )( O ) Ψ( ξ ) . Here we have used the fact that, by the product rule, | ξ i | ≥ . (10.0.2)for ξ i ∈ V ◦ i ( F ). (cid:3) Lemma 10.3.
Let constants c > / > ǫ > be given. For / ǫ < r ≤ c and an integer N > max( d / d / − , r + d / d / − there exists a Schwartz function Ψ ∈ S ( V ( A F )) (depending on ǫ, c ) such that Z G γ ( A F ) \ G ( A F ) | f ( γ g ) | Z N ( A F ) \ SL ( A F ) (cid:12)(cid:12) ρ (( I , h, ( − ) h ( − )) g ) f ( y )Φ((0 , x ) hp ( g )) | x | r (cid:12)(cid:12) d × xdhdg ≤ Ψ( y ) | y | − N | y | − d / − ǫ Y v max ( | y | v , | y | v ) − r − d / ǫ . The function I ( f ⊗ Φ)( y, s ) defines a holomorphic function of s in the strip + ǫ < Re( s ) < c for each y and admits the same bound with r = Re( s ) . Proof.
Let S be a finite set of places including the infinite places such that f S = b SX , f S = V ( b O S ) is fixed by ρ (SL ( b O S )) and Φ S = ( b O ) S . Assume moreover that ψ v is unramifiedand F v is absolutely unramified for v S . Using Lemma 7.5 and propositions 8.3 and 9.5,for any given integers N > N > max( d / d / − , r + d / d / − ∞ ∈ S ( V ( A ∞ F )) and a positive constant C depending on ǫ and c , such that the integral isbounded by a constant depending on N , N, ǫ, c timesΨ ∞ ( y ) Y v |∞ max( | y | v , − N | y | − Nv max ( | y | v , | y | v ) − r − d / − d / × Y v ∤ ∞ ζ v (2 r ) ζ v (2 r + d / − | y | − d / v | y | − d / v max ( | y ⊗ y | v , | y | v ) − d / − r × ( C v + max(ord v ( y ⊗ y ) − ord v ( y ) ,
0) + ζ v (2 r + d / − C v ∈ { C, } and C v = 0 for almost all v . Since 1 / ǫ < r , we have C v + max(ord v ( y ⊗ y ) − ord v ( y ) ,
0) + ζ v (2 r + d / − ≪ min( | y ⊗ y | v / | y | v , − ǫ ζ v (2 r + d / − | y | − ǫv | y | − ǫv max( | y ⊗ y | v , | y | v ) ǫ ζ v (2 r + d / − . for all finite v . Here the implied constant is equal to 1 for q v sufficiently large in a senseindependent of y . Thus (10.0.3) is bounded by a constant depending on c and ε timesΨ ∞ ( y ) Y v |∞ max( | y | v , − N | y | − Nv max ( | y | v , | y | v ) − r − d / − d / × Y v ∤ ∞ | y | − d / − ǫv | y | − d / − ǫv max ( | y ⊗ y | v , | y | v ) − d / − r + ǫ . (10.0.4)For a finite place v , if Φ ∨ ( y ) = 0 then | y | v ≤ C ′ v for some constant C ′ v ≥
1, which is 1 foralmost all v , and hencemax ( | y ⊗ y | v , | y | v ) ≥ C ′− v | y | v max( | y | v , | y | v ) . Thus (10.0.4) is bounded by a constant timesΨ ∞ ( y ) Y v |∞ max( | y | v , − N | y | − Nv max ( | y | v , | y | v ) − r − d / − d / × Y v ∤ ∞ | y | − d / − d / − rv | y | − d / − ǫv max ( | y | v , | y | v ) − d / − r + ǫ . (10.0.5)The desired inequality follows from max( | y | v , | y | v ) ≥ | y | v . (cid:3) To study the sum of the boundary terms involving I i , we require the following: Lemma 10.4.
Let Λ ⊂ O × be a finite index subgroup. There is a constant C Λ ≥ dependingon Λ such that for all α ∈ F ×∞ , there exists u ∈ Λ such that C − ( N α ) [ Fv : R ][ F : Q ] < | uα | v < C Λ ( N α ) [ Fv : R ][ F : Q ] SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 59 for all v |∞ .Proof. Let r , r be the number of real and complex places of F respectively. Letlog ∞ : F ×∞ −→ R r + r t (log | t | v ) v |∞ be the usual logarithm map and let L := { x ∈ R r + r | X v |∞ x v = 0 } be the usual trace zero hyperplane. Then log ∞ ( α ) − log ∞ (cid:16) ( N α ) F : Q ] (cid:17) ∈ L . By Dirichlet’sunit theorem and the fact that Λ ⊂ O × is of finite index, one can choose u ∈ Λ so thatlog ∞ ( u ) + log ∞ ( α ) − log ∞ (cid:16) ( N α ) F : Q ] (cid:17) lies in a fundamental domain for the full rank latticelog ∞ (Λ) ⊂ L . It follows that there is a constant c Λ ≥ α ∈ F ×∞ , thereexists u ∈ Λ such that − c Λ < log | u | v + log | α | v − log (cid:16) ( N α ) [ Fv : R ][ F : Q ] (cid:17) < c Λ for all v |∞ . Taking exponentials implies the statement of the lemma. (cid:3) Proposition 10.5. If f satisfies (6.0.2) and r ≫ , the sum X ξ ∈ Y ( F ) Z G γ ( A F ) \ G ( A F ) | f ( γ g ) |× Z N ( A F ) \ SL ( A F ) Z A × F | ρ (( I , h, ( − ) h ( − )) g ) f ( ξ )Φ((0 , x ) hp ( g )) | | x | r d × xdhdg is finite. Therefore, X ξ ∈ Y ( F ) I ( f ⊗ Φ)( ξ, s ) defines a holomorphic function for Re( s ) ≫ . Moreover, it extends to a meromorphicfunction of C , holomorphic except for possible simple poles at s = 0 and s = 1 . One has Res s =1 X ξ ∈ Y ( F ) I ( f ⊗ Φ)( ξ, s )= Vol( F × \ A × F ) ) b Φ(0)2 Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [ G γ ] X ξ ∈ V ( F ) ρ ( hg ) f ( ξ ) dhdg. and Z G γ ( A F ) \ G ( A F ) | f ( γ g ) | Z [ G γ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ξ ∈ V ( F ) ρ ( hg ) f ( ξ ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dg (10.0.6) is absolutely convergent. The corresponding assertions for the integrals I ( f ⊗ Φ) and I ( f ⊗ Φ) are valid by symmetry.
