On the smooth transfer for Guo-Jacquet relative trace formulae
aa r X i v : . [ m a t h . R T ] N ov On the smooth transfer for Guo-Jacquet relativetrace formulae
Chong Zhang
Abstract
We establish the existence of smooth transfer for Guo-Jacquet relativetrace formulae in p -adic case. This kind of smooth transfer is a key steptowards a generalization of Waldspurger’s result on central values of L-functions of GL . History
The periods of automorphic forms play an important role in thestudy of automorphic representations and related number theoretic problems.For example, people believe that periods of automoprhic forms can characterizethe Langlands functoriality of automorphic representations. Recently, Y. Sakel-laridis and A. Venkatesh [SV] developed an ambitious program, the so-calledrelative Langlands program, on this aspect. There are several powerful tools tostudy periods. The theory of relative trace formula is one of them, which wasfirst studied by H. Jacquet. In [Ja1], Jacquet reproved a remarkable result ofJ.-L. Waldspurger [Wa1] on central values of L-functions of GL by comparingrelative trace formulae on different groups. In [Gu1], J. Guo and Jacquet madea conjecture (see Conjecture 1.1) generalizing Waldspurger’s result to higherrank cases.To be precise, let k be a number field, A its ring of adeles. Consider G =GL n and H = GL n × GL n embedded into G diagonally, which are reductivegroups over k . Let k ′ be a quadratic field extension of k , η the quadraticcharacter of A × /k × attached to k ′ by class field theory. Let Z be the center of G .When we say a cuspidal representation π , we always mean that π is irreducibleand automorphic. For a cuspidal representation π of G ( A ), we consider thelinear forms ℓ H and ℓ H ,η on π defined by periods: ℓ H ( φ ) := Z H ( k ) Z ( A ) \ H ( A ) φ ( h ) d h, ℓ H ,η ( φ ) := Z H ( k ) Z ( A ) \ H ( A ) φ ( h ) η ( h ) d h, where φ ∈ π and η ( h ) := η (det h ). We say that π is H -distinguished (resp.( H, η )-distinguished) if ℓ H = 0 (resp. ℓ H ,η = 0). On the other hand, fora quaternion algebra D over k containing k ′ , let G ′ = G ′ D = GL n (D) and H ′ = GL n ( k ′ ), both viewed as reductive groups defined over k . View H ′ as asubgroup of G ′ in the natural way and identify the center of G ′ with Z . For acuspidal representation π ′ of G ′ ( A ), consider the linear form ℓ H ′ on π ′ definedby ℓ H ′ ( φ ) := Z H ′ ( k ) Z ( A ) \ H ′ ( A ) φ ( h ) d h, φ ∈ π ′ .
1e say that π ′ is H ′ -distinguished if ℓ H ′ = 0.Denote by X ( k ′ , k ) the set of quaternion algebras D over k containing k ′ . Fora cuspidal representation π of G ( A ), denote by X ( k ′ , k ; π ) the subset of X ( k ′ , k )such that the Jacquet-Langlands correspondence π ′ D := JL( π ) of π exists as acuspidal representation of G ′ D ( A ).Motivated by Waldspurger’s result in the case n = 1, the following conjecturewas made in [Gu1]. Conjecture 1.1 ((Guo-Jacquet)) . Fix a cuspidal representation π of G ( A ).1. Fix a quaternion algebra D in X ( k ′ , k ; π ). Suppose that π ′ D is H ′ -distinguished.Then π is both H -distinguished and ( H , η )-distinguished.2. Suppose that n is odd and π is both H -distinguished and ( H , η )-distinguished.Then there exists D ∈ X ( k ′ , k ; π ) such that π ′ D is H ′ -distinguished.Moreover, when n is even, with more restriction, the direction (ii) of Con-jecture 1.1 should also hold. We refer the reader to [FM, Conjecture 3] and[FMW, Conjecture 1.5] for more information.The periods defined above can be used to study the central value L ( , π k ′ ) = L ( , π ) L ( , π ⊗ η ) where π k ′ is the base change of π to G ( A k ′ ). It was shown in[FJ] that if π is both H -distinguished and ( H , η )-distinguished then L ( , π k ′ ) =0. One also expects that there exists a relation between this L-value and theperiod ℓ H ′ on π ′ .In [Gu1], a relative trace formula approach called Guo-Jacquet relative traceformulae today, which is a natural extension of Jacquet’s method in [Ja1], wasproposed to attack the above conjecture. The first step, that is, the fundamentallemma for unit Hecke functions, has also been established by [Gu1]. The smoothtransfer can be viewed as the second step on the geometric side of Guo-Jacquetrelative trace formulae. Since we only focus on the smooth transfer, which isa local issue, we will not recall the precise form of Guo-Jacquet relative traceformulae, which is a global issue. We refer the reader to [Gu1] or [FMW] formore details.Very recently, B. Feigon, K. Martin and D. Whitehouse [FMW] obtainedsome partial results on Conjecture 1.1, by using a simple form of Guo-Jacquettrace formulae. They showed the existence of smooth transfer for Bruhat-Schwartz functions satisfying certain specific properties. Of course, one hasto show the existence of smooth transfer for the full space of Bruhat-Schwartzfunctions, if one aims to prove Conjecture 1.1 completely. Due to our result,one can remove some conditions of the results in [FMW], as [FMW, Remark6.2] states.There is also a generalization of Waldspurger’s result in another direction:the so called Gan-Gross-Prasad conjecture [GGP] and the refined version of itby Ichino-Ikeda [II] in the case of orthogonal groups and by N. Harris [Ha] in thecase of unitary groups. Recently, W. Zhang ([Zhw1],[Zhw2]) has made a greatadvance towards the global Gan-Gross-Prasad conjecture for unitary groups byusing the relative trace formula developed by Jacquet and S. Rallis. One of hisachievements is his proof of the smooth transfer conjecture in p -adic case forthe Jacquet-Rallis relative trace formula. His method is close to that of [Ja2].The several remarkable successes on the Gan-Gross-Prasad conjecture, both inlocal and global directions, will shed some light on the problem considered here.2 esults of this article In this article, we establish the existence of smoothtransfer in p -adic case for Guo-Jacquet relative trace formulae. Let us brieflyexplain what the smooth transfer means. From now on, let F be a p -adicfield, which is a completion of k at a finite place. Let E be a quadratic fieldextension of F and D a quaternion algebra over F containing E . Notice thatsuch quaternion algebras are parameterized by F × / N E × , where N is the normmap from E × to F × . When we want to emphasize the dependence of D on ǫ ∈ F × / N E × , we write D ǫ . Let η be the quadratic character of F × associatedto E/F . We define ( G , H ) and ( G ′ , H ′ ) over F in the same way as the globalsituation. Write G = G ( F ) , H = H ( F ) , G ′ = G ′ ( F ) and H ′ = H ′ ( F ).The group H × H (resp. H ′ × H ′ ) acts on G (resp. G ′ ) by left and righttranslations. With respect to this action, we can talk about the notion of H × H -or H ′ × H ′ -regular semisimple (cf. § G or G ′ respectively. Denoteby G rs and G ′ rs the set of the regular and semisimple elements in G and G ′ respectively. Then there is a natural injection (cf. Proposition 5.1)[ G ′ rs ] ֒ → [ G rs ]from the set of H ′ × H ′ -orbits in G ′ rs to the set of H × H -orbits in G rs . We saythat x ∈ G rs matches y ∈ G ′ rs and write x ↔ y if the orbit of y goes to thatof x under this injection. We say that x ∈ G rs comes from G ′ rs if there exists y ∈ G ′ rs such that x ↔ y . If x ↔ y , their stabilizers denoted by ( H × H ) x and( H ′ × H ′ ) y are isomorphic. Fix a Haar measure on H and a Haar measure on( H × H ) x for each x ∈ G rs . Note that η | ( H × H ) x = 1. For each f ∈ C ∞ c ( G ),define the orbital integral of f at x to be O η ( x, f ) = Z ( H × H ) x \ H × H f ( h − xh ) η (det h ) d h d h . We can define a transfer factor κ (cf. Defintion 5.7) which is a function on G rs so that κ ( · ) O η ( · , f ) only depends on the H × H -orbits in G rs . Similarly, fix aHaar measure on H ′ . We fix the Haar measure on ( H ′ × H ′ ) y for each y ∈ G ′ rs so that it is compatible with that on ( H × H ) x if x ↔ y . For each f ′ ∈ C ∞ c ( G ′ ),define the orbital integral of f ′ at y to be O ( y, f ′ ) = Z ( H ′ × H ′ ) y \ H ′ × H ′ f ′ ( h − yh ) d h d h . For f ∈ C ∞ c ( G ) and f ′ ∈ C ∞ c ( G ′ ), we say that f and f ′ are smooth transfer ofeach other if κ ( x ) O η ( x, f ) = ( O ( y, f ′ ) , if there exists y ∈ G ′ rs such that x ↔ y, , otherwise . Denote by C ∞ c ( G ) the subspace of elements f in C ∞ c ( G ) satisfying that O η ( x, f ) =0 for any x ∈ G rs that does not come from G ′ rs .Our main result is the following theorem. Theorem 1.2.
For each f ′ ∈ C ∞ c ( G ′ ) , there exists f ∈ C ∞ c ( G ) that is a smoothtransfer of f ′ . Conversely, for each f ∈ C ∞ c ( G ) , there exists f ′ ∈ C ∞ c ( G ′ ) thatis a smooth transfer of f . θ on G such that H = G θ is the subgroup of G fixedby θ . Let S := G/H be the p -adic symmetric space associated to ( G , H ). Thegroup H acts on S by the conjugate action. There is a symmetrization map s : G → G ι , where ι is the anti-involution on G defined by ι ( g ) = θ ( g − ) and G ι is the subgroup fixed by ι . The symmetrization map is given by s ( g ) = gι ( g ).Via the map s , we view S as a subset of G ι ( F ). An element g ∈ G is H × H -regular semisimple if and only if x = s ( g ) ∈ S is H -regular semisimple. Denoteby S rs the subset of regular semisimple elements in S . Let q : C ∞ c ( G ) → C ∞ c ( S )be the natural surjection map defined by( qf )( x ) = Z H f ( gh ) d h if x = s ( g ). Let x = s ( g ) ∈ S be regular semisimple. Then its stabilizer H x isisomorphic to ( H × H ) g . We choose the same Haar measure on H as before andthe Haar measure on H x compatible with that on ( H × H ) g . For e f ∈ C ∞ c ( S ),define the orbital integral of e f at x to be O η ( x, e f ) = Z H x \ H e f ( h − xh ) η (det h ) d h. We define a transfer factor on S rs so that κ ( x ) = κ ( g ) if x = s ( g ). Then, by aroutine computation, we have κ ( g ) O η ( g, f ) = κ ( x ) O η ( x, e f )for each f ∈ C ∞ c ( G ), e f = qf ∈ C ∞ c ( S ) and x = s ( g ) ∈ S rs . Thus, the study oforbital integrals for C ∞ c ( G ) with respect to H × H -action is equivalent to thatof orbital integrals for C ∞ c ( S ) with respect to H -action. Similarly, the study oforbital integrals for C ∞ c ( G ′ ) with respect to H ′ × H ′ -action is equivalent to thatof orbital integrals for C ∞ c ( S ′ ) with respect to H ′ -action, where S ′ := G ′ /H ′ isthe p -adic symmetric space associated to ( G ′ , H ′ ).There is a natural injection (cf. Proposition 5.1)[ S ′ rs ] ֒ → [ S rs ]from the set of H ′ -orbits in S ′ rs to the set of H -orbits in S rs . We say that x ∈ S rs matches y ∈ S ′ rs and write x ↔ y if the orbit of y goes to that of x . Similarly,for f ∈ C ∞ c ( S ) and f ′ ∈ C ∞ c ( S ′ ), we can define the notion of smooth transfer forthem (see § s and s ′ of G / H and G ′ / H ′ at the identity respectively. Thenotion of smooth transfer in this version is determined by the orbital integralswith respect to adjoint actions of H and H ′ on s ( F ) and s ′ ( F ) respectively. Werefer the reader to § § s rs ( F )] k [ ǫ ∈ F × / N E × [ s ′ ǫ, rs ( F )] , where s ′ ǫ is the Lie algebra associated to ( G ′ ǫ = GL n (D ǫ ) , H ′ ) and [ s ′ ǫ, rs ( F )](resp. [ s rs ( F )]) is the set of H ′ - (resp. H )-regular semisimple orbits. The abovetwo sets are equal if and only if n = 1. Even worse, the elliptic parts of theabove two sets are equal if and only n is odd. These phenomenons are unlikeother cases of relative trace formulae. Now suppose that we are in the globalsetting. The second fact is that if X is a global element in s rs ( k ) which does notcome from s ′ rs ( k ) then there exist at least two places v , v such that X doesnot come from s ′ rs ( k v ) or s ′ rs ( k v ). This is unlike the case of endoscopic transferand prevents us to use global method to prove Theorem 8.1 which asserts thatthe orbital integral O η ( X, b f ) = 0 for X ∈ s rs ( F ) not coming from s ′ rs ( F ) where f ∈ C ∞ c ( s ( F )) is a smooth transfer of some element in C ∞ c ( s ′ ( F )) and b f is itsFourier transform. Instead we will use a pure local argument, which is due tothe referee, to show Theorem 8.1.To prove Theorem 5.16, we have to show the representability of the Fouriertransform of orbital integrals as distributions (see Theorem 6.1), exhibit “limitformulae” for the kernel functions (see Proposition 7.1) as Waldspurger did in[Wa2], and also prove analogues of some results (see Proposition 7.6 and Theo-rem 8.4) in [Wa3]. These results, which are on harmonic analysis on certain p -adic symmetric spaces, maybe appear in the literature for the first time. We ex-pect that the techniques developed in this paper should be probably generalizedto treat some other similar open questions concerning relative trace formulae forsymmetric pairs. Actually, we do successfully generalize this method to provethe existence of smooth transfer for other relative trace formula in [Zhc]. Herewe mention some cases of symmetric pairs where our results in § § E be a quadratic field extension of a p -adic field F . The first classof symmetric pairs are “inner forms” of ( G , H ) or ( G ′ , H ′ ). Now let D be a cen-tral division algebra over F . Let G = GL m (D) and H = GL m (D) × GL m (D).Then ( G , H ) is the symmetric pair considered in [Zhc]. We can also considerthe symmetric pair ( G , H ) = (GL m (D) , GL m (D ⊗ F E )), or, more generally,the symmetric pair ( G , H ) = (GL m (D) , GL m (D ′ )) where D is a central simplealgebra over F containing E and D ′ is the centralizer of E in D. The secondclass of symmetric pairs are Galois symmetric pairs. Now let H be a connectedreductive group over F , and G = Res E/F ( H E ) the Weil restriction of the base5hange of H to E . Then ( G , H ) is called a Galois symmetric pair. Structure of this article In §
2, we introduce some notations and conventionsthat are frequently used in the paper.In §
3, since ( G , H ) and ( G ′ , H ′ ) are symmetric pairs, we collect some basicnotions and results on symmetric pairs. In particular, we recall the analyticLuna Slice Theorem which plays an pivotal role on the reduction steps of thesmooth transfer.In §
4, we study our specific symmetric pairs ( G , H ) and ( G ′ , H ′ ) more con-cretely. We give a complete description of all the descendants of the corre-sponding symmetric spaces and their Lie algebras. We also prove Propositions4.4 and 4.8, which are about two inequalities. These inequalities are crucial forbounding the orbital integrals later (see Theorem 6.11).In §
5, we introduce the main issue of this article, that is, the smooth transferat the level of symmetric spaces and its Lie algebra version. We explain whyTheorem 5.16 implies Theorem 5.13. We also prove the fundamental lemmain the Lie algebra version, which is crucial for our global approach to proveTheorem 5.16.In §
6, to prove Theorem 5.16, we pay more effort on studying the Fouriertransform of orbital integrals. One of the most important question is to show therepresentability, that is, the Fourier transform of an orbital integral consideredas a distribution can be represented by a locally integrable kernel function.We deal with this issue in this section. The representability itself is also afundamental question in harmonic analysis on p -adic symmetric spaces. § §
8, we finish the proof of Theorem 5.16, basing on the results of § We now introduce some notations and conventions, which are frequently usedin § § Fields
Let F be a non-archimedean local field of characteristic 0, with finiteresidue field. Fix an algebraic closure ¯ F , and denote by Γ F = Gal( ¯ F /F ) theabsolute Galois group. We denote by |·| F (resp. v F ) the absolute value (resp. thevaluation) of F , and extend them to ¯ F in the usual way. Let O F be the integerring of F and fix a uniformizer ̟ of O F . For a finite extension field L of F ,denote by N L/F and Tr
L/F the norm and trace maps respectively. Throughoutthis article, we fix a nontrivial additive unitary character ψ : F → C × . Varieties and groups
All the algebraic varieties and algebraic groups that weconsider are defined over F except in §
8. We always use bold letter to denote analgebraic group, italic letter to denote its F -rational points, and Fraktur letter6o denote its Lie algebra. For example, let G be a reductive group. We write G = G ( F ) and denote by g the Lie algebra of G . By a subgroup of G , wemean a closed F -subgroup. We write N G ( · ) for the normalizer and Z G ( · ) forthe centralizer of a certain set in G , and write Z for the center of G . For analgebraic variety X , X = X ( F ) is equipped with the natural topology inducedfrom F . Thus, X is a locally compact totally disconnected topological space.Sometimes we treat finite dimensional vector spaces defined over F as algebraicvarieties over F . Heights
Let G be a reductive group and G = G ( F ). Following Harish-Chandra, we define a height function k · k on G valued in R ≥ . If T is asub-torus of G and T = T ( F ), denote by k · k T \ G the induced height functionon G . The precise definitions and some important properties of height functionsare well discussed in [Ko, § ℓ -spaces For a group H acting on a topological X and for a subset ω ⊂ X ,we denote by ω H the set { h · x : x ∈ ω, h ∈ H } , and by cl( ω ) the closure of ω in X . For an element x ∈ X , we denote by H x the stabilizer of x in H .For a locally compact totally disconnected topological space X , we denoteby C ∞ c ( X ) the space of locally constant and compactly supported C -valuedfunctions, and by D ( X ) the space of distributions on X . For f ∈ C ∞ c ( X ), wedenote by Supp( f ) its support. Suppose that H (an ℓ -group) acts on X . Then H acts on C ∞ c ( X ) by( h · f )( x ) = f ( h − · x ) , where h ∈ H, f ∈ C ∞ c ( X ) , x ∈ X, and acts on D ( X ) by h h · T, f i = h T, h · f i , where T ∈ D ( X ) , f ∈ C ∞ c ( X ) . For a locally constant character η : H → C × , we say that a distribution T ∈D ( X ) is ( H, η )-invariant if h · T = η ( h ) T for each h ∈ H . We denote by D ( X ) H,η the space of (
H, η )-invariant distributions on X . If X is a finite dimensionalspace and the Fourier transform f b f on C ∞ c ( X ) has already been defined, for T ∈ D ( X ), we denote by b T its Fourier transform, which is a distribution on X defined by b T ( f ) = T ( b f ). Fourier transforms
Let G be a reductive group, g its Lie algebra. Fix anondegenerate symmetric bilinear form h , i on g ( F ), which is invariant underconjugation. For each subspace f of g ( F ) on which the restriction of h , i isnondegenerate, we always equip this subspace with the self-dual Haar measurewith respect to the bi-character ψ ( h , i ). Define the Fourier transform f b f on C ∞ c ( f ) by b f ( X ) = Z f f ( Y ) ψ ( h X, Y i ) d Y. Then ˆˆ f ( X ) = f ( − X ). 7 eil index At last, we recall the definition of Weil index γ ψ associated to aquadratic space. Let q be a nondegenerate quadratic form on a finite dimensionalvector space V over F . If L ⊂ V is an O F -lattice, set i ( L ) = R L ψ ( q ( v ) /
2) d v and e L = { v ∈ V : ∀ ℓ ∈ L, ψ ( q ( v, ℓ )) = 1 } . It is well known that, if e L ⊂ L ,then | i ( L ) | = vol( L ) vol( e L ) , and i ( L ) | i ( L ) | − is independent of L . We denoteby γ ψ ( q ) the value i ( L ) | i ( L ) | − , assuming e L ⊂ L . Recall that γ ψ ( q ) is an 8throot of unity. In this section, we recall some basic theory and necessary results for generalsymmetric pairs. We refer the reader to [AG] and [RR] for most of the contents.