Proof.
We use the notation introduced at the beginning of Proposition 10.2. For each ξ ∈ ( V × V )( F ) there exists an α ∈ F × such that αξ ∈ ( V × V )( O ) and the greatest commondivisor gcd( αξ ) of the coefficients of αξ is a j for some 1 ≤ j ≤ k . Then the sum in theproposition is bounded by k X j =1 X ( ξ ,ξ ) ∈ V ( F ) × ( V ◦ × V ◦ )( F ) ∩ ( V × V )( O ) / O × gcd( αξ )= a j Z G γ ( A F ) \ G ( A F ) | f ( γ g ) |× Z N ( A F ) \ SL ( A F ) Z A × F | ρ (( I , h, ( − ) h ( − )) g ) f ( ξ , ξ )Φ((0 , x ) hp ( g )) | | x | r d × xdhdg. Here O × acts diagonally via scaling on V ( O ) × V ( O ). It suffices to fix a = a j and prove theconvergence of the corresponding summand. Choose ǫ sufficiently small. By Lemma 10.3and (10.0.2), there exists Φ ∈ S ( V ( A F )) such that the sum is bounded by a constant times X ( ξ ,ξ ) ∈ V ( F ) × ( V ◦ × V ◦ )( F ) ∩ ( V × V )( O ) / O × )gcd( ξ )= a Ψ( ξ ) | ξ | − d / − r + ǫ . To show the sum is finite for r ≫
1, it suffices to show X ξ ∈ ( V ◦ × V ◦ )( F ) ∩ ( V × V )( O ) / ( O × ) Y v |∞ | ξ | − rv is finite for r ≫
1. By Lemma 10.4, all but finitely many classes in
O ∩ F × / O × admitrepresentatives α such that | α | v ≥ v |∞ , and thus the sum is bounded by a constanttimes X ξ ∈ ( V × V )( O ) Y v |∞ max( | ξ | v , − r which is finite for r ≫
1. This completes the proof of the first claim of the proposition, andwe deduce that P ξ ∈ Y ( F ) I ( f ⊗ Φ)( ξ, s ) is holomorphic for Re( s ) ≫ P ξ ∈ Y ( F ) I ( f ⊗ Φ)( ξ, s ) into two sums of the form Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [SL ] X ξ ∈ V ( F ) Q ( ξ )=0 ρ (( I , h, ( − ) h ( − )) g ) f ( ξ ) × Z X δ ∈ B ( F ) \ SL ( F ) Φ( x (0 , δhp ( g )) | x | s d × xdhdg (10.0.7)where the unspecified integral is over | x | ≥ | x | ≤
1. The contribution of | x | ≥ s ) large and hence converges for all s . Using the Poisson summation formula on F , SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 61 the contribution of | x | ≤ Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [SL ] X ξ ∈ V ( F ) Q ( ξ )=0 ρ (( I , h, ( − ) h ( − )) g ) f ( ξ ) × Z | x |≤ X δ ∈ B ( F ) \ SL ( F ) b Φ( x − (1 , δ − t h − t p ( g ) − t ) | x | s d × x + b Φ(0) | x | s − − Φ(0) | x | s d × x ! Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [ G γ ] X ξ ∈ V ( F ) ρ ( hg ) f ( ξ ) dhdg. Taking a change of variable x → x − , one easily sees that the first term also defines an entirefunction. For Re( s ) ≫
1, the latter term isVol( F × \ ( A × F ) )2 b Φ(0) s − − Φ(0) s ! Z G γ ( A F ) \ G ( A F ) f ( γ g ) Z [ G γ ] X ξ ∈ V ( F ) ρ ( hg ) f ( ξ ) dhdg. Assuming that (10.0.6) is convergent, this term admits a meromorphic continuation to the s plane, holomorphic except at s ∈ { , } with poles and residues as specified. To obtain theconvergence of (10.0.6) one begins with Z G γ ( A F ) \ G ( A F ) | f ( γ g ) | Z [SL ] X ξ ∈ V ( F ) Q ( ξ )=0 | ρ (( I , h, ( − ) h ( − )) g ) f ( ξ ) |× Z X δ ∈ B ( F ) \ SL ( F ) Φ( x (0 , δhp ( g )) | x | s d × xdhdg instead of (10.0.7), argues as before, and then observes that one obtains an equality between(10.0.6) times Vol( F × \ ( A × F ) )2 (cid:16) b Φ(0) s − − Φ(0) s (cid:17) and a sum that converges for Re( s ) large. Theabsolute convergence statement follows. (cid:3) The L -theory We now discuss the L -theory. In this section, F denotes a local field of characteristiczero. We assume throughout this section that Y sm ( F ) ⊂ Y ( F ) is nonempty. It is then dense(in the Hausdorff topology). Lemma 11.1. If Z is a separated irreducible F -scheme of finite type with Z sm ( F ) = ∅ and Z ′ ⊂ Z is a Zariski open subscheme, then Z ′ ( F ) is open and dense in Z ( F ) .Proof. This follows from [Poo17, Remark 3.5.76]. (cid:3)
Thus Y ani ( F ) ⊂ Y ( F ) is dense.We first improve the bound in [GL19b, Propositions 7.1 and Proposition 8.2]: Proposition 11.2.
For > β ≥ and v ∈ V ◦ ( F ) , if F is nonarchimedean then Z G γ ( F ) \ G ( F ) | f ( γ g ) ρ ( g ) f ( v ) | dg ≪ f ,f Y i =1 | v i | β/ − d i / / . (11.0.1) The integral as a function of v has support in ω − N V ( O ) for some N ∈ Z . If F is archimedeanthen, given N > , there is a continuous seminorm ν β,N on S ( X ( F ) × V ( F )) such that Z G γ ( F ) \ G ( F ) | f ( γ g ) ρ ( g ) f ( v ) | dg ≤ ν β,N ( f ⊗ f ) Y i =1 max( | v i | , − N | v i | β/ − d i / / . The function I ( f ) satisfies the same bound and support constraint.Proof. Assume for the moment that F is nonarchimedean. By Lemma 3.5 the integral isbounded by a constant depending on β and f times Z | γ g |≤ c | γ g | − β | ρ ( g ) f | ( v ) dg (11.0.2)for some c >
1. Using the Iwasawa decomposition we can write the integral as Z m ( t,a ) ≤ c m ( t, a ) − β e f ( a − v ) Y i =1 | a i | − d i / ! d × adt, (11.0.3)where e f and m ( t, a ) are defined as in (8.0.4) and (5.2.2) respectively. Observe that m ( t, a ) = | a a a | max (cid:0) | t | , | a | − , | a | − , | a | − (cid:1) . (11.0.4)Thus (11.0.3) is equal to Z m ( t,a ) ≤ c max( | t | , | a | − , | a | − , | a | − ) − β e f ( a − v ) Y i =1 | a i | β − d i / ! d × adt. (11.0.5)We will bound this integral in a moment. Assuming that the integral is convergent, it isclear that the support of this function is contained in a compact subset of V ( F ). Since Z F max( | t | , | a | − , | a | − , | a | − ) − β dt ≪ β min( | a | , | a | , | a | ) − β ≤ | a a a | / − β/ , (11.0.6)the integral (11.0.5) is bounded by a constant times Z ( F × ) e f ( a − v ) Y i =1 | a i | β/ − d i / / ! d × a. Taking a change of variables a i a i ̟ ord( v i ) , the above is bounded by a constant times Y i =1 | v i | β/ − d i / / Z q − N ≤| a i | Y i =1 | a i | β/ − d i / / ! d × a for some N depending on f . This converges as d i ≥ i . SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 63
Now assume F is archimedean. By Lemma 3.5 and the argument above, there is a con-tinuous seminorm ν β,N ′ on S ( X ( F )) such that the integral is bounded by ν β,N ′ ( f ) times Z ( F × ) × F max (cid:0) | t | , | a | − , | a | − , | a | − (cid:1) − β max( m ( t, a ) , − N ′ e f ( a − v ) Y i =1 | a i | β − d i / ! dtd × a. (11.0.7)Here m ( t, a ) and e f are defined as above. Choose N ′ so that N ′ > N . Using the inequality m ( t, a ) ≥ max( | a | ,
1) max( | a | ,
1) max( | a | , , we deduce that for any M > ν ′ N,M on S ( V ( F )) such that(11.0.7) is bounded by ν ′ N,M ( f ) times Z ( F × ) × F max (cid:0) | t | , | a | − , | a | − , | a | − (cid:1) − β Y i =1 max( | a i | , − N max( | a − i v i | , − M | a i | β − d i / ! dtd × a. The proposition then follows from (11.0.6) (which is still valid for F archimedean) andLemma 9.1. (cid:3) Let Ω V be the top degree form on V ( F ) such that | Ω V | is the Haar measure. We endow Y sm ( F ) with the unique positive measure dy = | Ω Y | such that d ( Q − Q ) ∧ d ( Q − Q ) ∧ Ω Y is Ω V on V ( F ). We observe that Y sm ( F ) ⊂ Y ( F ) is dense, and hence we can form the L space L ( Y ( F )) := L ( Y ( F ) , dy ) := L ( Y sm ( F ) , dy ) . (11.0.8)We observe that for r ∈ F × , one has r d + d + d Ω V = Ω V ( rv )= d ( Q − Q )( rv ) ∧ d ( Q − Q )( rv ) ∧ Ω Y ( rv )= r d ( Q − Q ) ∧ d ( Q − Q ) ∧ Ω Y ( rv ) . Thus d ( ry ) = | r | d + d + d − dy. (11.0.9) Proposition 11.3.