Fix a reductive group H and an affine variety X with an action by H , bothdefined over F . Write H = H ( F ) and X = X ( F ). Then the categoricalquotient X / H of X by H exists. In fact, X / H = Spec( O ( X ) H ). Let π denotethe natural maps X → X / H and X → ( X / H )( F ).Let x ∈ X . We say that x is • H -semisimple or H -semisimple if H x is Zariski closed in X (or equiva-lently, Hx is closed in X for the analytic topology), • H -regular or H -regular if the stabilizer H x has minimal dimension.We usually say semisimple or regular without mentioning H if there is no confu-sion. Denote by X rs (resp. X ss ) the set of regular semisimple (resp. semisimple)elements in X .If X is an F -rational finite dimensional representation of H , say a point x ∈ X nilpotent if 0 ∈ cl( Hx ). Let N denote the set of nilpotent elements in X , which is called the null-cone of X . Note that N = π − ( π (0)).An open subset U ⊂ X is called saturated if there exists an open subset V ⊂ ( X / H )( F ) such that U = π − ( V ).For x ∈ X a semisimple element, we denote by N XHx,x the normal space of Hx at x . Then the stabilizer H x acts naturally on the vector space N XHx,x . Wecall ( H x , N XHx,x ) the sliced representation at x , or the descendent of ( H, X ) at x . Then we have the following analytic Luna Slice Theorem (cf. [AG, Theorem2.3.17]): there exist • an open H -invariant neighborhood U x of Hx in X with an H -equivariantretract p : U x → Hx , • and an H x -equivariant embedding ψ : p − ( x ) ֒ → N XHx,x with an opensaturated image such that ψ ( x ) = 0.Write Z x = p − ( x ) and N x = N XHx,x . We call ( U x , p, ψ, Z x , N x ) an analyticLuna slice at x . Let y ∈ p − ( x ) and z := ψ ( y ). Then we have (cf. [AG,Corollary 2.3.19]): • ( H x ) z = H y , 8 N XHy,y ≃ N N x H x z,z as H y spaces, • y is H -semisimple if and only if z is H x -semisimple. A symmetric pair is a triple ( G , H , θ ) where H ⊂ G are reductive groups, and θ is an involution of G such that H = G θ is the subgroup of fixed points.For a symmetric pair ( G , H , θ ) we define an anti-involution ι : G → G by ι ( g ) = θ ( g − ). Set G ι = { g ∈ G ; ι ( g ) = g } and define a symmetrization map s : G → G ι , s ( g ) = gι ( g ) . By this symmetrization map we can view the symmetric space S := G/H asa subset of G ι ( F ). We consider the action of H × H on G by left and righttranslation and the action of H on G ι by conjugation.Let θ act by its differential on g = Lie( G ). Write h = Lie( H ). Thus, h = { X ∈ g : θ ( X ) = X } . Put s = { X ∈ g : θ ( X ) = − X } , on which H acts by adjoint action. We also call s the Lie algebra of S forsimplicity, though, in fact s is not a Lie algebra. We always write X h = h − · X = Ad( h − ) X for h ∈ H and X ∈ s . There exists a G -invariant θ -invariantnondegenerate symmetric bilinear form h , i on g . In particular, g = h ⊕ s is anorthogonal direct sum with respect to h , i .Let ( G , H , θ ) be a symmetric pair. Let g ∈ G be H × H -semisimple, and x = s ( g ). Then the triple ( G x , H x , θ | G x ) is still a symmetric pair, and we have(cf. [AG, Proposition 7.2.1]) • x is semisimple (both as an element of G and with respect to the H -action), • H x ≃ ( H × H ) g and s x ≃ N GHgH,H as H x -spaces, where s x is the centralizerof x in s ( F ).A symmetric pair obtained in this way is called a descendant of ( G , H , θ ). Notethat s x can be identified with the Lie algebra of G x / H x . Weyl integration formula
Let ( G , H , θ ) be a symmetric pair. Denote by s rs the regular and semisimple locus in s with respect to the H -action. We calla torus T of G θ -split if θ ( t ) = t − for all t ∈ T . Fix a Cartan subspace c of s , which by definition is a maximal abelian subspace of s consisting of H -semisimple elements. We always assume that a Cartan subspace is F -rationalwhen we mention it. Then there is an F -rational θ -split torus denoted by T − whose Lie algebra is c . Denote by c reg the H -regular locus in c . Let T be thecentralizer of c in H , which is a torus. Write t = Lie( T ).For X ∈ c reg ( F ), we now introduce the factor | D s ( X ) | F . Consider themorphism β : ( T \ H ) × c −→ s , ( h, X ) X h , , X ). The Jacobian of the differential d β at (1 , X ) is equalto | D s ( X ) | F := | det(ad( X ); h / t ⊕ s / c ) | F . Denote by S c the set of roots of T − in g ( ¯ F ). For any α ∈ S c , since c ⊂ s ,we have θ ( α ) = − α . Therefore θ interchanges the root subspaces g α and g − α .Fix a set of positive abstract roots in S c , and choose a basis { E , E , ..., E k } ofroot vectors for the direct sum of g α with α >
0. Set g = ⊕ α ∈ S c g α so that g = t ⊕ c ⊕ g . Then over ¯ F , • { E , E , ..., E k } ∪ { θ ( E ) , θ ( E ) , ..., θ ( E k ) } is a basis for g ; • { E − θ ( E ) , E − θ ( E ) , ..., E k − θ ( E k ) } is a basis for s := s ∩ g ; • { E + θ ( E ) , E + θ ( E ) , ..., E k + θ ( E k ) } is a basis for h := h ∩ g .Under the adjoint action, elements of c map h to s and vice versa. There isan involution ̺ on g whose +1-eigenspace is ⊕ α> g α and whose − ⊕ α< g α . Then ̺ interchanges s and h , and ̺ commutes with ad( X ) for X in c ( F ). Thus we have | D s ( X ) | F = | det( ̺ ◦ ad( X ); h / t ) | F = | det( ̺ ◦ ad( X ); s / c ) | F . For a Cartan subspace c , let M be its normalizer in H , W c := M/T be itsWeyl group. The map ( T \ H ) × c reg ( F ) −→ s rs ( F )obtained from β by restriction is a local isomorphism of p -adic manifolds andits image, denoted by s c rs , is open in s ( F ). The fiber of β through ( h, X ) ∈ ( T \ H ) × c reg ( F ) has | W c | elements. We have s rs ( F ) = G c s c rs , where the union runs over a (finite) set of representatives c for the set of H -conjugacy classes of F -rational Cartan subspaces in s . Then, for f ∈ C ∞ c ( s ( F )),we have the following Weyl integration formula (cf. [RR, page 106]) Z s ( F ) f ( X ) d X = X c | W c | Z c reg ( F ) | D s ( X ) | F Z T \ H f ( X h ) d h d X. The null-cone
Denote by N the null-cone of s ( F ) with respect to the H -action. Then, by [AG, Theorem 7.3.8], N is also the set of nilpotent elements(considered as elements in g ) in s ( F ). It is known that N consists of finitelymany H -orbits. Denote by N q the union of all H -orbits in N of dimension ≤ q ,which is closed in N q +1 .Fix X = 0 in N . Denote by X H the H -orbit of X , and h X the centralizerof X in h ( F ). Write r = dim h X . Then X H is of dimension d − r where d = dim h ( F ), and is open in N d − r . Lemma 3.1.
There exists a group homomorphism φ : SL ( F ) → G such that d φ (cid:18)(cid:18) (cid:19)(cid:19) = X , d φ (cid:18)(cid:18) (cid:19)(cid:19) =: Y , φ (cid:18)(cid:18) t t − (cid:19)(cid:19) =: D t ( X ) , with Y ∈ s ( F ) and D t ( X ) ∈ H . roof. See [AG, Lemma 7.1.11].We write d( X ) = d φ (cid:18)(cid:18) − (cid:19)(cid:19) , which is in h ( F ). Actually, we oftenwrite d = d( X ) when there is no confusion. For any X ∈ s ( F ), we denote by s X (resp. g X ) the centralizer of X in s ( F ) (resp. g ( F )). Lemma 3.2.
We have s Y ⊕ [ X , h ( F )] = s ( F ) , s X ⊕ [ Y , h ( F )] = s ( F ) . Proof.
We have the following decompositions (cf. [HC1, page 73]) g Y ⊕ [ X , g ( F )] = g X ⊕ [ Y , g ( F )] = g ( F ) . From the decomposition g ( F ) = h ( F ) ⊕ s ( F ) , we see that g X = h X ⊕ s X , g Y = h Y ⊕ s Y , since[ X , h ( F )] ⊂ s ( F ) , [ X , s ( F )] ⊂ h ( F ) , [ Y , h ( F )] ⊂ s ( F ) , [ Y , s ( F )] ⊂ h ( F ) . Thus we have( h Y ⊕ s Y ) M ([ X , s ( F )] ⊕ [ X , h ( F )]) = h ( F ) ⊕ s ( F ) , and ( h X ⊕ s X ) M ([ Y , s ( F )] ⊕ [ Y , h ( F )]) = h ( F ) ⊕ s ( F ) . Taking the s -parts of the above identities, we prove the assertions of the lemma.Let Γ be the Cartan subgroup of H with the Lie algebra F · d( X ). Let ξ be the rational character of Γ defined by X γ = ξ ( γ ) X , Y γ = ξ − ( γ ) Y , which is not trivial. Let r ′ = dim s Y . The following lemma essentially is avariant of [HC1, Lemma 34], and the proof is also similar to that of [HC1,Lemma 34]. Lemma 3.3.
We can choose a basis Y = U , U , ..., U r ′ for s Y and rationalcharacters ξ , ξ , ..., ξ r ′ of Γ such that1. ξ i = ξ λ i , λ i ≥ ,2. ad( − d) U i = λ i U i ,3. U γi = ξ − i ( γ ) U i , for all ≤ i ≤ r ′ . Set m = 12 X ≤ i ≤ r ′ λ i = 12 Tr (cid:0) ad( − d) | s Y (cid:1) . Symmetric pairs II: specific cases
Now we focus on the symmetric pairs concerned in this article. The notationsintroduced here will be used without mention from now on. ( G , H ) Let G = GL n and H = GL n × GL n , both defined over F . H is viewed as asubgroup of G by embedding it into G diagonally. Let ǫ = (cid:18) n − n (cid:19) anddefine an involution θ on G by θ ( g ) = ǫgǫ . Then H = G θ , and the Lie algebra s associated to ( G , H , θ ) is s ( F ) = (cid:26)(cid:18) AB (cid:19) : A, B ∈ gl n ( F ) (cid:27) ≃ gl n ( F ) ⊕ gl n ( F ) . If we identify s ( F ) with gl n ( F ) ⊕ gl n ( F ), then H acts on s ( F ) by( h , h ) · ( A, B ) = ( h Ah − , h Bh − ) . Recall that we write X h = h − · X for h ∈ H, X ∈ s ( F ). We fix a nondegeneratesymmetric bilinear form h , i on g ( F ) defined by h X, Y i = tr( XY ) , for X, Y ∈ g ( F ) . Then h , i is both G -invariant and θ -invariant.Since H ( F, H ) is trivial, we have S = S ( F ) where S := G/H and S := G / H . We identify S with its image in G ι ( F ) by the the symmetrization map s . When we want to emphasize the index n , we write G n , H n , θ n and s n . Descendants
Now we describe all the H -semisimple elements x of S and s ( F )and the descendants ( H x , s x ) at x . The results below also hold when F = k isa number field. Proposition 4.1.
1. Each semisimple element x of S is H -conjugate to anelement of the form x ( A, n , n ) := A A − m n − n A + m A n
00 0 0 0 0 − n , with n = m + n + n , A ∈ gl m ( F ) being semisimple without eigenvalues ± and unique up to conjugation. Moreover, x ( A, n , n ) is regular if andonly if n = n = 0 and A is regular in gl n ( F ) .2. Let x = x ( A, n , n ) in S be semisimple. Then the descendant ( H x , s x ) isisomorphic to the product (as a representation) (GL m ( F ) A , gl m ( F ) A ) × ( H n , s n ) × ( H n , s n ) . Here GL m ( F ) A and gl m ( F ) A are the centralizers of A in GL m ( F ) and gl m ( F ) respectively, and GL m ( F ) A acts on gl m ( F ) A by conjugation. roof. See [JR, Proposition 4.1] or [Gu1, Proposition 1.1] for the first assertion.The second assertion can be proved by a direct computation.
Proposition 4.2.
1. Each semisimple element X of s ( F ) is H -conjugate toan element of the form X ( A ) = m
00 0 0 0 A with A ∈ GL m ( F ) being semisimple and unique up to conjugation. More-over, X ( A ) is regular if and only if m = n and A ∈ GL n ( F ) is regular.2. Let X = X ( A ) in s ( F ) be semisimple. Then the descendant ( H X , s X ) isisomorphic to the product (as a representation) (GL m ( F ) A , gl m ( F ) A ) × ( H n − m , s n − m ) . Proof.
See [JR, Propositions 2.1 and 2.2].
The null-cone
Fix X = 0 in the null-cone N of s ( F ). Let ( X , d , Y ) be an sl -triple as before. Recall d = d( X ). Lemma 4.3.
We have dim s Y = dim h X = r .Proof. It follows from Lemma 3.2 and the relationdim h X + dim[ X , h ( F )] = dim h ( F ) = dim s ( F ) . In [JR, Lemma 3.1], h X is well studied, and an upper bound for Tr (cid:0) ad(d) | h X (cid:1) is given there. By a minor modification of the discussion in [JR, § s Y . For our purpose, we want to compare r + m with n + n ,where r = dim s Y and m = Tr (cid:0) ad( − d) | s Y (cid:1) . The following inequalities willbe used in § Proposition 4.4.
We have the relations1. r ≥ n ,2. r + m > n + n .Proof. Write Y = Y for short. Let V = V ⊕ V , where V i = F n , ≤ i ≤
1. We identify g ( F ) = Hom( V, V ), h ( F ) = Hom( V , V ) ⊕ Hom( V , V ) and s ( F ) = Hom( V , V ) ⊕ Hom( V , V ). Given Y , there is a decomposition V = W ⊕ W ⊕ · · · ⊕ W k , where each W i is an indecomposable F [ Y ]-submodule.We can choose a generator z i of W i such that z i is in either V or V . Definedeg( z i ) = 0 if z i ∈ V , otherwise deg( z i ) = 1. Write w i = dim W i . There is anisomorphism from s Y to some space Z = ⊕ ≤ i,j ≤ k S ij . Now we describe S ij precisely. An element b ij ∈ S ij is in F [ X ] / ( X w j ) of theform: 13. b ij ( X ) = P max { w j − w i , }≤ ℓ 1. For ≤ i ≤ k , if w i = 2 p i or p i + 1 , we have r ii = p i , m ii = p i . 2. For ≤ i < j ≤ k , we have the following table. w i , w j δ i δ j m ij r ij w i = 2 p i , w j = 2 p j p i p j p i , p j ) w i = 2 p i , w j = 2 p j − p i p j − p i , p j ) 2 min( p i , p j ) w i = 2 p i , w j = 2 p j + 1 , w i < w j ± p i p j p i w i = 2 p i , w j = 2 p j + 1 , w i > w j p i p j + 2( p i − p j ) − p j + 1 w i = 2 p i , w j = 2 p j + 1 , w i > w j − p i p j p j + 1 w i = 2 p i + 1 , w j = 2 p j + 1 1 2 p i p j p i , p j ) w i = 2 p i + 1 , w j = 2 p j + 1 − p i p j + 2 sup( p i , p j ) 2 min( p i , p j ) + 2Now we continue to prove the proposition.(1) The first inequality of the proposition can be read off from the abovelist. It is not hard to see that r = n if and only if Y n = 0 and Y n − = 0.(2) For the second inequality, compare with the proof of [JR, Lemma 3.1].We denote by u the number of indices i such that w i is odd and δ i = 1, whichis equal to the number of indices j such that w j is odd and δ j = − 1. Then n = u + X ≤ i ≤ k p i , where w i = 2 p i or 2 p i + 1. Thus n + n u + u (cid:18) u + 12 (cid:19) X ≤ i ≤ k p i + X ≤ i ≤ k p i + 2 X ≤ i 14n the other hand r + m = X ≤ i ≤ k ( r ii + m ii ) + X ≤ i Now we describe all the H ′ -semisimple elements x of S ′ and s ′ ( F ) and the descendants at x . The results below also hold when F = k is anumber field. Proposition 4.6. 1. Each semisimple elements y of S ′ is H ′ -conjugate to n element of the form y ( A, n , n ) = A γB n − n B A n 00 0 0 0 0 − n , with A ∈ gl m ( F ) being semisimple and unique up to conjugation such that A − m ∈ γ N(GL m ( E )) and B ∈ GL m ( E ) is a matrix unique up to twistedconjugation such that A − m = γB ¯ B , AB = BA , and n = m + n + n .Moreover, y ( A, n , n ) is regular if and only if n = n = 0 and A isregular in gl n ( F ) .2. Let y = y ( A, n , n ) in S ′ be semisimple. Then the descendant ( H ′ y , s ′ y ) isisomorphic to the product (as a representation) (GL m ( E ) A ∩ GL σ,m ( E ) B , gl m ( E ) A ∩ gl σm ( E ) B ) × ( H ′ n , s ′ n ) × ( H ′ n , s ′ n ) . Here GL σ,m ( E ) B := (cid:8) h ∈ GL m ( E ) : hB = B ¯ h (cid:9) , gl σm ( E ) B := (cid:8) X ∈ gl m ( E ) : X ¯ B = B ¯ X (cid:9) , and GL σ,m ( E ) B acts on gl σm ( E ) B by σ -twisted conjugation.Proof. See [Gu1, Proposition 1.2] for the first assertion. The second assertioncan be proved by a direct computation. Proposition 4.7. 1. Each semisimple element Y of s ′ ( F ) is H ′ -conjugateto an element of the form Y ( A ) = γB 00 0 0 0¯ B where A ∈ GL m ( F ) is semisimple and unique up to conjugation such that A ∈ γ N(GL m ( E )) and B ∈ GL m ( E ) is a matrix unique up to twistedconjugation such that A = γB ¯ B . Moreover, Y ( A ) is regular if and only if A ∈ GL n ( F ) is regular.2. Let Y = Y ( A ) in s ′ ( F ) be semisimple. Then the descendant ( H ′ Y , s ′ Y ) isisomorphic to the product (as a representation) (GL σ,m ( E ) B , gl σm ( E ) B ) × ( H ′ n − m , s ′ n − m ) . Proof. See [Gu2, Lemma 2.1] for the first assertion. The second assertion canbe proved by a direct computation. 17 he null-cone Fix X = 0 in the null-cone N ′ of s ′ ( F ). Let ( X , d , Y ) bean sl -triple as before. By the same proof as Lemma 4.3, we havedim s ′ Y = dim h ′ X . Write r = dim s ′ Y and m = Tr (cid:16) ad( − d) | s ′ Y (cid:17) . We still want to compare r + m with n + n , which is easier in this case. Proposition 4.8. We have r + m > n + n and m ′ < n where m ′ = Tr (cid:16) ad( − d) | h ′ Y (cid:17) .Proof. Write Y = (cid:18) γA ¯ A (cid:19) . If we change ( X , d , Y ) to be any triple in the H ′ -orbit of ( X , d , Y ), the numbers r and m are unchanged. By [Gu2, Lemma2.2], we can choose A to be of the Jordan normal form. At the same time, wecan also choose d to be in gl n ( F ). In such situation, it is easy to see that thereis a d-equivariant isomorphism s ′ Y ≃ h ′ Y . Thus r = r ′ and m = m ′ , where r ′ = dim h ′ Y . Since g ′ Y = h ′ Y ⊕ s ′ Y , we have m + m ′ = (cid:0) n − r − r ′ (cid:1) . Thuswe get m = (cid:0) n − r (cid:1) and r + m = n + r . The inequality r ≥ n impliesthe lemma. In this section, we introduce the main object of this article: the smooth transferbetween Schwartz functions on different symmetric spaces. By several reductionsteps, we explain why Theorem 5.16 implies Theorem 5.13 in details. Matching of orbits We first recall the matching between semisimple orbitsin symmetric spaces S and S ′ , and then give the definition of matching betweensemisimple orbits in Lie algebras s ( F ) and s ′ ( F ). These definitions of matchingorbits also hold when F = k is a number field. Proposition 5.1. 1. For each semisimple element y of S ′ , there exists h ∈ H ( E ) such that hyh − belongs to S . This establishes an injection of the H ′ -semisimple orbits in S ′ into the H -semisimple orbits in S , which car-ries the orbit of y ( A, n , n ) in S ′ to the orbit of x ( A, n , n ) in S .2. For each semisimple element Y of s ′ ( F ) , there exists h ∈ H ( E ) such that hY h − belongs to s ( F ) . This establishes an injection of the H ′ -semisimpleorbits in s ′ ( F ) into the H -semisimple orbits in s ( F ) , which carries theorbit of Y ( A ) in s ′ ( F ) to the orbit of X ( A ) in s ( F ) .Proof. See [Gu1, Proposition 1.3] for the first assertion. The second assertioncan be proved in the same way. Definition 5.2. 1. We say that y ∈ S ′ ss (resp. Y ∈ s ′ ss ( F )) matches x ∈ S ss (resp. X ∈ s ss ( F )) and write x ↔ y (resp. X ↔ Y ) if the above mapsends the orbit of y (resp. Y ) to the orbit of x (resp. X ).18. We say that x ∈ S ss (resp. X ∈ s ss ( F )) comes from S ′ ss (resp. s ′ ss ( F )) ifthere exists y ∈ S ′ ss (resp. Y ∈ s ′ ss ( F )) such that x ↔ y (resp. X ↔ Y ).We denote by S ss , (resp. s ss ( F ) ) the subset of elements in S ss (resp. s ss ( F )) coming from S ′ ss (resp. s ′ ss ( F )). Remark . Denote by Q (resp. Q ′ ) the categorical quotient S / H (resp. S ′ / H ′ ), and by q (resp. q ′ ) the categorical quotient s / H (resp. s ′ / H ′ ). Themaps in Proposition 5.1 induce natural maps Q ′ ֒ → Q , and q ′ ֒ → q . Actually, Q is isomorphic to the affine space A n , and the quotient map π : S →Q is given by (cid:18) A BC D (cid:19) (cid:0) tr ∧ i BC (cid:1) , i = 1 , , ..., n. The natural map Q ′ ֒ → Q is induced by S ′ −→ Q , (cid:18) A γB ¯ B ¯ A (cid:19) (cid:0) tr ∧ i γB ¯ B (cid:1) , i = 1 , ..., n. Similarly, q is isomorphic to the affine space A n , and the quotient map π : s → q is given by (cid:18) AB (cid:19) (cid:0) tr ∧ i AB (cid:1) , i = 1 , , ..., n. The natural map q ′ ֒ → q is induced by s ′ −→ q , (cid:18) γB ¯ B ¯0 (cid:19) (cid:0) tr ∧ i γB ¯ B (cid:1) , i = 1 , ..., n. Remark . A semisimple element x = x ( A, n , n ) in S ss comes from S ′ ss if andonly if A − m ∈ γ N(GL m ( E )) where m = n − n − n . A semisimple element X = X ( A ) in s ss ( F ) comes from s ′ ss ( F ) if and only if A ∈ γ N(GL m ( E )). Remark . Suppose that x ∈ S ss and y ∈ S ′ ss match. We want to compare( H x , s x ) with ( H ′ y , s ′ y ). It suffices to assume that x = x ( A, n , n ) and y = y ( A, n , n ). Thus, by Propositions 4.1 and 4.6, we have( H x , s x ) ≃ (GL m ( F ) A , gl m ( F ) A ) × ( H n , s n ) × ( H n , s n ) , and( H ′ y , s ′ y ) ≃ (GL m ( E ) A ∩ GL σ,m ( E ) B , gl m ( E ) A ∩ gl σm ( E ) B ) × ( H ′ n , s ′ n ) × ( H ′ n , s ′ n )with A − m = γB ¯ B and AB = BA . By the proof of Lemma 5.26 below, wesee that (GL m ( E ) A ∩ GL σ,m ( E ) B , gl m ( E ) A ∩ gl σm ( E ) B ) essentially is an innerform of (GL m ( F ) A , gl m ( F ) A ). The other factors in the descendants are relatedin a similar manner as ( H, s ) and ( H ′ , s ′ ) are. For X ∈ s ss ( F ) and Y ∈ s ′ ss ( F )such that X ↔ Y , by Propositions 4.2 and 4.7 and Lemma 5.26, the factors ofthe descendants ( H X , s X ) and ( H ′ Y , s ′ Y ) have the similar relations as above. Remark . It is obvious that the maps in Proposition 5.1 send regular semisim-ple orbits to regular semisimple ones. We denote by S rs , (resp. s rs ( F ) ) thesubset of elements in S rs (resp. s rs ( F )) coming from S ′ rs (resp. s ′ rs ( F )). Supposethat x ∈ S rs (resp. x ∈ s rs ( F )) and y ∈ S ′ rs (resp. y ∈ s ′ rs ( F )) match. Then bythe above remark, we see that H x is an inner form of H ′ y . Since they are torus,we have H x ≃ H ′ y . ransfer factors To state our results on smooth transfer, we need to definetransfer factors for the symmetric pair ( G , H , θ ) and its descendants. In general,the transfer factor is defined as follows (cf. [Zhw1, Definition 3.2]). Definition 5.7. Let a reductive group H act on an affine variety X , bothdefined over F . Let η be a quadratic character of H . Suppose that for allregular semisimple x ∈ X = X ( F ), the character η is trivial on the stabilizer H x . Then a transfer factor is a smooth function κ : X rs → C × such that κ ( x h ) = η ( h ) κ ( x ) for any h ∈ H . Definition 5.8. For convenience, we give an explicit definition of various trans-fer factors in our situation as follows: • type ( H, S ): for x = (cid:18) A BC D (cid:19) ∈ S regular semisimple, define κ ( x ) := η (det( B )); • type ( H m , s m ): for X = (cid:18) AB (cid:19) ∈ s m ( F ) regular semisimple, define κ ( X ) := η (det( A )); • type (GL m ( F ) A , gl m ( F ) A ): we define κ to be the constant function withvalue 1.In the cases (1) and (2), η is the non-trivial quadratic character on F × associatedto E , while in the case (3) η is the trivial character. In all the cases, it is easyto see that η is trivial on the stabilizers H x . Smooth transfer Now we give the definition of smooth transfer. First, wefix Haar measures on H and H ′ . Notice that, for x ∈ S rs (resp. x ∈ s rs ( F ))and y ∈ S ′ rs (resp. y ∈ s ′ rs ( F )) such that x ↔ y , their stabilizers H x and H ′ y are isomorphic to each other (see Remark 5.6), and we fix such an isomorphism.Fix a Haar measure on H x for each x ∈ S rs (resp. x ∈ s rs ( F )). We fix a Haarmeasure on H ′ y for each y ∈ S ′ rs (resp. y ∈ s ′ rs ( F )) which is compatible withthat of H x if x ↔ y . Definition 5.9. For x ∈ S rs (resp. x ∈ s rs ( F )) and f ∈ C ∞ c ( S ) (resp. f ∈C ∞ c ( s ( F ))), define the orbital integral of f at x to be O η ( x, f ) := Z H x \ H f ( x h ) η ( h ) d h. For y ∈ S ′ rs (resp. y ∈ s ′ rs ( F )) and f ′ ∈ C ∞ c ( S ′ ) (resp. f ′ ∈ C ∞ c ( s ′ ( F ))), definethe orbital integral of f ′ at y to be O ( y, f ) := Z H ′ y \ H ′ f ( x h ) d h. Definition 5.10. 1. For f ∈ C ∞ c ( S ) and f ′ ∈ C ∞ c ( S ′ ), we say that f and f ′ are smooth transfer of each other if for each x ∈ S rs κ ( x ) O η ( x, f ) = ( O ( y, f ′ ) , if there exists y ∈ S ′ rs such that x ↔ y, , otherwise . We denote by C ∞ c ( S ) the subspace of elements f in C ∞ c ( S ) satisfying that O η ( x, f ) = 0 for any x in S rs but not in S rs , .20. For f ∈ C ∞ c ( s ( F )) and f ′ ∈ C ∞ c ( s ′ ( F )), we say that f and f ′ are smoothtransfer of each other if for each X ∈ s rs ( F ) κ ( X ) O η ( X, f ) = ( O ( Y, f ′ ) , if there exists Y ∈ s ′ rs ( F ) such that X ↔ Y, , otherwise . We denote by C ∞ c ( s ( F )) the subspace of elements f in C ∞ c ( s ( F )) satisfyingthat O η ( X, f ) = 0 for any X in s rs ( F ) but not in s rs ( F ) . Remark . The definition of smooth transfer depends on the Haar measureson H and H ′ , but the existence of smooth transfer does not depend on them.Sometimes, we will write transfer in place of smooth transfer for short. Remark . For semisimple x ∈ S and semisimple y ∈ S ′ such that x ↔ y ,by Remark 5.5, we can define the notion of smooth transfer between elementsin C ∞ c ( s x ( F )) and those in C ∞ c ( s ′ y ( F )), determined by the orbital integrals withrespect to the action of H x on s x ( F ), the action of H ′ y on s ′ y ( F ), and the transferfactor κ defined as above. Similarly, for semisimple X ∈ s ( F ) and semisimple Y ∈ s ′ ( F ) such that X ↔ Y , we can also define the notion of smooth transferbetween elements in C ∞ c ( s X ( F )) and those in C ∞ c ( s ′ Y ( F )).Our main theorems are as follows. Theorem 5.13. For each f ′ ∈ C ∞ c ( S ′ ) , there exists f ∈ C ∞ c ( S ) that is a smoothtransfer of f ′ . Conversely, for each f ∈ C ∞ c ( S ) , there exists f ′ ∈ C ∞ c ( S ′ ) thatis a smooth transfer of f . Theorem 5.14. For each f ′ ∈ C ∞ c ( s ′ ( F )) , there exists f ∈ C ∞ c ( s ( F )) thatis a smooth transfer of f ′ . Conversely, for each f ∈ C ∞ c ( s ( F )) , there exists f ′ ∈ C ∞ c ( s ′ ( F )) that is a smooth transfer of f . In the later subsections, we will show that Theorem 5.14 implies Theorem5.13. Lemma 5.15. To prove Theorem 5.14, it suffices to prove it for the case s = s ǫ when ǫ = 1 .Proof. Let s ′ ( F ) = (cid:26) Y ( B ) = (cid:18) B ¯ B (cid:19) : B ∈ gl n ( E ) (cid:27) . Choose a representative γ ∈ F × of the nontrivial element in F × / N E × . Let s ′ γ ( F ) = (cid:26) Y γ ( B ) = (cid:18) γB ¯ B (cid:19) : B ∈ gl n ( E ) (cid:27) . Identify H ′ with GL n ( E ). Then there is a natural H ′ -equivariant isomorphism j : s ′ ( F ) ∼ → s ′ γ ( F ) , Y ( B ) Y γ ( B ) , which implies the lemma. 