One has S ( Y ( F )) < L ( Y ( F )) and the inclusion is continuous if F isarchimedean.Proof. We have S ( Y ( F )) < C ∞ ( Y sm ( F )) by propositions 8.1 and 9.3, thus Z Y sm ( F ) | f ( y ) | dy = Z Y ani ( F ) | f ( y ) | dy. We will therefore bound the integral on the right.
Let f ∈ S ( Y ( F )) and let > β >
0. Assume first that F is nonarchimedean. ByProposition 11.2, for some c ∈ Z one has Z Y ani ( F ) | f ( y ) | dy ≪ f Z Y ani ( F ) ∩ ̟ − c V ( O ) Y i =1 | y i | β/ − d i +4 / ! dy. Using the homogeneity property (11.0.9) this is bounded by a constant depending on c times ζ (2 β ) Z { y ∈ Y ani ( F ):1 ≤| y | < } Y i =1 | y i | β/ − d i +4 / ! dy. Here we could just write | y | = 1, but we have written 1 ≤ | y | < F is archimedean. Fix N > β . Then by Proposition 11.2 there is acontinuous seminorm v β on S ( Y ( F )) such that Z Y ani ( F ) | f ( y ) | dy ≤ ν β ( f ) ∞ X j =1 Z { y ∈ Y ani ( F ):2 − j ≤| y | < − j } Y i =1 | y i | β/ − d i +4 / ! dy + ν β ( f ) ∞ X j =0 Z { y ∈ Y ani ( F ):2 j ≤| y | < j +1 } | y | − N Y i =1 | y i | β/ − d i +4 / ! dy. Using the homogeneity property (11.0.9) again, we see that this is ν β ( f ) ∞ X j =1 − βj + ∞ X j =0 (2 β − N ) j ! Z { y ∈ Y ani ( F ):1 ≤| y | < } Y i =1 | y i | β/ − d i +4 / ! dy. Thus for F arbitrary, we are reduced to showing that Z { y ∈ Y ani ( F ):1 ≤| y | < } Y i =1 | y i | β/ − d i +4 / ! dy is finite. By symmetry, it suffices to show that the integral Z { y ∈ Y ani ( F ):max( | y | , | y | ) ≤| y | , ≤| y | < } Y i =1 | y i | β/ − d i +4 / ! dy (11.0.10)is finite. After a change of variables, we can and do assume that Q i ( v i ) = v t c i v where c i = (cid:18) c i ... c idi (cid:19) is diagonal. Write v i = ( v i , . . . , v id i ). For any 1 ≤ j ≤ d , 1 ≤ k ≤ d , we have (cid:12)(cid:12)(cid:12)(cid:16) ∂ v j ( Q −Q ) ∂ v k ( Q −Q ) ∂ v j ( Q −Q ) ∂ v k ( Q −Q ) (cid:17)(cid:17) = (cid:12)(cid:12)(cid:0) c j v j − c k v k c k v k (cid:1)(cid:12)(cid:12) = | c j c k || v j || v k | . Then for any j and k as above we have Y i =1 | y i | β/ − d i +4 / ! dy = (cid:0)Q i =1 | y i | β/ − d i +4 / (cid:1) | c j c k || y j y k | dy dy dy dy j dy k (11.0.11) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 65 outside a set of measure zero with respect to dy . Here dy i dy i := dy i . . . dy id , etc. and thevalues of y j and y k are given implicitly in terms of the other entries of y (this is possibleoutside a set of measure zero with respect to dy ). We can and do assume that j and k arechosen so that | y j | = | y | and | y k | = | y | . Therefore, setting t = | y | and t = | y | , we seethat (11.0.10) is bounded by a constant times Z Z (cid:18)Z ( x ,x ) ∈ F d − × F d − , | x |≤ t , | x |≤ t t β/ − d +1 / t β/ − d +1 / dx dx (cid:19) dt dt < ∞ . (cid:3) Let K ≤ Sp ( O ) be a compact open subgroup and let L ( X P ( F )) K denote the space offunctions on X P ( F ) that are right K -invariant and square-integrable. Lemma 11.4.
For f ∈ L ( X P ( F )) K we have | f ( x ) | ≤ || f || | x | meas( K ) for any x ∈ X P ( F ) .Proof. For f ∈ L ( X P ( F )) K we have the estimate || f || = X γ ∈ X P ( F ) /K | f ( γ ) | meas( K ) δ P ( γ ) = X γ ∈ X P ( F ) /K | f ( γ ) | meas( K ) | γ | . (cid:3) Proposition 11.5.
For F nonarchimedean and f ∈ S ( V ( F )) , the map I ( · ⊗ f ) extends toa continuous map I ( · ⊗ f ) : L ( X P ( F )) K −→ L ( Y ani ( F )) . Proof.