21 ourier transform Define the Fourier transform f b f on C ∞ c ( s ( F )) (resp. C ∞ c ( s ′ ( F ))) with respect to the fixed bilinear form h , i and the additive charac-ter ψ . The following theorem is the key point in proving the existence of smoothtransfer. Theorem 5.16. There exists a nonzero constant c ∈ C such that if f ∈C ∞ c ( s ( F )) and f ′ ∈ C ∞ c ( s ′ ( F )) are smooth transfer of each other then b f and c b f ′ are also smooth transfer of each other. In the later subsections, we will prove the following main result of this sec-tion. Proposition 5.17. Theorem 5.16 implies Theorem 5.14. In this subsection, we prove the following fundamental lemma (Lemma 5.18).This is an important example of the smooth transfer and also a crucial lemmafor us to prove Theorem 5.16 by using global method.Now assume that γ = 1. Thus, G ′ is isomorphic to G . Suppose that F is of odd residual characteristic and E is unramified over F . We choose theHaar measures on H and H ′ so that vol( H ( O F )) = 1 and vol( H ′ ( O F )) = 1respectively.Let f ∈ C ∞ c ( s ( F )) and f ′ ∈ C ∞ c ( s ′ ( F )) be the characteristic functions of thestandard lattices L = gl n ( O F ) ⊕ gl n ( O F ) , L ′ = gl n ( O E )respectively. Lemma 5.18. f and f ′ are smooth transfer of each other.Remark . The group version of the above fundamental lemma was provedin [Gu1] (cf. [Gu1, Theorem]). Proof. Let X ∈ s rs ( F ). It suffices to consider X of the form (cid:18) n A (cid:19) with A ∈ GL n ( F ) being regular semisimple. Then we have κ ( X ) O η ( X, f ) = Z H X ( F ) \ H ( F ) f ( h − h , h − Ah ) η ( h h ) d h d h = Z (GL n ( F ) A \ GL n ( F )) × GL n ( F ) f ( h , h − h − Ah ) η ( h ) d h d h . Let K = GL n ( O F ) and K ′ = GL n ( O E ). For r = ( r i,j ) ∈ gl n ( ¯ F ), put | r | =max i,j | r i,j | F . Then for r, t ∈ gl n ( F ), the value f ( r, t ) = 0 if and only if | r | ≤ , | t | ≤ 1. Let Φ A be the characteristic function of the set of ( r, t ) ∈ GL n ( F ) × GL n ( F ) satisfying | r | ≤ , | t | ≤ | det( rt ) | F = | det A | F . Then Φ A belongsto C ∞ c (GL n ( F ) × GL n ( F )) and is bi- K -invariant both for the variables r and t .Let Ψ A be the function on GL n ( F ) defined byΨ A ( g ) = Z GL n ( F ) Φ A ( h, h − g ) η ( h ) d h. A belongs to C ∞ c (GL n ( F )), and is bi- K -invariant (that is, Ψ A is a Heckefunction). We have κ ( X ) O η ( X, f ) = Z GL n ( F ) A \ GL n ( F ) Ψ A ( g − Ag ) d g. If Y = (cid:18) B ¯ B (cid:19) ∈ s ′ rs ( F ), we have O ( Y, f ′ ) = Z GL σ,n ( E ) B \ GL n ( E ) f ′ ( h − B ¯ h ) d h. Let Ψ B be the characteristic function of the set of r ∈ GL n ( E ) satisfying | r | ≤ | det r | F = | det B | F . Then Ψ B belongs to C ∞ c (GL n ( E )), and is bi- K ′ -invariant. We have O ( Y, f ′ ) = Z GL σ,n ( E ) B \ GL n ( E ) Ψ B ( h − B ¯ h ) d h. Denote by bc : H (GL n ( E ) , K ′ ) −→ H (GL n ( F ) , K )the base change map between the two spaces of Hecke functions. Then, in fact, itwas shown in [Gu1, Corollary 3.7] (can be read off from the proof of Proposition3.7 loc. cit.) that Ψ A = 0 if A / ∈ N(GL n ( E )), and Ψ A = bc(Ψ B ) if A = B ¯ B .Recall that Ψ A = bc(Ψ B ) implies that κ ( X ) O η ( X, f ) = O ( Y, f ′ ) , if X ↔ Y. Hence the lemma follows. The main aim of this subsection is to reduce Theorem 5.13 to Theorem 5.14.The reduction steps here are almost the same as those in [Zhw1, Section 3]. Descent of orbital integrals The following proposition essentially is [Zhw1,Proposition 3.11], whose proof is also valid here. Proposition 5.20. Let X be any one of S, S ′ , s ( F ) or s ′ ( F ) . Let x ∈ X besemisimple and ( U x , p, ψ, Z x , N x ) an analytic Luna slice at x . Then there existsa neighborhood ξ ⊂ ψ ( p − ( x )) of in N x satisfying the following properties: • for each f ∈ C ∞ c ( X ) , there exists f x ∈ C ∞ c ( N x ) such that for all regularsemisimple z ∈ ξ with z = ψ ( y ) we have Z H y \ H f ( y h ) η ( h )d h = Z H y \ H x f x ( z h ) η ( h )d h ; • and conversely, for each f x ∈ C ∞ c ( N x ) , there exists f in C ∞ c ( X ) such thatabove equality holds for any regular semisimple z ∈ ξ .Here H = H ′ and η = when X is S ′ or s ′ ( F ) . eduction to local transfer Recall that we denote by Q (resp. Q ′ ) thecategorical quotient S / H (resp. S ′ / H ′ ), and by q (resp. q ′ ) the categoricalquotient s / H (resp. s ′ / H ′ ). By Remark 5.3, we always view Q ′ and q ′ as closedsubsets of Q and q respectively. Let X be any one of S , S ′ , s or s ′ , and Q thequotient Q , Q ′ , q or q ′ of X . Let Q ( F ) rs be the regular semisimple locus in Q ( F ). Since H ( F, H ) = H ( F, H ′ ) = 1, the natural map π : X ( F ) → Q ( F ) isa surjection. For x ∈ Q ( F ) rs , the fiber π − ( x ) consists of precisely one orbit. Definition 5.21. Let X and Q be as above. Write X = X ( F ) and Q = Q ( F ).1. Let Φ be a function on Q rs which vanishes outside a compact set of Q rs .For x ∈ Q , we say that Φ is a local orbital integral around x , if thereexists a neighborhood U of x and a function f ∈ C ∞ c ( X ) such that for all y ∈ U rs and z with π ( z ) = y we haveΦ( y ) = κ ( z ) O η ( z, f ) . 2. For f ∈ C ∞ c ( X ), define a function π ∗ ( f ) on Q rs to be: π ∗ ( f )( x ) = κ ( y ) O η ( y, f ) , for x ∈ Q rs , y ∈ π − ( x ) . Here κ = and η = when X is S ′ or s ′ .The following result is [Zhw1, Proposition 3.8]. Proposition 5.22. Let Φ be a function on Q rs which vanishes outside a compactset Ξ of Q . If Φ is a local orbital integral at each x ∈ Ξ , it is an orbital integral.Namely there exists f ∈ C ∞ c ( X ) such that for all y ∈ Q rs , and z with π ( z ) = y we have Φ( y ) = κ ( z ) O η ( z, f ) . Definition 5.23. For x ∈ Q ( F ) (resp. x ∈ q ( F )), we say that local transferaround x exists, if for each f ′ ∈ C ∞ c ( S ′ ) (resp. f ′ ∈ C ∞ c ( s ′ ( F ))), there exists f ∈ C ∞ c ( S ) (resp. f ∈ C ∞ c ( s ( F )) ) such that in a neighborhood U of x , thefollowing equality holds: π ∗ ( f ) = π ∗ ( f ′ ) on U ∩ Q ( F ) rs (resp. U ∩ q ( F ) rs ) , and conversely for each f ∈ C ∞ c ( S ) (resp. f ∈ C ∞ c ( s ( F )) ), there exists f ′ ∈C ∞ c ( S ′ ) (resp. f ′ ∈ C ∞ c ( s ′ ( F ))) satisfying the above equality. Corollary 5.24. To prove Theorems 5.13 and 5.14, it suffices to prove theexistence of local transfer around all elements of Q ( F ) and q ( F ) .Proof. This is a direct consequence of Proposition 5.22. Reduction to local transfer around zeroLemma 5.25. To prove the existence of local transfer around an element z in Q ( F ) (resp. q ( F ) ), it suffices to prove the existence of smooth transfer for thesliced representations ( H x , s x ) and (cid:0) H ′ y , s ′ y (cid:1) where x in S ss (resp. s ss ( F ) ) and y in S ′ ss (resp. s ′ ss ( F ) ) are such that x ↔ y and π ( x ) = π ( y ) = z . roof. This result partially follows from Proposition 5.20 and the fact that for f ′ ∈ C ∞ c ( S ′ ) (resp. C ∞ c ( s ′ ( F ))) and f ∈ C ∞ c ( S ) (resp. C ∞ c ( s ( F ))) the functions π ∗ ( f ′ ) and π ∗ ( f ) are locally constant on Q ( F ) rs (resp. q ( F ) rs ). It remains toprove Lemma 5.29 ahead, which shows the compatibility of the transfer factorsunder the semisimple descent. Lemma 5.26. 1. Given semisimple A ∈ gl m ( F ) such that A − m = γB ¯ B , AB = BA with B ∈ GL m ( E ) , the smooth transfer exists for the slicedrepresentations (GL m ( F ) A , gl m ( F ) A ) and (GL m ( E ) A ∩ GL σ,m ( E ) B , gl m ( E ) A ∩ gl σm ( E ) B ) . 2. Given semisimple A ∈ GL m ( F ) such that A = γB ¯ B with B ∈ GL m ( E ) ,the smooth transfer exists for the sliced representations (GL m ( F ) A , gl m ( F ) A ) and (GL σ,m ( E ) B , gl σm ( E ) B ) Proof. Firstly, we prove the second assertion. We can assume that γ = 1 and A is of the form diag( A , A , ..., A k ) such thatGL m ( F ) A = k Y i =1 GL m i ( F i ) , where F i = F [ A i ] is a field and A i is in the center of GL m i ( F i ). For each 1 ≤ i ≤ k , let L i = E ⊗ F F i . Since A ∈ N(GL m ( E )), there exists B i ∈ GL m i ( L i ) suchthat A i = N( B i ) for each i . We can choose B to be diag( B , B , ..., B k ). ThenGL σ,m i ( L i ) B i is an inner form of GL m i ( F i ), and GL σ,m ( E ) B = Q ki =1 GL σ,m i ( L i ) B i .For X ∈ gl σm ( E ) B , it is easy to see that X ¯ B ∈ gl σ,m ( E ) B , where gl σ,m ( E ) B = (cid:8) Y ∈ gl m ( E ) : Y B = B ¯ Y (cid:9) , which is the Lie algebra of GL σ,m ( E ) B . For X ∈ gl σm ( E ) B and h ∈ GL σ,m ( E ) B ,we have h − X ¯ h ¯ B = h − X ¯ Bh . Therefore, timing ¯ B on right, we get a GL σ,m ( E ) B -equivariant isomorphism gl σm ( E ) B −→ gl σ,m ( E ) B , where GL σ,m ( E ) B acts on gl σ,m ( E ) B by conjugation. Since the existence ofsmooth transfer between GL m ( F i ) and its inner forms is known, we completesthe proof.The first assertion is proved in the same way. By the above discussion, weknow that the smooth transfer holds for(GL m ( F ) A − m , gl m ( F ) A − m ) and (GL σ,m ( E ) B , gl σm ( E ) B ) . We can choose some scalar λ ∈ F so that A + λ ∈ GL m ( F ). Then A + λ ∈ GL m ( F ) A − m and A + λ ∈ GL σ,m ( E ) B . Hence(GL σ,m ( E ) B ) A + λ = GL m ( E ) A ∩ GL σ,m ( E ) B is an inner form of (GL m ( F ) A − m ) A + λ = GL m ( F ) A . The rest of the proof is the same as that of the first assertion.25 roposition 5.27. To prove the existence of local transfer around all elementsof Q ( F ) or q ( F ) , it suffices to prove the existence of local transfer around zeroof q ( F ) .Proof. By Lemma 5.25, it suffices to prove the existence of smooth transferfor the sliced representations ( H x , s x ) and ( H ′ y , s ′ y ) where x ↔ y . By Remark5.5 and Lemma 5.26, it suffices to prove the existence of smooth transfer for( H m , s m ) and ( H ′ m , s ′ m ), that is, the existence of local transfer around zero of q ( F ). Corollary 5.28. Theorem 5.14 implies Theorem 5.13. Explicit analytic Luna slices We now describe explicit analytic Luna slicesat semisimple elements of S or s ( F ). We refer the reader to [JR, page 76] forthe discussions on s , and to [JR, § S .First let X ∈ s ( F ) be semisimple. Write s ( F ) = s X ⊕ s ⊥ X , where s ⊥ X is theorthogonal complement of s X in s ( F ) with respect to h , i . Set Z = n ξ ∈ s X : det (cid:16) [ad( X + ξ ) ] | s ⊥ X (cid:17) = 0 o , which is a non-empty open set of s X and invariant under H X . Let Z X = { X + ξ : ξ ∈ Z } . Consider the map φ : H × Z X −→ s ( F ) , ( h, X + ξ ) Ad h ( X + ξ ) , which is everywhere submersive. Let U X be the image of φ , which is an open H -invariant set in s ( F ). Then Z X and U X are what we want, and ψ is thenatural map: ψ : Z X −→ s X , X + ξ ξ. Next let x ∈ S be semisimple. Write x = s ( g ) for some g ∈ G , where s isthe symmetrization map. Consider the map φ : H × G x × H −→ G, ( h, ξ, h ′ ) hξgh ′ . Let Z ′ be the set of ξ such that φ is submersive at (1 , ξ, ξ in G x such that det (cid:0) [ − Ad s ( ξg )] | g ⊥ x (cid:1) = 0 . Let W ′ = { X ∈ s x : det( + X ) det( − X ) = 0 } , which is an open neighborhood of 0 in s x . Consider the Cayley transform λ : W ′ −→ G x , X ( − X )( + X ) − , and denote by V the image of W ′ under λ . Put Z = Z ′ ∩ V and W = λ − ( Z ).Let U x be the image of φ ( H × Z × H ) under the symmetrization map s , and Z x the image of φ ( × Z × ) under s . Then Z x and U x are what we want.The lemma below follows from the above construction and a direct compu-tation by choosing x = x ( A, n , n ) and X = X ( A ) in a standard form. Weomit the proof here. Lemma 5.29. Let x ∈ S (resp. X ∈ s ( F ) ) be semisimple. Then we may choosean H x -invariant (resp. H X -invariant) neighborhood of x (resp. X ) such thatfor any regular semisimple y in this neighborhood, κ ( y ) is equal to a non-zeroconstant times κ ( ψ ( y )) . .4 Proof of Proposition 5.17 Now we can prove Proposition 5.17 with the help of the following results. Theorem 5.30. Denote by N the null-cone of s ( F ) , by N ′ the null-cone of s ′ ( F ) .1. Let T ∈ D ( s ( F )) H,η be such that Supp( T ) ⊂ N and Supp( b T ) ⊂ N . Then T = 0 .2. Let T ∈ D ( s ′ ( F )) H ′ be such that Supp( T ) ⊂ N ′ and Supp( b T ) ⊂ N ′ . Then T = 0 .Proof. The first assertion is proved in [JR, Proposition 3.1] when η is the trivialcharacter. The same proof goes through for the quadratic character η . Thesame proof is also valid for the second assertion, noting the relation m ′ < n inProposition 4.8.The following corollary is a direct consequence of the above theorem (cf.[Zhw1, Corollary 4.20]). Corollary 5.31. 1. Let C = T T ker( T ) where T runs over all ( H, η ) -invariantdistributions on s ( F ) . Then each f ∈ C ∞ c ( s ( F )) can be written as f = f + f + b f , with f ∈ C and f i ∈ C ∞ c ( s ( F ) − N ) , i = 1 , .2. Let C = T T ker( T ) where T runs over all H ′ -invariant distributions on s ′ ( F ) . Then each f ∈ C ∞ c ( s ′ ( F )) can be written as f = f + f + b f , with f ∈ C and f i ∈ C ∞ c ( s ′ ( F ) − N ′ ) , i = 1 , .Proof of Proposition 5.17. Now we assume that Theorem 5.16 is true. First weconsider the converse direction: given f ∈ C ∞ c ( s ( F )) , we want to show itssmooth transfer exists in C ∞ c ( s ′ ( F )). For a general element f in C ∞ c ( s ( F )), wesay that f ′ ∈ C ∞ c ( s ′ ( F )) is a smooth transfer of f if O ( y, f ′ ) = κ ( x ) O η ( x, f ) , x ↔ y, for each y ∈ s ′ rs ( F ). We can and do assume that: there exists a nonzero c ∈ C such that if f ′ is a smooth transfer of f ∈ C ∞ c ( s ( F )) then c b f ′ is a smooth transferof b f . This assumption is proved in Theorem 8.2. Basing on this assumption, wewill show the following stronger form of Theorem 5.14: for each f ∈ C ∞ c ( s ( F )),there exists f ′ ∈ C ∞ c ( s ′ ( F )) that is a smooth transfer of f . We use inductionargument to show this result. Suppose that the stronger form of Theorem 5.14holds for C ∞ c ( s m ( F )) and C ∞ c ( s ′ m ( F )) for every m < n . Thus, by Corollary5.24 and Lemma 5.26, for each f ∈ C ∞ c ( s ( F ) − N ), its smooth transfer exists.Therefore, by Corollary 5.31, it suffices to show the existence of smooth transferfor b f with f ∈ C ∞ c ( s ( F ) − N ), which is guaranteed by the assumption.For the other direction in Theorem 5.14 the proof is the same.27 Representability For X ∈ s rs ( F ), it is more convenient to consider the normalized orbital integral I η ( X, f ) := | D s ( X ) | F O η ( X, f ) , f ∈ C ∞ c ( s ( F )) . Similarly, for Y ∈ s ′ rs ( F ), we consider the normalized orbital integral I ( Y, f ′ ) := | D s ′ ( Y ) | F O ( Y, f ′ ) , f ′ ∈ C ∞ c ( s ′ ( F )) . If X ↔ Y it is not hard to see that | D s ( X ) | F = | D s ′ ( Y ) | F . Hence it doesnot matter if we consider the smooth transfer with respect to the normalizedorbital integrals instead of the orbital integrals introduced before. The Fouriertransform of the normalized orbital integral I ηX is defined to be b I η ( X, f ) = I η ( X, b f ) . For Y ∈ s ′ rs ( F ), we define b I Y similarly.To prove Theorem 5.16, we first need to study the Fourier transform oforbital integrals. In this section, we prove the following fundamental theoremon the representability of b I ηX and b I Y . Theorem 6.1. 1. For each X ∈ s rs ( F ) , there exists a locally constant H -invariant function b i ηX defined on s rs ( F ) which is locally integrable on s ( F ) ,such that for any f ∈ C ∞ c ( s ( F )) we have b I η ( X, f ) = Z s ( F ) b i ηX ( Y ) κ ( Y ) f ( Y ) | D s ( Y ) | − / F d Y. 2. For each X ∈ s ′ rs ( F ) , there exits a locally constant H ′ -invariant function b i X defined on s ′ rs ( F ) which is locally integrable on s ′ ( F ) , such that for any f ∈ C ∞ c ( s ′ ( F )) we have b I ( X, f ) = Z s ′ ( F ) b i X ( Y ) f ( Y ) | D s ′ ( Y ) | − / F d Y. We also write b i η ( X, Y ) (resp. b i ( X, Y )) instead of b i ηX ( Y ) (resp. b i X ( Y )), whichis viewed as a function on s rs ( F ) × s rs ( F ) (resp. s ′ rs ( F ) × s ′ rs ( F )). Then it is nothard to see that b i η ( X, Y ) (resp. b i ( X, Y )) is locally constant on s rs ( F ) × s rs ( F )(resp. s ′ rs ( F ) × s ′ rs ( F )), ( H, η )-invariant (resp. H ′ -invariant) on the first variableand H -invariant (resp. H ′ -invariant) on the second variable. Our method toprove Theorem 6.1 follows that of of [HC1] and [HC2]. Some of our treatmentalso follows that of [Ko]. We only prove the assertion for b I ηX . The assertion for b I X can be proved in the same way and is left to the reader. In this subsection we reduce the question of the representability of b I ηX to thatfor elliptic elements X ∈ s rs ( F ). For X ∈ s rs ( F ), we say that X is elliptic if itsstabilizer H X is an elliptic torus. Thus, if X = (cid:18) AB (cid:19) , X is elliptic if andonly AB is elliptic in GL n ( F ) in the usual sense.28or convenience, we suppose that X ∈ s rs ( F ) is of the form (cid:18) n A (cid:19) . Fromnow on, we also suppose that X is not elliptic, or equivalently, A is not elliptic.Then there exists a proper Levi subgroup M of GL n such that A ∈ M . Let P be a proper parabolic subgroup of GL n such that M is a Levi component of P . Let U be the unipotent subgroup of P . Set m = Lie( M ), p = Lie( P )and u = Lie( U ). Then p = m ⊕ u , and gl n = p ⊕ ¯ u where ¯ u is the Liealgebra of the unipotent subgroup ¯ U opposite to U .Write s = s + ⊕ s − , where s + = (cid:26)(cid:18) B (cid:19) : B ∈ gl n (cid:27) , s − = (cid:26)(cid:18) C (cid:19) : C ∈ gl n (cid:27) . Identify s + (resp. s − ) with gl n . Under this identification, let r + ⊂ s + (resp. r − ⊂ s − ) be the subspace that corresponds to m , n + ⊂ s + (resp. n − ⊂ s − )the subspace that corresponds to u , ¯ n + ⊂ s + (resp. ¯ n − ⊂ s − ) the subspacethat corresponds to ¯ u . Set r = r + ⊕ r − , n = n + ⊕ n − and ¯ n = ¯ n + ⊕ ¯ n − . Then s = r ⊕ n ⊕ ¯ n and X ∈ r ( F ). Notice that r is isomorphic to a product of s n i with P n i = n . Also notice that n ⊥ = r ⊕ n and ( r ⊕ n ) ⊥ = n under the fixedpairing h· , ·i on s .We call a subspace f of s a proper Levi subspace if f is of the form r as abovefor some r .Let P = P × P , which is a parabolic subgroup of H = GL n × GL n .There is a Levi decomposition P = MU and p = m ⊕ u , with M = M × M , U = U × U , m = m ⊕ m and u = u ⊕ u . Notice that ( M , r ) ≃ Q ( H n i , s n i )for some ( H n i , s n i ). We fix an open compact subgroup K of H such that H = M U K and η | K is trivial. Recall that we write M = M ( F ) and U = U ( F ).Here we choose the Haar measure on H so that vol( K ) = 1, and choose Haarmeasures on M and U so that for any f ∈ C ∞ c ( H ), Z H f ( h ) d h = Z M Z U Z K f ( muk ) d m d u d k. We choose the Haar measure on Lie algebra u ( F ) compatible with that on U under the exponential map, and choose Haar measures on r ( F ) , n ( F ) , ¯ n ( F )according to the above identifications.For f ∈ C ∞ c ( s ( F )), we define f r ∈ C ∞ c ( r ( F )) to be f r ( Y ) := Z n ( F ) f ( Y + Z ) d Z, define e f ∈ C ∞ c ( s ( F )) to be e f ( Y ) = Z K f ( Y k ) d k, and define f ( r ) ∈ C ∞ c ( r ( F )) to be f ( r ) := (cid:16) e f (cid:17) r . Definition 6.2. Let T r be a distribution on r ( F ). We define the distribution i sr ( T r ) on s ( F ) to be: i sr ( T r )( f ) := T r ( f ( r ) ) , for f ∈ C ∞ c ( s ( F )) . § 1] or [Ko, § M acts on r by the adjoint action, which is induced from the action of H on s . Denote by r rs the regular semisimple locus of r with respect to the actionof M . If Y is in s rs ( F ), then it is also in r rs ( F ). If Y ∈ r rs ( F ), put | D r ( Y ) | F = | det(ad( Y ); m / t ⊕ r / c ) | , where c is the Cartan space of r containing Y and t is the Lie algebra of the cen-tralizer of Y in M . The normalized orbital integral I η,MX ( f ′ ), for f ′ ∈ C ∞ c ( r ( F )),is defined to be | D r ( X ) | F Z H X \ M f ′ ( X m ) η ( m ) d m. Then I η,MX is a distribution on r ( F ). In the proposition below, we write I η,HX instead of I ηX to distinguish it from I η,MX . Proposition 6.3. 1. Suppose that T r is an ( M, η ) -invariant distribution on r ( F ) , then i sr ( T r ) is an ( H, η ) -invariant distribution on s ( F ) .2. We have i sr ( I η,MX ) = I η,HX . 3. Suppose that T r is an ( M, η ) -invariant distribution on r ( F ) , which is rep-resented by a function Θ r which is locally constant on r rs ( F ) and locallyintegrable on r ( F ) . In other words, for any f ∈ C ∞ c ( r ( F )) , T r ( f ) = Z r rs ( F ) Θ r ( Y ) κ ( Y ) f ( Y ) | D r ( Y ) | − / F d Y. Then the distribution i sr ( T r ) is represented by the function Θ s ( Y ) = X Y ′ Θ r ( Y ′ ) , where Y ′ runs over a finite set of representatives for the M -conjugacyclasses of elements in r ( F ) which are H -conjugate to Y . The function Θ s is locally constant on s rs ( F ) and locally integrable on s ( F ) , and, for any f ∈ C ∞ c ( s ( F )) , i sr ( T r )( f ) = Z s rs ( F ) Θ s ( Y ) κ ( Y ) f ( Y ) | D s ( Y ) | − / F d Y. 4. The map f f ( r ) commutes with the Fourier transform, and therefore i sr ( b T r ) = \ i sr ( T r ) .Proof. (1) For f ∈ C ∞ c ( s ( F )) and h ∈ H , define h f ∈ C ∞ c ( s ( F )) by h f ( Y ) = f ( Y h ). To prove (i), it suffices to observe the following relation: for p = mu ∈ P and Y ∈ r ( F ), we have( p f ) r ( Y ) = Z n ( F ) p f ( Y + Z ) d Z = Z n ( F ) f ( Y p + Z p ) d Z = Z n ( F ) f ( Y m + Z p ) d Z = | det(Ad( p ); n ) | F Z n ( F ) f ( Y m + Z ) d Z. 30t is easy to verify that | det(Ad( p ); n ) | F = | det(Ad( p ); u ) | F = δ P ( p ) , where δ P is the modulus character of P . Therefore ( p f ) r ( Y ) = δ P ( p ) f r ( Y m ).The rest arguments are routine.(2). Write T = H X for simplicity. For f ∈ C ∞ c ( s ( F )), Z T \ H f ( X h ) η ( h ) d h = Z T \ M Z U Z K f ( X muk ) η ( m ) d k d u d m = Z T \ M Z U e f ( X mu ) η ( m ) d u d m. Write Y = X m . Notice that the map α : U −→ n , u u − Y u − Y is an isomorphism of algebraic varieties, whose Jacobian is | det( ̺ ◦ ad( Y ); u ) | F . Also note that | det( ̺ ◦ ad( Y ); u ) | F = | det( ̺ ◦ ad( Y ); u ⊕ ¯ u ) | / F = | det( ̺ ◦ ad( Y ); h / t ) | / F | det( ̺ ◦ ad( Y ); m / t ) | / F = | D s ( Y ) | / F | D r ( Y ) | / F . Therefore I η,H ( X, f ) = Z T \ M Z n ( F ) | D r ( X m ) | / F e f ( X m + Z ) η ( m ) d Z d m = | D r ( X ) | / F Z T \ M f ( r ) ( X ) η ( m ) d m = I η,M ( X, f ( r ) ) . The assertion (iii) is a consequence of Weyl integration formula, and theassertion (iv) is obvious.Let s ell be the open subset of elliptic regular semisimple elements in s ( F ). Lemma 6.4. Suppose that φ ∈ C ∞ c ( s ell ) . Then φ ( r ) and ( b φ ) ( r ) are identicallyzero for every proper Levi subspace r of s . Moreover, for any regular semisimpleelement X of s ( F ) lying in r ( F ) , we have I η ( X, φ ) = b I η ( X, φ ) = 0 . Proof. The vanishing of φ ( r ) is obvious. The vanishing of ( b φ ) ( r ) is a consequenceof Proposition 6.3 (iv). The vanishing of the orbital integrals follows from thefirst assertion and Proposition 6.3 (ii).31et c be an elliptic Cartan subspace of s , which means that one (any) elementof c reg ( F ) is elliptic. Let T be the centralizer of c in H , and Z the center of G which is also contained in T . Then Z \ T is compact since c is elliptic. Now werequire that vol( Z \ T ) = 1 here, which does not matter.Let s c rs = ( c reg ( F )) H and φ ∈ C ∞ c ( s c rs ). We define the distribution I φ ∈D ( s ( F )) H,η to be I φ ( f ) = Z Z \ H Z s ( F ) f ( Y ) φ ( Y h ) η ( h ) d Y d h. This distribution is well defined: Z Z \ H Z s ( F ) | f ( Y ) φ ( Y h ) | d Y d h = Z Z \ H d h Z c ( F ) | D s ( Y ) | F d Y Z Z \ H | f ( Y h ′ ) | · | φ ( Y h ′ h ) | d h ′ ! = Z c ( F ) | D s ( Y ) | F d Y Z ( Z \ H ) × ( Z \ H ) | f ( Y h ′ ) | · | φ ( Y h ) | d h ′ d h ! = Z c reg ( F ) I ( Y, | f | ) · I ( Y, | φ | ) d Y < ∞ , since I ( Y, | φ | ) ∈ C ∞ c ( c reg ( F )). Here I ( · , f ) is the normalized orbital integralwithout twisting η . We also define the distribution I b φ ∈ D ( s ( F )) H,η to be I b φ ( f ) = Z Z \ H Z s ( F ) f ( Y ) b φ ( Y h ) η ( h ) d Y d h. We have the relation Z s ( F ) f ( Y ) b φ ( Y h ) d Y = Z s ( F ) b f ( Y ) φ ( Y h ) d Y. Thus Z Z \ H η ( h ) d h Z s ( F ) f ( Y ) b φ ( Y h ) d Y ! = Z Z \ H η ( h ) d h Z s ( F ) b f ( Y ) φ ( Y h ) d Y ! , by the absolute convergence of the latter one, which shows that I b φ is well definedand I b φ = b I φ . In summary, we have the following lemma. Lemma 6.5. Let c be an elliptic Cartan subspace of s and φ ∈ C ∞ c ( s c rs ) . Then b I φ = I b φ In the next subsection, we will reduce Theorem 6.6 to the following theoremwhose proof will be given in §§ Theorem 6.6. Let c be an elliptic Cartan subspace of s and φ ∈ C ∞ c ( s c rs ) . Then b I φ is represented by a locally integrable function on s ( F ) which is locally constanton s rs ( F ) . .2 Proof of Theorem 6.1 To show the representability of the Fourier transform of orbital integrals, weneed the following relative version of Howe’s finiteness theorem (Theorem 6.7).Let us introduce some notation. If ω is a compact set in s ( F ), put J ( ω ) η = { T ∈ D ( s ( F )) H,η : Supp( T ) ⊂ cl( ω H ) } . Let L ⊂ s ( F ) be a lattice (a compact open O F -submodule). Denote by C c ( s ( F ) /L )the space of f ∈ C ∞ c ( s ( F )) which is invariant under translation by L . Let j L : J ( ω ) η → C c ( s ( F ) /L ) ∗ be the composition of the maps: j L : J ( ω ) η ֒ → D ( s ( F )) res −→ C c ( s ( F ) /L ) ∗ , where C c ( s ( F ) /L ) ∗ is the vector space dual to C c ( s ( F ) /L ) and res is the restric-tion map. Then Howe’s finiteness theorem is the following. Theorem 6.7. For any lattice L and any compact set ω in s ( F ) , we have dim j L ( J ( ω ) η ) < + ∞ . Proof. It was shown in [RR, Theorem 6.1] that Howe’s