Assume y ∈ Y ani ( F ). By Lemma 11.4 for any f ∈ S ( X ( F )) the integral I ( f ⊗ f )( y )is bounded by meas( K ) − times k f k Z G γ ( F ) \ G ( F ) | γ g | − | ρ ( g ) f | ( y ) dg. The proposition thus follows from the argument proving Proposition 11.2 in the special case β = 0. (cid:3) The Fourier transform
Let F be a number field and let S be a finite set of places of F . In this section we provethe following theorem: Theorem 12.1.
Assume Y sm ( A F ) is nonempty. Let S be a finite set of places of F . Thereis a unique C -linear isomorphism F Y : S ( Y ( F S )) → S ( Y ( F S )) such that F Y ◦ I = I ◦ F X . Itis continuous if S consists of infinite places. In particular there is a commutative diagram S ( X ( F S ) × V ( F S )) S ( X ( F S ) × V ( F S )) S ( Y ( F S )) S ( Y ( F S )) I F X I F Y We should pause to explain why this theorem is not obvious. For simplicity assume that S = { v } is a nonarchimedian place. Let C := S ( X ( F v ) × V ( F v )) SL ( F v ) denote the space of coinvariants. It is clear that the map I factors through C and yields asurjection C → S ( Y ( F v )). Since F X is equivariant under the action of SL ( F v ) < Sp ( F v ), itis clear that F X descends to define an automorphism of C . However, it is not clear that themap C → S ( Y ( F v )) is injective. For instance, there are several orbits of SL ( F v ) on X ( F v ),but the map I depends only on the restriction of a function in S ( X ( F v ) × V ( F v )) to one ofthese orbits. This could probably be sorted out through local arguments, but because wehave already developed the global theory to a certain extent, we prefer to make use of it.Now that one has Theorem 12.1, many of the prior results in this paper can be statedmore transparently. For example, by Lemma 5.2 we have Corollary 12.2.
For any place v of F and f ∈ S ( Y ( F v )) , one has that F Y ( L ( h ) f ) = | λ ( h ) | P i =1 d i / − L (cid:18) hλ ( h ) (cid:19) F Y ( f ) . (cid:3) Unlike in other sections in this paper, we have not abbreviated F v by F . We really requireboth F and F v in this section because we will use a local-global argument to prove Theorem12.1. The local global argument is fairly simple, and we invite the reader to skip to the proofof Theorem 12.1 to see the basic idea.There is a somewhat hidden assumption on the base change Y F S of Y to F S in the statementof Theorem 12.1. Namely, we are assuming that Y F S is the base change to F S of the schemecut out of a triple of quadratic spaces over the number field F by the simultaneous values ofthree quadratic forms. Since every characteristic zero local field is a localization of a numberfield, and every quadratic form over a local field is the localization of a quadratic form overthe corresponding number field [O’M00, Theorem 72:1], this is no loss of generality. Theorem 12.3. If Y sm ( F v ) = ∅ for all v , then Y sm ( F ) is nonempty and has dense imagein Y sm ( F S ) .Proof. Since we have assumed dim V i ≥ Q i is nondegenerate for each i , this is a directconsequence of [CTS82, Corollaire in § (cid:3) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 67
Let γ ∈ Sp ( Z ) be the representative for the open orbit in X P of G given in (4.0.12). Inthe following, for a place v of F , the notation γ G ( F v ) will refer to the open orbit of G ( F v )in X P ( F v ). Moreover, for a place v of F , let S ,v := \ w ∈ Γ ρ ( w ) − S ( V ◦ ( F v )) < S ( Y ( F v ))where Γ is the subgroup of SL ( F ) generated by ( − ). This coincides with the definitionof S ,v from (1.0.1). Lemma 12.4.
Let v be a place where Q iv splits for all ≤ i ≤ . Suppose there exists y ∈ Y ani ( F v ) such that I ( F X ( f ) ⊗ f )( y ) = 0 for all f ⊗ f ∈ C ∞ c ( γ G ( F v )) ⊗ S ,v . Then I ( F X ( f ) ⊗ f ) = 0 for any such f ⊗ f .Proof. Given y ∈ Y ani ( F v ), choose h ∈ H ( F v ) such that λ ( h ) h − y = y . Let L ′ ( h ) be definedas in (5.1.5). We observe that L ′ ( h ) preserves C ∞ c ( γ G ( F v )) ⊗ S ,v (see (5.1.3) and (5.1.2)).Thus by Lemma 5.2, for f ⊗ f ∈ C ∞ c ( γ G ( F v )) ⊗ S ,v we have I ( F X ( f ) ⊗ f )( y ) = L (cid:18) hλ ( h ) (cid:19) I ( F X ( f ) ⊗ f )( y )= | λ ( h ) | − P i =1 d i / I ( F X ( L ′ ( h ) f ⊗ L ( h ) f ))( y )= 0by hypothesis Since I ( F X ( f ) ⊗ f ) is continuous on Y sm ( F v ) by propositions 8.1 and 9.3,and Y ani ( F v ) ⊂ Y sm ( F v ) is dense by Lemma 11.1, we deduce the lemma. (cid:3) Lemma 12.5.