finiteness theorem holdsin a more general setting when η = . It is not hard to check that it still holdswhen η is our quadratic character.The following variant of Howe’s theorem is often used, and we refer thereader to [Ko, § 26] for more details. Let b j L : J ( ω ) η → D ( L ) be the compositionof the maps b j L : J ( ω ) η ֒ → D ( s ( F )) F −→ D ( s ( F )) res −→ D ( L ) , where F denotes the Fourier transform. Theorem 6.8. For any lattice L and any compact set ω in s ( F ) , dim b j L ( J ( ω ) η ) < + ∞ . Proof. See [Ko, Theorem 26.3]. Corollary 6.9. Let ω be compact, and let V be a subspace of J ( ω ) η . Let L beany lattice in s ( F ) . Then b j L ( V ) = b j L (cl( V )) .Proof. See [Ko, Proposition 26.1]. Proof of Theorem 6.1. By Proposition 6.3, it suffices to show that b I ηX can berepresented when X lies in c reg ( F ) for some elliptic Cartan subspace c of s .Then Theorem 6.1 follows from Theorem 6.6, Lemma 6.10 and the fact that s ( F ) = S lattice L . Lemma 6.10. Let X ∈ c reg ( F ) be an elliptic element and ω a compact openneighborhood of X in c reg ( F ) . Then given a lattice L in s ( F ) , there exists φ ∈ C ∞ c ( ω H ) such that b I ηX and b I φ have the same restriction to L . roof. The proof is similar as that of [Ko, Lemma 26.5]. We first show that I ηX lies in the closure of the linear space I ω := { I φ : φ ∈ C ∞ c ( ω H ) } , which is a subspace of J ( ω ) η . It suffices to show that: if I φ ( f ) = 0 for all φ ∈ C ∞ c ( ω H ) then I ηX ( f ) = 0. Note that I φ ( f ) = Z ω I ηY ( f ) · I ηY ( φ ) d Y. We may shrink ω so that every function ϕ ∈ C ∞ c ( ω ) arises as Y I ηY ( φ ) forsome φ ∈ C ∞ c ( ω ). Thus I ηX ( f ) = 0 if I φ ( f ) = 0 for all φ ∈ C ∞ c ( ω H ). ByCorollary 6.9, we see that b j L ( I ηX ) ∈ b j L ( I ω ) for any lattice L . In other words,given a lattice L , there exists a φ ∈ C ∞ c ( ω H ) such that b I ηX and b I φ have the samerestriction to L . In this subsection, we will show the boundness of the normalized orbital integralsalong a Cartan subspace (Theorem 6.11), which is crucial for proving Theorem6.6. We follow the same line as the proof of [HC1, Theorem 14], where thereare no Shalika germs involved. Theorem 6.11. 1. Let c be a Cartan subspace of s and f ∈ C ∞ c ( s ( F )) . Then sup X ∈ c reg ( F ) | I η ( X, f ) | < + ∞ . 2. Let c ′ be a Cartan subspace of s ′ and f ′ ∈ C ∞ c ( s ′ ( F )) . Then sup X ∈ c ′ reg ( F ) | I ( X, f ′ ) | < + ∞ . We will prove only the first assertion with respect to s . The second assertioncan be proved in the same way. We use inductive method to prove this theorem.In the case n = 1, our case essentially is the Gan-Gross-Prasad conjecture forunitary groups of rank 1. Thus Theorem 6.11 follows from the discussions in[Zhw1, § C ∞ c ( s m ( F )) for every m < n . Lemma 6.12. Fix a compact set ω of s ( F ) and a Cartan subspace c . Then theset of all X ∈ c ( F ) such that X ∈ cl( ω H ) is relative compact in c ( F ) .Proof. It suffices to assume ω is closed. Consider the closed inclusion i :( c /W )( F ) → ( s / H )( F ) where W is the Weyl group of c , and the natural map π : s ( F ) → ( s / H )( F ). Then π ( ω ) and thus i − ( π ( ω )) is compact. The lemmafollows from the fact that the map c ( F ) → ( c /W )( F ) is a proper map betweenlocally compact Hausdorff spaces. Corollary 6.13. For f ∈ C ∞ c ( s ( F )) , I η ( X, f ) = 0 for X ∈ c reg ( F ) lying outsidea compact subset of c ( F ) . We first prove Theorem 6.11 in the following situation.34 emma 6.14. Let f be in C ∞ c ( s ( F ) − N ) . Then I η ( · , f ) is bounded on c reg ( F ) .Proof. By Lemma 6.12 and Corollary 6.13, it suffices to prove: given X ∈ c ( F ),we can choose a neighborhood V of X in c ( F ) such thatsup X ∈ V ′ | I η ( X, f ) | < + ∞ , where V ′ = V ∩ s rs ( F ) . When X = 0, using the descent of orbital integrals (Proposition 5.20), wereduce to considering the orbital integrals for C ∞ c ( s X ) with respect to the actionof H X . Since X = 0, ( H X , s X ) is of the form(GL m ( F ) A , gl m ( F ) A ) × ( H n − m , s n − m ( F ))for some semisimple A in GL n ( F ) and some integer 0 < m ≤ n . Then the resultfollows from the inductive hypothesis on n − m and the bound of the usualorbital integrals for C ∞ c ( gl m ( F ) A ) by Harish-Chandra. When X = 0, sinceSupp( f ) ∩ N = ∅ , we can find a neighborhood V of X such that I η ( X, f ) = 0on V ′ .Now let s be the set of Y ∈ s ( F ) such that: there exists an open neighbor-hood ω of Y in s ( F ) so that sup X ∈ c reg ( F ) | I η ( X, f ) | < ∞ for all f ∈ C ∞ c ( s ( F )) withSupp( f ) ⊂ ω . Since N is closed in s ( F ), Lemma 6.14 implies that s ( F ) −N ⊂ s .To prove Theorem 6.11, it remains to show that N ⊂ s . We need some prepa-ration below.Fix X = 0 in N . Let ( X , d( X ) , Y ) be an sl -triple as in Lemma 3.1.Consider the map ψ : H × s Y −→ s ( F ) , ( h, U ) ( X + U ) h . By the same discussion as that of [HC1, Part VI, § ψ is everywheresubmersive. Set ω = ψ ( H × s Y ), which is an open and H -invariant subset of s ( F ). Since ψ is everywhere submersive, we have a surjective linear map C ∞ c ( H × s Y ) −→ C ∞ c ( ω ) , α f α such that Z ω f α ( X ) p ( X ) d X = Z H × s Y α ( h, u ) p (cid:0) ( X + U ) h (cid:1) d h d U for every locally integrable function p on ω .Let Γ be the Cartan subgroup of H with the Lie algebra F · d( X ). Pleaserefer to Lemma 3.3 and Proposition 4.4 for the notations below. Put t = ξ ( γ )and write U γ = ξ ( γ ) U γ − for U ∈ s Y , γ ∈ Γ. We have( X + U γ ) γh = ( X + tU γ − ) γh = t ( X + U ) h . For γ ∈ Γ and α ∈ C ∞ c ( H × s Y ), define α ′ ∈ C ∞ c ( H × s Y ) to be α ′ ( h, U ) = α ( γ − h, U γ − ) . emma 6.15. Fix γ ∈ Γ and α ∈ C ∞ c ( H × s Y ) . Then f α ( t − X ) = | t | n − r − mF f α ′ ( X ) , X ∈ ω. Proof. Choose any function α in C ∞ c ( ω ). We have Z s ( F ) f α ( t − X ) p ( X ) d X = | t | n F Z s ( F ) f α ( X ) p ( tX ) d X = | t | n F Z H × s Y α ( h, U ) p (cid:0) t ( X + U ) h (cid:1) d h d U = | t | n F Z H × s Y α ( h, U ) p (cid:0) ( X + U γ ) γh (cid:1) d h d U = | t | n F Z H × s Y α ( γ − h, U γ − ) p (cid:0) ( X + U ) h (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) d U γ − d U (cid:12)(cid:12)(cid:12)(cid:12) F d h d U. It remains to compute the Jacobian (cid:12)(cid:12)(cid:12) d U γ − d U (cid:12)(cid:12)(cid:12) F . Choose a basis U , ..., U r of s Y as in Lemma 3.3. Write U = P ≤ i ≤ r a i U i . Then U γ − = t − U γ = t − X i a i U γi = t − X i a i ξ i ( γ − ) U i . . Hence (cid:12)(cid:12)(cid:12)(cid:12) d U γ − d U (cid:12)(cid:12)(cid:12)(cid:12) F = | t | − rF Y ≤ i ≤ r | t | − λi F = | t | − r − mF , which implies the lemma.For X ∈ c reg ( F ), there is a unique distribution τ ηX on s Y such that I η ( X, f α ) = τ ηX ( β α ) where β α ( U ) = Z H α ( h, U ) η ( h ) d h, α ∈ C ∞ c ( H × s Y ) . For f ∈ C ∞ c ( ω ), define f ′ ∈ C ∞ c ( ω ) to be f ′ ( X ) = f ( t − X ). It is easy to seethat I η ( X, f ′ ) = | t | (2 n − n ) F I η ( t − X, f ) . Now fix α ∈ C ∞ c ( H × s Y ), and set f = f α , f ′ = f ′ α , β = β α and β ′ = β α ′ . Notethat f ′ = | t | n − r − mF f α ′ . We have β ′ ( U ) = Z H α ( γ − h, U γ − ) η ( h ) d h = η ( γ ) β ( t − U γ ) , U ∈ s Y . So we obtain | t | (2 n − n ) F I η ( t − X, f ) = | t | n − r − mF I η ( X, f α ′ ) = | t | n − r − mF τ ηX ( β ′ ) , I η ( t − X, f ) = | t | n + n − r − mF ( I η ( X, f ) + τ ηX ( β ′ − β )) . (1)By Proposition 4.4, we know n + n − r − m < X ∈ N and suppose X = 0.We want to construct an open neighborhood ω of X such that I η ( · , f ) isbounded on c reg ( F ) as soon as Supp( f ) ⊂ ω . Recall that we denote by N q the union of all H -orbits in N of dimension ≤ q , and notice that X ∈ N n − r and N n − n = N . So we can choose an open neighborhood ω of X in ω suchthat ω ∩ N n − r ⊂ X H , and can assume ω = ω H . By [HC1, Lemma 37], wecan choose an open neighborhood U of zero in s Y such that X + U ⊂ ω and( X + U ) ∩ X H = { X } .Fix γ ∈ Γ such that η ( γ ) = 1 and | t | F = | ξ ( γ ) | F > 1. Choose an openneighborhood U of zero in U such that t − U γ ∪ t U γ − ⊂ U . Put N ∗ = N − { } . Lemma 6.16. N ∗ ⊂ s .Proof. We induct on r = dim s Y for X ∈ N ∗ . Put ω = ( X + U ) H , which isan open invariant neighborhood of X . Consider the surjective map H × U −→ ω , ( h, U ) ( X + U ) h , which is everywhere submersive. Consider the surjective linear map C ∞ c ( H × U ) −→ C ∞ c ( ω ) , α f α , which is the restriction of the map C ∞ c ( H × s Y ) → C ∞ c ( ω ) as before. Let f ∈ C ∞ c ( ω ) and choose α ∈ C ∞ c ( H × U ) such that f = f α . Set β = β α , β ′ = β α ′ and f ′ = f ′ α as before. Then β − β ′ ∈ C ∞ c ( U ), and 0 / ∈ Supp( β − β ′ ). Define α ( h, U ) = α ( h ) ( β ( U ) − β ′ ( U )) for h ∈ H, U ∈ U , where α ∈ C ∞ c ( H ) and R H α ( h ) η ( h ) d h = 1. For X ∈ c reg ( F ), we have I η ( X, f α ) = τ ηX ( β α ) = τ ηX ( β − β ′ ) , and Supp( f α ) ∩ N n − r = ∅ . Now we start the induction on r = dim s Y .First assume that r = n . Note that r = n is the initial step. In such case wehave n + n − r − m = − n and I η ( t − X, f ) = | t | − n F ( I η ( X, f ) + τ ηX ( β ′ − β )) , by (1). Put c = | t | − n F < 1. Since Supp( f α ) ∩ N = ∅ ( N = N n − n ), by Lemma6.14, we have a = sup X ∈ c reg ( F ) | τ ηX ( β ′ − β ) | < + ∞ . Iteration gives I η ( t − d X, f ) = | t | − dn F I η ( X, f ) + X ≤ k ≤ d | t | − kd F τ ηt k − d X ( β ′ − β ) , ( d ≥ , or I η ( X, f ) = c d I η ( t d X, f ) + X ≤ k ≤ d c d τ ηt k X ( β ′ − β ) , ( d ≥ . d → + ∞ I η ( t d X, f ) = 0, we get | I η ( X, f ) | ≤ a X ≤ k< ∞ c k ≤ a c − c . Now assume r > n . Since Supp( f α ) ∩ N n − r = ∅ , by the inductive hypoth-esis and Lemma 6.14, I η ( X, f α ) is bounded on c reg ( F ) and so is τ ηX ( β − β ′ ).Applying the same argument as the case r = n , we complete the proof of thelemma.Applying the same arguments as those of [HC1, Part VI § Lemma 6.17. ∈ s . At last, Theorem 6.11 follows from Lemma 6.14, Lemma 6.16 and Lemma6.17. Now we continue to prove Theorem 6.6. Let c be an elliptic Cartan subspaceof s and φ ∈ C ∞ c ( s c rs ). For simplicity, we write φ = c φ , and denote by Θ thedistribution I φ , that is, for f ∈ C ∞ c ( s ( F )),Θ( f ) := Z Z \ H Z s ( F ) f ( Y ) φ ( Y h ) η ( h ) d Y d h. Our goal is to prove that the distribution Θ can be represented by a locallyintegrable function on s ( F ) which is locally constant on s rs ( F ). We follow thestrategy of the proof of [HC1, Theorem 16].For t ≥ 1, let Ω t denote the set of all h ∈ H such that 1 + log k h k Z \ H ≤ t .Then Ω t is a compact set modulo Z . Let Φ t denote the characteristic functionof Ω t . Then we haveΘ( f ) = lim t → + ∞ Z Z \ H Φ t ( h ) Z s ( F ) f ( Y ) φ ( Y h ) η ( h ) d Y d h = lim t → + ∞ Z s ( F ) f ( Y )Θ t ( Y ) d Y, where Θ t ( Y ) = Z Z \ H Φ t ( h ) φ ( Y h ) η ( h ) d h. We will first show that lim t → + ∞ Θ t ( Y ) exists for all Y ∈ s rs ( F ), and then will givean estimation on Θ t to apply Lebesgue’s Theorem. Lemma 6.18. Given a compact subset ω of s ( F ) , we can choose c ≥ suchthat k h k T \ H ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) for h ∈ H, X ∈ c reg ( F ) such that X h ∈ ω .Proof. The proof is the same as that of [Ko, Lemma 20.3].38e choose a compact set ω ⊂ s ( F ) such that Supp( φ ) ⊂ ω, Supp( f ) ⊂ ω .Fix a Cartan subspace c ⊂ s . Let T be the centralizer of c in H , and A the maximal split torus in T . Notice that A consists of elements of the formdiag( a, a ) where a ∈ A for some split torus A contained in GL n ( F ). Let ω c be the set of X ∈ c ( F ) such that X h ∈ ω for some h ∈ H . Then ω c is compact.For X ∈ ω c , h ∈ H , set φ X ( h ) = φ ( X h ) η ( h ) . Note that φ X has the following properties:(i) Supp( φ X ) ⊂ C X for some subset C X ⊂ H which is compact modulo A and φ X ( ah ) = φ X ( h ) for h ∈ H, a ∈ A ;(ii) if P ′ is a proper parabolic subgroup in GL n ( F ) with Levi decomposition P ′ = M ′ U ′ , and A ′ ⊂ A where A ′ is the center of M ′ , then Z U ′ φ X ( uh ) d u = 0 , for each h ∈ H, where U ′ = U ′ × U ′ is a unipotent subgroup of H .Let K ′ be an open subgroup of K ′ = GL n ( O F ) such that k k k = 1 , η ( k ) = 1for all k ∈ K ′ . Here we choose the Haar measure on H so that vol( K ′ × K ′ ) = 1.Fix an open compact subgroup K ′ of GL n ( F ) such that K ′ ⊂ ( ¯ U ∩ K ′ )( M ∩ K ′ )( U ∩ K ′ )for any parabolic subgroup P ′ = M ′ U ′ in P ( A ), where we denote by P ( A ) theset of all parabolic subgroups P ′ = M ′ U ′ of GL n ( F ) such that A is the centerof M ′ . Set K = K ′ × K ′ ⊂ H . For an element y ∈ H , put K ( y ) = K ∩ K y .Set k C X k T \ H = sup h ∈ C X k h k T \ H . The following lemma is an analogue of [HC1, Theorem 20], and we omit thedetails of the proof since it is the same as that of [HC1, Theorem 20] Lemma 6.19. There exists a number c ≥ with the following property. Let y ∈ H , and Ω = Ω( C X , y ) be the set of h ∈ H such that k h k Z \ H ≤ c (1 + log k C X k T \ H )(1 + log k y k T \ H ) . Then Z K ( y ) φ X ( ykh ) d k = 0 unless h ∈ Ω . Now suppose that X ∈ c reg ( F ) and y ∈ H are such that X y ∈ ω . Then X ∈ ω c . By Lemma 6.18, there is a positive