Let v be a finite place where Q iv splits for ≤ i ≤ . For a given y ∈ Y ani ( F v ) ,there exists f ⊗ f ∈ C ∞ c ( γ G ( F v )) ⊗ S v such that I ( F X ( f ) ⊗ f )( y ) = 0 .Proof. Choose f ′ ⊗ f ∈ S ( X ( F v ) × V ( F v )) such that I ( f ′ ⊗ f ) = 0. For example, we couldtake f ′ to be the characteristic function of a sufficiently small neighborhood of γ in γ G ( F v ).Choose a compact open subgroup K ≤ Sp ( O ) such that f ′ is fixed by K . Finally, choose f n ∈ C ∞ c ( γ G ( F v )) K indexed by n ∈ Z > such thatlim n →∞ f n = F − X ( f ′ )in L ( X P ( F )) K . Then since F X is an isometry of L ( X P ( F )) K , we have F X ( f n ) → f ′ in L ( X P ( F )) K . Since I ( · ⊗ f ) : L ( X P ( F )) K −→ L ( Y ani ( F ))is well-defined and continuous by Proposition 11.5, we deduce that I ( F X ( f n ) ⊗ f ) → I ( f ′ ⊗ f )in L ( Y ani ( F )) and hence I ( F X ( f n ) ⊗ f ) = 0 for n large enough. The statement thusfollows from Lemma 12.4. (cid:3) Proof of Theorem 12.1.
Suppose that I ( f S ) = 0. It suffices to show that I ( F X ( f S )) = 0.Let v and v be finite places of F not in S and suppose that f v ∈ S ( X ( F v ) × V ( F v ))is chosen so that F X ( f v ) ∈ C ∞ c ( γ G ( F v ) × V ( F v )) and I ( F X ( f v )) ∈ C ∞ c ( Y sm ( F v )) andthat f v ∈ C ∞ c ( γ G ( F v )) ⊗ S v . Moreover choose f v v S ∈ S ( X ( A Sv v F ) × V ( A Sv v F )). Thenapplying Theorem 6.1, we obtain0 = X y ∈ Y sm ( F ) I ( F X ( f S f v f v f Sv v ))( y ) . In particular, since F X ( f v ) ∈ C ∞ c ( γ G ( F v ) × V ( F v )) and f v ∈ C ∞ c ( γ G ( F v )) ⊗ S v , allof the boundary terms in the formula vanish.We observe that Y ( F ) is discrete in Y ( A F ). We claim that we can choose f v f v f Sv v sothat the right hand side of the equality above is reduced to a single term y ∈ Y ani ( F ) and I ( F X ( f v f v f Sv v ))( y ) = 0. Indeed, using the argument in the proof of Lemma 5.7 we canchoose f v so that I ( F X ( f v )) is any function in C ∞ c ( Y sm ( F v )). Combining this with Lemma5.7, the computation of the basic function in Proposition 7.1, and Lemma 12.5, we deducethe claim.The claim implies that I ( F X ( f S ))( y ) = 0 for all y ∈ Y ani ( F ). Since Y ani ( F ) is densein Y sm ( F S ) by Lemma 11.1 and Theorem 12.3, we can use the continuity of I ( F X ( f S ))(propositions 8.1 and 9.3) to deduce that I ( F X ( f S )) = 0, completing the proof. (cid:3) The Fourier transform F X,ψ and I := I ψ (see (5.2.3)) depend on a choice of additivecharacter ψ . Hence so does F Y . We write F Y,ψ when we need to indicate this dependence.Thus F Y,ψ is determined by the relation F Y,ψ ◦ I ψ = I ψ ◦ F X,ψ . (12.0.1) Corollary 12.6.
For f ∈ S ( Y ( F S )) , we have F Y ( f )( v ) = f ( v ) , (12.0.2) F Y,ψ ( f ) = F Y,ψ ( f )(12.0.3) F Y,ψ = F Y,ψ . (12.0.4) Proof.
The first equation (12.0.2) is immediate from Proposition 3.12. As for (12.0.3), by(3.1.19) and Lemma 3.10, for any f ∈ S ( X ( F )) one has F X,ψ ( f ) = F X,ψ (cid:0) L ( ε ) f (cid:1) = F X,ψ ( f ) . Moreover, we claim that for f ∈ S ( V ( F )) one has ρ ψ ( g ) f = ρ ψ ( g ) f (12.0.5)for all g ∈ SL ( F ). By the second corollary to [Wei64, Th´eor`eme 2], the Weil index γ ( Q i ) := γ ( Q i , ψ ) satisfies the relation γ ( Q i , ψ ) = γ ( Q i , ψ ). Using this fact, one checks (12.0.5)by checking it on the same set of generators for SL ( F ) traditionally used to define the SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 69
Weil representation (see [GL19b, § f ⊗ f ∈ S ( X ( F ) × V ( F )) one has I ψ ( F X,ψ ( f ) ⊗ f ) = I ψ ( F X,ψ ( f ) ⊗ f ) = I ψ ( F X,ψ ( f ) ⊗ f ) . To show (12.0.4), as S ( Y ( F )) is independent of the character ψ , by (12.0.2) it suffices toshow F Y,ψ ◦ F
Y,ψ ( f ) = f for functions f of the form I ψ ( f ⊗ f ). Using lemmas 3.8, 5.2 and5.6, we compute F Y,ψ ◦ F
Y,ψ ( I ψ ( f ⊗ f ))= F Y,ψ ( I ψ ( F X,ψ ( f ) ⊗ f ))= γ ( Q , ψ ) γ ( Q , ψ ) F Y,ψ ◦ L ( − I ψ ( L ( ε ) R ( − I I ) F X,ψ ( f ) ⊗ f )= γ ( Q , ψ ) γ ( Q , ψ ) F Y,ψ ◦ I ψ ( L ( ε ) R ( − I I ) F X,ψ ( f ) ⊗ L ( − f )= γ ( Q , ψ ) γ ( Q , ψ ) I ψ ◦ F X,ψ ( L ( ε ) R ( − I I ) F X,ψ ( f ) ⊗ L ( − f )= γ ( Q , ψ ) γ ( Q , ψ ) I ψ ( L ( ǫ ) R ( I − I ) F X,ψ ◦ F
X,ψ ( f ) ⊗ L ( − f )= γ ( Q , ψ ) γ ( Q , ψ ) I ψ ( R ( I − I ) f ⊗ L ( − f )= I ψ ( L ( ε ) R ( − I ) f ⊗ f )= I ψ ( f ⊗ f ) . Here the last equality follows from the fact that ε [ P, P ]( F ) = ( − I )[ P, P ]( F ). (cid:3) We now explain how to deduce Theorem 1.2 from Theorem 6.1 and Theorem 12.1. Forthe reader’s convenience, we restate Theorem 1.2.