constant c , only depending on ω and c , such that1 + log k y k T \ H ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) . Set ω ′ c = ω c ∩ c reg ( F ). Then for any X ∈ ω ′ c we can choose a subset C X of H such that(1) Supp( φ X ) ⊂ C X and C X is compact modulo A ;392) 1 + log k C X k T \ H ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) .Let Ω X ( X ∈ ω ′ c ) be the set of h ∈ H such that1 + log k h k Z \ H ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) , where c = c · c with c as in Lemma 6.19. Let Φ X denote the characteristicfunction of Ω X . Then we haveΘ t ( X y ) = Z Z \ H Φ t ( h ) φ ( X yh ) η ( h ) d h = Z Z \ H Φ t ( h ) Z K φ ( X ykh ) η ( h ) d k d h. Note that k kh k = k h k for k ∈ K . By Lemma 6.19 we have Z K φ ( X ykh ) d k = Z K φ X ( ykh ) d k = 0 , unless: 1 + log k k h k Z \ H ≤ c (1 + log k C X k T \ H )(1 + log k y k T \ H ) , where k runs over a set of representatives of K /K ( y ) in K . Since k kh k = k h k and c (1 + log k C X k T \ H )(1 + log k y k T \ H ) ≤ cc (cid:0) { , | D s ( X ) | − F } ) (cid:1) , the integral R K φ ( X ykh ) = 0 unless h ∈ Ω X . Thus, if t ≥ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) , we get Θ t ( X y ) = Z Z \ H Φ t ( h )Φ X ( h ) η ( h ) Z K φ ( X ykh ) d k d h = Z Z \ H Φ X ( h ) Z K φ ( X ykh ) d k d h = Z Z \ H Z K φ ( X ykh ) η ( h ) d k d h. Therefore lim t → + ∞ Θ t ( X y ) exists for X y ∈ ω ∩ s rs ( F ). By enlarging ω , lim t → + ∞ Θ t ( X )exists for all X ∈ s rs ( F ).Now we estimate Θ t ( X ). All the notations are the same as above. We have | Θ t ( X y ) | ≤ Z Z \ H Φ X ( y − h ) | φ ( X h ) | d h = Z A \ H | φ ( X h ) | d h Z Z \ A Φ X ( y − ah ) d a. Recall that φ ( X h ) η ( h ) = φ X ( h ) = 0 unless h ∈ C X . Suppose h ∈ C X . We canassume log k h k ≤ log k C X k and log k y k = log k y k T \ H . Then Φ X ( y − ah ) = 0unless y − ah ∈ Ω X . Since1 + log k a k Z \ H ≤ (1 + log k h k ) (cid:0) k y − ah k Z \ H (cid:1) (1 + log k y k ) , 40e have Φ X ( y − ah ) = 0 unless1 + log k a k Z \ H ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) , where c = c c . Therefore Z Z \ A Φ X ( y − ah ) d a ≤ Z k a k Z \ H ≤ c ( { , | D s ( X ) | − F } ) ) d a ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) ℓ where c is a positive constant, independent of the choice of X ∈ ω ′ c , and ℓ = dim Z \ A . This shows that | Θ t ( X y ) | ≤ c (cid:0) { , | D s ( X ) | − F } ) (cid:1) ℓ Z A \ H | φ ( X h ) | d h. Notice that Theorem 6.11 also holds when η = . Then we have:sup X ∈ ω ′ c | D s ( X ) | F Z A \ H | φ ( X h ) | d h < + ∞ . Hence | Θ t ( X y ) | ≤ c | D s ( X ) | − F (cid:0) { , | D s ( X ) | − F } ) (cid:1) ℓ for all X ∈ c ( F ) and y ∈ H such that X h ∈ ω ′ = ω ∩ s rs ( F ). Since there areonly finitely many non-conjugate Cartan subspaces in s , there exists a constant c such that | Θ t ( X ) | ≤ c | D s ( X ) | − F (cid:0) { , | D s ( X ) | − F } ) (cid:1) ℓ for all X ∈ ω ′ and all t ≥ X 7→ | D s ( X ) | − F (cid:0) { , | D s ( X ) | − F } ) (cid:1) ℓ is locally integrable on s ( F ). Then Theorem 6.6 follows from Lebesgue’s Theo-rem. Lemma 6.20. There exists ǫ > such that the function | D s ( X ) | − ǫF is locallyintegrable on c ( F ) for any Cartan subspace c of s .Proof. See [Zhw1, Lemma 4.3]. In this subsection, we obtain formulae for b i η ( X, Y ) ( X, Y ∈ s rs ( F )) and b i ( X, Y )( X, Y ∈ s ′ rs ( F )) at “infinity”, which are analogues of [Wa2, VIII.1 Proposi-tion]. The proof of [Wa2, VIII.1 Proposition] is very technical. Here we modifyWaldspurger’s proof a little to make it available in our situation.41 tatement Let c be a Cartan subspace of s , and T − the maximal θ -splittorus in G whose Lie algebra is c . Let T be the centralizer of c in H , and write t = Lie( T ). For X, Y ∈ c reg ( F ), define a bilinear form q X,Y on h ( F ) / t ( F ) by q X,Y ( Z, Z ′ ) = h [ Z, X ] , [ Y, Z ′ ] i , where the pairing h· , ·i is the one as before. One can check that the form q X,Y isnondegenerate and symmetric. One can also verify that q X,Y = q Y,X . We write γ ψ ( X, Y ) = γ ψ ( q X,Y ) for simplicity. Recall that, by conventions, T = T ( F ), H = H ( F ).Let c ′ be a Cartan subspace of s ′ . Similarly, we denote by T ′− the maximal θ -split torus in G ′ whose Lie algebra is c ′ , by T ′ the centralizer of c ′ in H ′ , andby t ′ the Lie algebra of T ′ . For X, Y ∈ c ′ reg ( F ), we also define a nondegenerate,bilinear and symmetric form q X,Y on h ′ ( F ) / t ′ ( F ) in the same way as above.The following formulae depend on the choices of the Haar measures on T and H (also on T ′ and H ′ ). Here we equip H or T with the Haar measure sothat the exponential map preserve the measure in a neighborhood of 0 in h ( F )or t ( F ). We make the similar choices for the Haar measures on T ′ and H ′ . Proposition 7.1. Let the notations be as above.1. Let X ∈ s rs ( F ) and Y ∈ c reg ( F ) . Then there exists N ∈ N such that if µ ∈ F × satisfying v F ( µ ) < − N , we have the equality b i η ( µX, Y ) = κ ( Y ) X h ∈ T \ H, h · X ∈ c η ( h ) γ ψ ( µh · X, Y ) ψ ( h µh · X, Y i ) , and b i η ( X, µY ) = κ ( µY ) X h ∈ T \ H, h · X ∈ c η ( h ) γ ψ ( µh · X, Y ) ψ ( h µh · X, Y i ) . 2. Let X ∈ s ′ rs ( F ) and Y ∈ c ′ reg ( F ) . Then there exists N ∈ N such that if µ ∈ F × satisfying v F ( µ ) < − N , we have the equality b i ( µX, Y ) = b i ( X, µY ) = X h ∈ T ′ \ H ′ , h · X ∈ c ′ γ ψ ( µh · X, Y ) ψ ( h ( µh · X, Y i ) . In particular, the above expression is zero if X is not conjugate to anyelement of c ( F ) (or c ′ ( F )). Proof of Proposition 7.1 We now prove the formula for b i η ( µX, Y ). Theformula for b i η ( X, µY ) can be deduced from it. We leave the proof of the formulaefor b i ( µX, Y ) and b i ( X, µY ) to the reader. They can be proved in the same way.Firstly, we introduce some notations. Let q (resp. p ) be the unique comple-ment of t (resp. c ) in h (resp. s ) which is stable under the adjoint action of T .Denote by S c the set of roots of T − in g ( ¯ F ). For each subspace f ⊂ g ( F ) suchthat the restriction of h· , ·i to f is nondegenerate and for each O F -lattice L ⊂ f ,set e L = { ℓ ∈ f : ∀ ℓ ′ ∈ L, ψ ( h ℓ ′ , ℓ i ) = 1 } . We denote by L c the O F -lattice of c ( F ) such that e L c = { Z ∈ c ( F ) : ∀ α ∈ S c , v F ( α ( Z )) ≥ } . O F -lattices L p ⊂ p ( F ), L t ⊂ t ( F ) and L q ⊂ q ( F ). Set L s = L c ⊕ L p , L h = L t ⊕ L q , L = L s ⊕ L h .For simplicity, write d = dim F ( g ( F )) = 4 n . Denote by F [ U ] d the set ofmonic polynomials of degree d with coefficients in F . For P ∈ F [ U ] d , write P ( U ) = d X i =0 s i ( P ) U d − i . For a ∈ Z and P , P ∈ F [ U ] d , we write P ≡ P mod ̟ a O F if v F ( s i ( P ) − s i ( P )) ≥ a for each i = 0 , , ..., d . For each Z ∈ g ( F ), denote by P Z thecharacteristic polynomial of ad( Z ) acting on g ( F ). Then P Z ∈ F [ U ] d .Fix an integer c ∈ N satisfying the following conditions.1. For each a ∈ N , a ≥ c , we have • ̟ a L h ⊂ V h and ̟ a L ⊂ V g ; • K a := exp( ̟ a L h ) is a subgroup of K = GL n ( O F ) × GL n ( O F ), and η | K a = 1; • the action of K a stabilizes L s (hence stabilizes e L s ).2. For each a ∈ N , a ≥ c , and each Z ∈ ̟ a L h , we have • (exp Z ) · Y − Y − [ Z, Y ] ∈ ̟ a − c L s ; • (exp Z ) · Y − Y − [ Z, Y ] − [ Z, [ Z, Y ]] ∈ ̟ a − c L s .3. Denote by C ( X ) the set of X ′ ∈ c ( F ) satisfying that there exists h ∈ H such that h · X ′ = X , which is a finite set. We require that: • if a ∈ N , a ≥ c , X ′ , X ′′ ∈ C ( X ), and γ ∈ K a satisfying γ · X ′ = X ′′ ,then X ′ = X ′′ ; • for each X ′ ∈ C ( X ), denote by e L X ′ q the dual of L q in q ( F ) withrespect to the form q X ′ ,Y ; then require ̟ c e L X ′ q ⊂ ̟ − c L q .4. If Z ∈ p ( F ) satisfying [ Y, Z ] ∈ e L h , then Z ∈ ̟ − c e L p .5. For each h ∈ H , denote by c ( h ) the unique element of Z such that X h ∈ ̟ − c ( h ) e L s − ̟ − c ( h )+1 e L s . Since X ∈ s rs ( F ), the set { c ( h ) , h ∈ H } has a lower bound. We requirethat • for each h ∈ H , c ( h ) ≥ − c .6. Fix a basis B of g ( ¯ F ) formed of basis of c ( ¯ F ) and t ( ¯ F ), and root vectorsassociated to S c . We require that • for each Z ∈ e L p , the coefficients of the matrix representation of ad( Z )with respect to the basis B are of valuation ≥ − c ; • for each i = 0 , , ..., d , v F ( s i ( P X )) ≥ − c .43. There exists an open compact set Ω ⊂ c reg ( F ) such that if Z ∈ c reg ( F )satisfying P Z ≡ P X mod ̟ c O F , then Z ∈ Ω.The integer c is fixed from now on. We also fix an open compact Ω satisfyingcondition (vii). The following lemma actually is [Wa2, VIII.3 Lemme], andwhose proof can be applied in our situation. Lemma 7.2. There exists c ′ ∈ N , c ′ ≥ c , such that if a ∈ N , a ≥ c ′ , and Z ∈ Ω + ̟ a + c ′ e L p , then there exists γ ∈ K a such that γ · Z ∈ Ω . From now on, we fix an integer c ′ as in the above lemma. Set N = 2( d + 8) c + 6 c ′ + 12 . (1)Let µ ∈ F × be such that v F ( µ ) < − N . Choose m ∈ N such that • the functions Y ′ b i η ( µX, Y ′ ), Y ′ 7→ | D s ( Y ′ ) | F and κ ( Y ) are constanton Y + ̟ m L s ; • for each X ′ ∈ C ( X ), µX ′ ∈ ̟ − m e L s .Let f be the characteristic function of Y + ̟ m L s , and f ′ be the characteristicfunction of ̟ − m e L s . Then we have b I η ( µX, f ) = Z s ( F ) b i η ( µX, Y ′ ) κ ( Y ′ ) f ( Y ′ ) | D s ( Y ′ ) | − / F d Y ′ = vol( ̟ m L s ) | D s ( Y ) | − / F κ ( Y ) b i η ( µX, Y ) . (2)On the other hand, it is easy to verify that b f ( Y ′ ) = vol( ̟ m L s ) ψ ( h Y, Y ′ i ) f ′ ( Y ′ ) . Hence b I η ( µX, f ) = | D s ( µX ) | / F vol( ̟ m L s ) Z T \ H f ′ ( µX h ) ψ (cid:0) h Y, µX h i (cid:1) η ( h ) d h. Set a = [ − v F ( µ ) / − c − c ′ − . (3)By (1), a ≥ c . Fix a set of representatives Γ in H for the double coset T \ H/K a .By condition (iii), we can suppose that if there exist h ∈ Γ and h ′ ∈ T hK a suchthat X h ′ ∈ c ( F ), then X h ∈ c ( F ). Then we have b i η ( µX, Y ) = | D s ( µX ) D s ( Y ) | / F κ ( Y ) X h ∈ Γ vol( T \ T hK a ) f ′ ( µX h ) η ( h ) i ( h ) , where i ( h ) = Z K a ψ ( h Y, µX hγ i ) d γ. Fix h ∈ Γ. Choose b ∈ N such that • ( c + c ( h ) − v F ( µ )) / ≤ b ≤ c ( h ) − v F ( µ ) − − c ; • if c ( h ) ≤ c, b ≤ (cid:26) − ( d + 2) c − − v F ( µ ) , − c − c ′ − − a − v F ( µ ) , (4)44hich implies b ≥ a . Fix a set of representatives Γ ′ of K a /K b . Then we have i ( h ) = X g ∈ Γ ′ i ( h, g ) , where i ( h, g ) = Z K b ψ (cid:0) h Y, µX hgγ i (cid:1) d γ. Fix g ∈ Γ ′ , and set X ′ = X hg . Then i ( h, g ) = Z ̟ b L h ψ ( h exp Z · Y, µX ′ i ) d Z. Notice that since K b stabilizes L s and e L s , then c ( hg ) = c ( h ). In particular, X ′ ∈ ̟ − c ( h ) e L s . By (4), we have ψ ( h Z, µX ′ i ) = 1for each Z ∈ ̟ b − c L s . Notice that b ≥ c . For Z ∈ ̟ b L h , by condition (ii), wehave ψ ( h exp Z · Y, µX ′ i ) = ψ ( h Y + [ Z, Y ] , µX ′ i )= ψ ( h Y, µX ′ i ) ψ ( h Z, [ Y, µX ′ ] i ) . Therefore we see that i ( h, g ) = 0 if [ Y, µX ′ ] / ∈ ̟ − b e L h . We make the followingclaim: ( ∗ ) if [ Y, µX ′ ] ∈ ̟ − b e L h , then X h ∈ c ( F ) . Now we prove this claim. Suppose [ Y, µX ′ ] ∈ ̟ − b e L h , in other words, [ Y, X ′ p ] ∈ µ − ̟ − b e L q , where X ′ = X ′ c + X ′ p is the decomposition of X ′ with respect to s = c ⊕ p . Thus, by condition (iv), X ′ p ∈ µ − ̟ − b − c e L p . (5)Moreover, by (4), X ′ p ∈ ̟ − c ( h )+1 e L p . By the definition of c ( h ) and that c ( hg ) = c ( h ), we deduce that X ′ c ∈ ̟ − c ( h ) e L c − ̟ − c ( h )+1 e L c . Set R = { α ∈ S c : v F ( α ( X ′ c )) < − c ( h ) + 1 } . The above relation and thedefinition of e L c imply that R = ∅ . Set r = R , we calculate the coefficient s r ( P X ′ ). This is a sum of products of the coefficients of the matrix representa-tions of ad X ′ c and ad X ′ p with respect to the basis B . By (4), (5) and condition(vi), the coefficients of ad X ′ p are of valuation ≥ − c ( h ) + 1. The same relationholds for the coefficients of ad X ′ c other than that of α ( X ′ c ) for α ∈ R . The term Q α ∈ R α ( X ′ c ), which occurs in s r ( P X ′ ), is of the valuation strictly less than thatof any other term. Thus v F ( s r ( P X ′ )) = v F ( Y α ∈ R α ( X ′ c )) < r ( − c ( h ) + 1) . Since X ′ is conjugate to X by the action of H , then P X ′ = P X . By condition(vi), we have − c < r ( − c ( h ) + 1) , c ( h ) ≤ c. (6)Let i ∈ { , , ..., d } . We now compare the coefficients s i ( P X ′ ) and s i ( P X ′ c ). Theirdifference is a sum of products of coefficients of the matrix representations ofad X ′ c and ad X ′ p with respect to the basis B , and at least one coefficient of ad X ′ p is involved in these products. By (6), the coefficients of ad X ′ c are of valuation ≥ − c ( h ) ≥ − c . By (4), (5), (6) and condition (vi), the coefficients of ad X ′ p areof valuation ≥ dc . Therefore v F (cid:0) s i ( P X ′ ) − s i ( P X ′ c ) (cid:1) ≥ − ( i − c + dc ≥ c. In other words, P X ′ c ≡ P X ′ mod ̟ c O F . Thus, by condition (vii), X ′ c ∈ Ω. By(4), (5) and (6), X ′ p ∈ ̟ a + c ′ e L p . By (1) and (3), a ≥ c ′ . By Lemma 7.2, thereexists γ ∈ K a such that γ · X ′ ∈ c ( F ). By the choice of Γ, we have X h ∈ c ( F ).Now we have finished the proof the