Theorem 12.7.
Let f ∈ S ( Y ( A F )) . Assume that there are finite places v , v of F suchthat f = f v v f v v where f v and F Y ( f v ) are restrictions of elements of S ,v and S ,v ,respectively. Then X y ∈ Y ( F ) f ( y ) = X y ∈ Y ( F ) F Y ( f )( y ) . Proof.
Given such f , we can choose for i = 1 , f v i ∈ C ∞ c ( γ G ( F v i ))such that I ( f v ⊗ f v ) = f v and I ( f v ⊗ f v ) = F Y ( f v ) . where f v | Y sm ( F v ) = F Y ( f v ). Indeed, we can take f v i to be a scalar multiple of thecharacteristic function of a sufficiently small neighborhood of γ in γ G ( F v i ). Moreover, choose f ′ v v ∈ S ( X ( A v v F ) × V ( A v v F )) such that I ( f ′ v v ) = f v v . To deduce the corollary, we now apply Theorem 6.1 to f ′ = ( f v ⊗ f v )( F − X ( f v ) ⊗ f v ) f ′ v v .Assumption (6.0.1) is clearly valid, and (6.0.3) is valid by our hypotheses on f v and F Y ( f v ).By construction, the boundary terms vanish and the theorem is proved. (cid:3) List of symbols b X basic function on X (3.1.15) b Y basic function on Y (7.1.2) C ∞ c ( X P ( F ) , K ) compactly supported K -finite smooth functions on X P ( F ) § ε m ( −
1) (3.1.17) f χ s local Mellin transform (3.1.1) | f | A,B,p seminorm (3.1.2) | f | A,B,p w , Ω ,D seminorm (3.1.4) F X Fourier transform on S ( X ( F )) § F Y Fourier transform on S ( Y ( F )) § h ( − ) i < SL ( F ) (6.0.3) G SL (4.0.11) g Lie algebra of Sp n § γ i representatives of X P ( F ) /G ( F ) (4.0.12) G γ i stabilizer of γ i in G § | g | the norm of Pl¨ucker embedding (3.1.7) H similitude group on Y (4.0.4) I integral operator attached to the representative γ (5.0.1) I integral operator attached to the representative Id = I (5.0.1) I i integral operator attached to the representative γ i (5.0.4) L ( m ) action by M ab on S ( X ( F )) (3.1.11) L ′ ( h ) action by H on S ( X ( F )) and S ( X ( F ) × V ( F )) (5.1.1) λ ( h ) similitude of h ∈ H (4.0.5) m an isomorphism M ab ( F ) → F × (3.1.13) M Levi subgroup of P (3.0.1) N unipotent radical of P (3.0.1) N standard maximal unipotent subgroup in SL § ω character of M § k characteristic function of [ P,P ]( F ) m ( ̟ k )Sp ( O ) (3.1.14) P Siegel parabolic § R action by GSp n ( F ) on S ( X ( F )) (3.1.11) SUMMATION FORMULA FOR TRIPLES OF QUADRATIC SPACES, II 71 S ( V ( F )) usual Schwartz space § S ( X ( F )) Schwartz space on X ( F ) § S ( X ( F ) × V ( F )) Schwartz space on X ( F ) × V ( F ) § S ( Y ( F )) Schwartz space on Y ( F ) § V, Q ) Q i =1 ( V i , Q i ) § V ′ { ( v , v , v ) ∈ V : no two v i are zero } (4.0.2) V ◦ { ( v , v , v ) ∈ V : v i = 0 } (4.0.1)( V i , Q i ) quadratic space of even dimension § X affine closure of X P (3.0.6) X P Braverman-Kazhdan space (3.0.3) Y { ( y , y , y ) ∈ V : Q ( y ) = Q ( y ) = Q ( y ) } . (4.0.3) Y ani anisotropic vectors in Y § Y sm smooth locus of Y § Y e Y / G m (4.0.10) Y i e Y i / G m (4.0.10) e Y vanishing locus of Q , Q , Q in V ◦ (4.0.6) e Y i { ( y , y , y ) ∈ V ◦ : Q i − ( y i − ) = Q i +1 ( y i +1 ) and Q i ( y i ) = 0 } (4.0.9) References [AG08] Avraham Aizenbud and Dmitry Gourevitch. Schwartz functions on Nash manifolds.
Int. Math.Res. Not. IMRN , (5):Art. ID rnm 155, 37, 2008. 6[BCR98] Jacek Bochnak, Michel Coste, and Marie-Fran¸coise Roy.
Real algebraic geometry , volume 36 of
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Ar-eas (3)] . Springer-Verlag, Berlin, 1998. Translated from the 1987 French original, Revised by theauthors. 7[BK99] Alexander Braverman and David Kazhdan. On the Schwartz space of the basic affine space.
SelectaMath. (N.S.) , 5(1):1–28, 1999. 2[BK00] A. Braverman and D. Kazhdan. γ -functions of representations and lifting. Geom. Funct. Anal. ,(Special Volume, Part I):237–278, 2000. With an appendix by V. Vologodsky, GAFA 2000 (TelAviv, 1999). 2[BK02] Alexander Braverman and David Kazhdan. Normalized intertwining operators and nilpotent el-ements in the Langlands dual group.
Mosc. Math. J. , 2(3):533–553, 2002. Dedicated to Yuri I.Manin on the occasion of his 65th birthday. 2, 5, 8, 9, 24[Con12] Brian Conrad. Weil and Grothendieck approaches to adelic points.
Enseign. Math. (2) , 58(1-2):61–97, 2012. 7[CTS82] Jean-Louis Colliot-Th´el`ene and Jean-Jacques Sansuc. Sur le principe de hasse et l’approximationfaible, et sur une hypoth`ese de schinzel.
Acta Arithmetica , 41(1):33–53, 1982. 6, 66[ES18] Boaz Elazar and Ary Shaviv. Schwartz functions on real algebraic varieties.
Canad. J. Math. ,70(5):1008–1037, 2018. 7 [GK19] Nadya Gurevich and David Kazhdan. Fourier transforms on the basic affine space of a quasi-splitgroup. arXiv e-prints , page arXiv:1912.07071, December 2019. 3[GL06] Stephen S. Gelbart and Erez M. Lapid. Lower bounds for L -functions at the edge of the criticalstrip. Amer. J. Math. , 128(3):619–638, 2006. 23[GL19a] Jayce R. Getz and Baiying Liu. A refined Poisson summation formula for certain Braverman-Kazhdan spaces.
Sci. China Math., accepted , 2019. 5, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 23,24, 26, 49[GL19b] Jayce R. Getz and Baiying Liu. A summation formula for triples of quadratic spaces.
Adv. Math. ,347:150–191, 2019. 2, 3, 6, 24, 26, 28, 29, 31, 35, 39, 61, 69[HK92] Michael Harris and Stephen S. Kudla. Arithmetic automorphic forms for the nonholomorphicdiscrete series of
GSp (2).
Duke Math. J. , 66(1):59–121, 1992. 28[Ike92] Tamotsu Ikeda. On the location of poles of the triple L -functions. Compositio Math. , 83(2):187–237,1992. 6, 10, 16, 17, 19[KM11] Toshiyuki Kobayashi and Gen Mano. The Schr¨odinger model for the minimal representation of theindefinite orthogonal group O( p, q ). Mem. Amer. Math. Soc. , 213(1000):vi+132, 2011. 3[Li18] Wen-Wei Li.
Zeta integrals, Schwartz spaces and local functional equations , volume 2228 of
LectureNotes in Mathematics . Springer, Cham, 2018. 9[Ngˆo20] Bao Chˆau Ngˆo. Hankel transform, Langlands functoriality and functional equation of automorphic L -functions. Jpn. J. Math. , 15(1):121–167, 2020. 2[O’M00] O. Timothy O’Meara.
Introduction to quadratic forms . Classics in Mathematics. Springer-Verlag,Berlin, 2000. Reprint of the 1973 edition. 66[Poo17] Bjorn Poonen.
Rational points on varieties , volume 186 of
Graduate Studies in Mathematics . Amer-ican Mathematical Society, Providence, RI, 2017. 61[Sak12] Yiannis Sakellaridis. Spherical varieties and integral representations of L -functions. Algebra NumberTheory , 6(4):611–667, 2012. 2, 6[Sha81] Freydoon Shahidi. On certain L -functions. Amer. J. Math. , 103(2):297–355, 1981. 22[Tat79] J. Tate. Number theoretic background. In
Automorphic forms, representations and L -functions(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 , Proc. Sympos.Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, R.I., 1979. 16[Wal92] Nolan R. Wallach. Real reductive groups. II , volume 132 of
Pure and Applied Mathematics . Aca-demic Press, Inc., Boston, MA, 1992. 10, 22[War72] Garth Warner.
Harmonic analysis on semi-simple Lie groups. I . Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. 14[Wei64] Andr´e Weil. Sur certains groupes d’op´erateurs unitaires.
Acta Math. , 111:143–211, 1964. 68
Department of Mathematics, Duke University, Durham, NC 27708
Email address : [email protected] Department of Mathematics, Duke University, Durham, NC 27708
Email address ::