claim.From now on, we suppose that X h ∈ c ( F ). Thus f ′ ( µX h ) = 1 by thecondition on f ′ . Notice that the multiplication by h − induces an isomorphismfrom T \ T hK a to T \ T K a . Now we have b i η ( µX, Y ) = κ ( Y ) | D s ( µX ) D s ( Y ) | / F vol( K a ) − vol( T \ T K a ) × X X ′ = X h ∈ C ( X ) η ( h ) j ( X ′ ) , (7)where j ( X ′ ) = Z K a ψ ( h Y, µX ′ γ i ) d γ = Z ̟ a L h ψ ( h exp Z · Y, µX ′ i ) d Z. Fix X ′ ∈ C ( X ). By (1) and (3), ψ ( h Y ′ , µX ′ i ) = 1 for Y ′ ∈ ̟ a − c L s . Since Y, X ′ ∈ c ( F ), then for any Z ∈ g ( F ), h [ Z, Y ] , X ′ i = h Z, [ Y, X ′ ] i = 0. Bycondition (ii), we have j ( X ′ ) = ψ ( h Y, µX ′ i ) Z ̟ a L h ψ (cid:18) h [ Z, [ Z, Y ]] , µX ′ i (cid:19) d Z = ψ ( h Y, µX ′ i ) Z ̟ a L h ψ (cid:18) h [ Z, Y ] , [ µX ′ , Z ] i (cid:19) d Z = ψ ( h Y, µX ′ i ) vol( ̟ a L t ) Z ̟ a L q ψ (cid:18) q µX ′ ,Y ( Z ) (cid:19) d Z. Since a ≤ − c − v F ( µ ) / j ( X ′ ) = vol( ̟ a L t )vol( ̟ a L q ) / vol( ̟ − a ˇ L q ) / γ ψ ( q µX ′ ,Y ) ψ ( h Y, µX ′ i ) , (8)where ˇ L q is the dual lattice of L q with respect to the form q µX ′ ,Y . There is arelation: vol( K a ) = vol( T \ T K a )vol( T ∩ K a ) = vol( T \ T K a )vol( ̟ a L t ) . (9)46y definitionˇ L q = { Z ∈ q ( F ) : ∀ Z ′ ∈ L q , ψ ( h [ Z, µX ′ ] , [ Y, Z ′ ] i ) = 1 } = { Z ∈ q ( F ) : ∀ Z ′ ∈ L q , ψ ( h [[ Z, µX ′ ] , Y ] , Z ′ ] i ) = 1 } = n Z ∈ q ( F ) : [[ Z, µX ′ ] , Y ] ∈ e L q o . In other words, (ad Y ) ◦ (ad µX ′ )( ˇ L q ) = e L q , and vol( ˇ L q ) = | D s ( Y ) D s ( µX ′ ) | − F vol( e L q ) . (10)On the other hand, we have the relationvol( L q )vol( e L q ) = 1 . (11)Then Proposition 7.1 follows. γ ψ ( X, Y ) For X, Y ∈ c reg ( F ) or c ′ reg ( F ), since γ ψ ( X, Y ) appears in the expression of b i η ( X, Y ) or b i ( X, Y ) as in Proposition 7.1, we need to know an explicit formulaof γ ψ ( X, Y ). In this subsection, we show a formula (see Proposition 7.3) of γ ψ ( X, Y ) for X, Y lying in a Cartan subspace of the Lie algebra associated to ageneral symmetric pair. This result is an analogue of [Wa2, VIII.5 Lemme].Now we introduce some notations. Assume that ( G , H , θ ) is a general sym-metric pair, as introduced in § 3. Let s be the Lie algebra associated to ( G , H , θ ),and c a Cartan subspace of s . Let T be the centralizer of c in H and write t = Lie( T ). Fix a G -invariant and θ -invariant nondegenerate symmetric bilin-ear form h , i on g ( F ). Then, for X, Y ∈ c reg ( F ), the bilinear form q X,Y on h ( F ) / t ( F ) defined by q X,Y ( Z, Z ′ ) = h [ Z, X ] , [ Y, Z ′ ] i is nondegenerate and symmetric. Write γ ψ ( X, Y ) = γ ψ ( q X,Y ). For any subspace f of g ( F ) such that the restriction of h , i on f is nondegenerate, we write γ ψ ( f )for the Weil index associated to ψ and the form h , i on f .Let T − be the maximal θ -split torus in G whose Lie algebra is c . Denoteby S c the set of roots of T − in g ( ¯ F ). Write Γ F for the absolute Galois groupGal( ¯ F /F ). Then Γ F acts on S c . For α ∈ S c , denote by m α its multiplicity in g ( ¯ F ). Since c ⊂ s , for α ∈ S c , we have θ ( α ) = − α and m α = m − α . For α ∈ S c ,denote by Γ ± α the stabilizer of { α, − α } in Γ F , by F ± α the fixed field of Γ ± α in ¯ F , and by S ∗ c a fixed set of representatives of orbits { α, − α } . Notice that, if X, Y ∈ c reg ( F ), α ( X ) α ( Y ) ∈ F ± α .For α ∈ S ∗ c , denote by ψ ′ the character ψ ◦ Tr F ± α /F of F ± α . Set γ F ± α ( α ( X ) α ( Y ) , ψ ′ ) = γ ψ ′ ( α ( X ) α ( Y ) q ) γ ψ ′ ( q )where q is the quadratic form on F ± α defined by q ( λ ) = λ .47 roposition 7.3. Let the notations be as above. Then, for X, Y ∈ c reg ( F ) , wehave γ ψ ( X, Y ) = γ ψ ( t ( F )) − γ ψ ( h ( F )) × Y α ∈ S ∗ c (cid:0) ( α ( X ) α ( Y ) , F ± α γ F ± α ( α ( X ) α ( Y ) , ψ ′ ) (cid:1) m α . where ( , ) F ± α is the Hilbert symbol on F ± α .Proof. Notice that for α ∈ S c we have m σα = m α for every σ ∈ Γ F . For eachroot space g α associated to α ∈ S c we can choose its basis { E α , ..., E m α α } sothat: (1) σ ( E iα ) = E iσα for each σ ∈ Γ F ; (2) θ ( E iα ) = E i − α ; (3) h E iα , E j − α i = δ ij .Consider the homomorphism τ : Y S ∗ c m α F ± α −→ g ( ¯ F ) , ( λ iα ) X α m α X i =1 X σ ∈ Γ / Γ ± α σ ( λ iα ) (cid:0) E iσα + E i − σα (cid:1) . In fact the image of τ lies in g ( F ) and τ defines an isomorphism Y S ∗ c m α F ± α ∼ −→ q ( F ) , where q is the unique complement of t in h which is stable under the adjointaction of T . For ( λ iα ) ∈ Q S ∗ c m α F ± α , we have q X,Y (cid:0) τ (cid:0) ( λ iα ) (cid:1)(cid:1) = X α ∈ S ∗ c m α X i =1 X σ ∈ Γ / Γ α σ ( λ iα ) h [ E iσα + E i − σα , X ] , [ Y, E iσα + E i − σα ] i = X α,i,σ σ ( λ iα ) ( − σα ( X ) σα ( Y )) h E iσα − E i − σα , E iσα − E i − σα i = X α,i,σ σ ( λ iα ) σα ( X ) σα ( Y ) h E iσα + E i − σα , E iσα + E i − σα i = X α ∈ S ∗ c m α X i =1 q X,Y,α ( λ iα ) , where q X,Y,α ( λ ) is the quadratic form on F ± α defined by q X,Y,α ( λ ) = Tr F ± α /F (cid:0) α ( X ) α ( Y ) λ (cid:1) . Therefore γ ψ ( X, Y ) = Y α ∈ S ∗ c γ ψ ( q X,Y,α ) m α . For α ∈ S ∗ c , let q ′ X,Y,α be the quadratic form on F ± α defined by: q ′ X,Y,α ( λ ) = 2 α ( X ) α ( Y ) λ . Then γ ψ ( q X,Y,α ) = γ ψ ′ ( q ′ X,Y,α ), and γ ψ ′ ( q ′ X,Y,α ) = ( α ( X ) α ( Y ) , F ± α γ F ± α ( α ( X ) α ( Y ) , ψ ′ ) γ ψ ′ ( q ′ α ) , q ′ α is the quadratic form on F ± α defined by q ′ α ( λ ) = 2 λ . Therefore γ ψ ( q X,Y,α ) = ( α ( X ) α ( Y ) , F ± α γ F ± α ( α ( X ) α ( Y ) , ψ ′ ) γ ψ ( q α ) , where q α ( λ ) = Tr F ± α /F (2 λ ) = Tr F ± α /F (cid:0) h E iα + E i − α , E iα + E i − α i λ (cid:1) . In summary, we deduce that γ ψ ( X, Y ) = Y α ∈ S ∗ c (cid:0) ( α ( X ) α ( Y ) , F ± α γ F ± α ( α ( X ) α ( Y ) , ψ ′ ) γ ψ ( q α ) (cid:1) m α . On the other hand, by the same argument as above, we can show that γ ψ ( q ( F )) = Y α ∈ S ∗ c γ ψ ( q α ) m α . Together with the obvious relation γ ψ ( q ( F )) = γ ψ ( t ( F )) − γ ψ ( h ( F )) , we complete the proof. To obtain the main result of this subsection, we need the following lemma. Lemma 7.4. Let X ∈ c reg ( F ) and Y ∈ c ′ reg ( F ) be such that X ↔ Y . Thenthere exists an element x ∈ GL n ( E ) such that Ad( x ) Y = X , and Ad( x ) inducesisomorphisms Ad( x ) : t ′ → t and Ad( x ) : c ′ → c over F .Proof. It suffices to prove this for X = (cid:18) n A (cid:19) and Y = (cid:18) γB ¯ B (cid:19) , where A ∈ GL n ( F ) is regular semisimple and A = γB ¯ B . Then we have c ( F ) = (cid:26)(cid:18) CAC (cid:19) : C ∈ gl n ( F ) , AC = CA (cid:27) , t ( F ) = (cid:26)(cid:18) D D (cid:19) : D ∈ gl n ( F ) , AD = DA (cid:27) , c ′ ( F ) = (cid:26)(cid:18) γP ¯ P (cid:19) : P ∈ gl n ( E ) , B ¯ P = P ¯ B (cid:27) , and t ′ ( F ) = (cid:26)(cid:18) Q 00 ¯ Q (cid:19) : Q ∈ gl n ( E ) , B ¯ Q = QB (cid:27) . Take x = (cid:18) n γB (cid:19) ∈ GL n ( E ). We claim that Ad( x ) satisfies the requiredcondition. By the above relation, it is easy to see that:1. Ad( x ) · (cid:18) γP ¯ P (cid:19) = (cid:18) γP B − AP B − (cid:19) , AP B − = P B − A ;49. Ad( x ) · (cid:18) Q 00 ¯ Q (cid:19) = (cid:18) Q Q (cid:19) , AQ = QA .Therefore we have to show that P B − ∈ gl n ( F ), Q ∈ gl n ( F ).Note that since A = B ¯ B , A commutes with B . It is easy to see that P and Q also commute with A . Hence P and Q commute with B , since A is regular.Therefore the relation B ¯ P = P ¯ B implies that P B − = ¯ P ¯ B − ; the relation B ¯ Q = QB implies that ¯ Q = Q , which concludes the proof.Now let X ∈ c reg ( F ) and Y ∈ c ′ reg ( F ) be such that X ↔ Y . Then wecan take an x ∈ GL n ( E ) as in the above lemma. For any V ∈ c ′ reg ( F ), put U = Ad( x ) V . Lemma 7.5. Let X, Y, U, V be as above. Then we have the following relations h X, U i = h Y, V i , and γ ψ ( X, U ) = γ ψ ( h ( F )) γ ψ ( h ′ ( F )) − γ ψ ( Y, V ) . Proof. The first relation follows directly from the above lemma. The second re-lation follows from the above lemma, Proposition 7.1 and the similar argumentsof equation (6) in [Wa1, page 96]. This subsection is devoted to showing that we can construct specific C ∞ c -functionssatisfying certain “good” matching conditions. Such functions will play an im-portant role in proving Theorem 5.16 by global method. The result below is ananalogue of [Wa3, Proposition in § Proposition 7.6. Let Y ∈ c ′ reg ( F ) ⊂ s ′ rs ( F ) and X ∈ c reg ( F ) ⊂ s rs ( F ) be suchthat X ↔ Y . Then there exist functions f ∈ C ∞ c ( s ( F )) and f ′ ∈ C ∞ c ( s ′ ( F )) satisfying the following conditions.1. If X ∈ Supp( f ) , there exists Y ∈ c ′ reg ( F ) such that X ↔ Y .2. If Y ∈ Supp( f ′ ) , Y is H ′ -conjugate to an element in c ′ reg ( F ) .3. f and f ′ are smooth transfer of each other.4. There is an equality κ ( X ) b I η ( X , f ) = c b I ( Y , f ′ ) = 0 , where c = γ ψ ( h ( F )) γ ψ ( h ′ ( F )) − .Proof. Let W c (resp. W c ′ ) be the Weyl group associated to c (resp. c ′ ), i.e. W c = N H ( c ) /Z H ( c ) (resp. W c ′ = N H ′ ( c ′ ) /Z H ′ ( c ′ )). Set C ( X ) = { X ∈ c reg ( F ) : X = i ( X ) for some i ∈ W c } , and C ( Y ) = { Y ∈ c ′ reg ( F ) : Y = i ( Y ) for some i ∈ W c ′ } . By Lemma 7.4, we fix an isomorphism ϕ : c ′ ( F ) → c ( F ) such that ϕ ( Y ) = X . Fix V ∈ c ′ reg ( F ) and U := ϕ ( V ) ∈ c reg ( F ) so that if X ∈ C ( X ) − X (resp. Y ∈ C ( Y ) − Y ), we have h X − X , U i 6 = 0 (resp. h Y − Y , V i 6 = 0),and moreover, κ ( U ) = κ ( X ). We make the following choices.50. Fix an integer r ≥ • ̟ r O F ⊂ F × ; • the sets i ((1 + ̟ r O F ) U ) (resp. i ((1 + ̟ r O F ) V )), for i ∈ W c (resp. i ∈ W c ′ ), are mutually disjoint.2. There exists an integer N such that if µ ∈ F × satisfying v F ( µ ) < − N , wehave • for each X ∈ C ( X ) − X (resp. Y ∈ C ( Y ) − Y ), the character α ψ ( ̟ r µα h X − X , U i ) (resp. α ψ ( ̟ r µα h Y − Y , Y i )) isnontrivial on O F .3. Fix N and µ ∈ F × with v F ( µ ) < − N such that • η ( µ )=1; • the condition (ii) above is satisfied; • the formulae of Proposition 7.1 hold for b i η ( X , i ( µU )) and b i ( Y , i ′ ( µV ))for all i ∈ W c and i ′ ∈ W c ′ .4. Set ω = µ (1 + ̟ r O F ) U and ω ′ = µ (1 + ̟ r O F ) V . Denote by d (resp. d ′ ) the F -vector space generated by U (resp. V ), and fix a complement e (resp. e ′ ) in c ( F ) (resp. c ′ ( F )), i.e. c ( F ) = d ⊕ e (resp. c ′ ( F ) = d ′ ⊕ e ′ ).If U ∈ c ( F ) (resp. V ∈ c ′ ( F )), denote by U d (resp. V d ′ ) its projection on d (resp. d ′ ). We choose open compact neighborhoods ω e and ω ′ e ′ of 0 in e and e ′ small enough so that: if we set ω = ω ⊕ ω e and ω ′ = ω ′ ⊕ ω ′ e ′ , then • the sets i ( ω ) (resp. i ′ ( ω ′ )), for i ∈ W c (resp. i ′ ∈ W c ′ ), are mutuallydisjoint; • ω ⊂ c reg ( F ) , ω ′ ⊂ c ′ reg ( F ), ϕ ( ω ′ ) = ω , and therefore, for U ∈ ω, V ∈ ω ′ , they match with each other if and only if ϕ ( V ) = U ; • for each X ∈ C ( X ) (resp. Y ∈ C ( Y )), and each U ∈ ω (resp. V ∈ ω ′ ), b i η ( X, U ) = b i η ( X, U d ) , b i ( Y, V ) = b i ( Y, V d ′ ); • the function κ is constant on ω , which hence equals to κ ( X ).Define a function f ω (resp. f ′ ω ′ ) on ω (resp. ω ′ ) by f ω ( U ) = ψ ( −h X , U d i ) , for U ∈ ω,f ′ ω ′ ( V ) = ψ ( −h Y , V d ′ i ) , for V ∈ ω ′ . Now we fix a function f ∈ C ∞ c ( s ( F )) (resp. f ′ ∈ C ∞ c ( s ′ ( F ))) such thatSupp( f ) ⊂ ω H , and κ ( U ) I η ( U, f ) = f ω ( U ) for each U ∈ ω, Supp( f ′ ) ⊂ ω ′ H ′ , and I ( V, f ′ ) = f ′ ω ′ ( V ) for each V ∈ ω ′ . Then, we have, for X ∈ s rs ( F ), κ ( X ) I η ( X, f ) = (cid:26) f ω ( U ) if X is H -conjugate to some U ∈ ω, , Y ∈ s ′ rs ( F ), I ( Y, f ′ ) = (cid:26) f ′ ω ′ ( V ) if Y is H ′ -conjugate to some V ∈ ω ′ , . Thus the assertions (i), (ii) and (iii) of the proposition follow from the aboveconstruction and Lemma 7.5.To prove the assertion (iv), we observe that κ ( X ) b I η ( X , f ) = κ ( X ) Z s ( F ) b i η ( X , U ) κ ( U ) f ( U ) | D s ( U ) | − / d U = | W c | − κ ( X ) Z c ( F ) b i η ( X , U ) κ ( U ) I η ( f, U ) d U = κ ( X ) Z ω b i η ( X , U ) f ω ( U ) d U = X i ∈ W c κ ( X ) Z ω η ( i ) κ ( U ) γ ψ ( i ( X ) , U ) ψ ( h i ( X ) , U i ) f ω ( U ) d U = X X ∈ C ( X ) vol( ω e ) κ ( X ) κ ( U ) Z ω γ ψ ( X, U d ) ψ ( h X − X , U d i ) d U d = X X ∈ C ( X ) vol( ω ) κ ( X ) κ ( U ) × Z O F γ ψ ( X, µ (1 + ̟ r α ) U ) ψ ( h X − X , µ (1 + ̟ r α ) U i ) d α. By condition (i), γ ψ ( X, µ (1 + ̟ r α ) U ) = γ ψ ( X, µU ) , for any α ∈ O F . If X = X , by condition (ii), Z ω ψ ( h X − X , µ (1 + ̟ r α ) U i ) d α = 0 . Therefore, κ ( X ) b I η ( X , f ) = vol( ω ) γ ψ ( X , µU ) = 0 . The same computation goes for b I ( Y , f ′ ) and we get b I ( Y , f ) = vol( ω ′ ) γ ψ ( Y , µV ) = 0 . Then the conclusion follows from Lemma 7.5 and vol( ω ) = vol( ω ′ ). In this section, we will prove Theorem 5.16. We divide this theorem into twoparts, i.e., Theorems 8.1 and 8.2 below. Theorem 8.1. If f is in C ∞ c ( s ( F )) , so is b f . heorem 8.2. There exists a nonzero constant c ∈ C satisfying that: if f ∈C ∞ c ( s ( F )) and f ∈ C ∞ c ( s ′ ( F )) are smooth transfer of each other, then κ ( X ) b I η ( X, f ) = c b I ( Y, f ′ ) for any X ∈ s rs ( F ) and Y ∈ s ′ rs ( F ) such that X ↔ Y . We will use a local method to prove Theorem 8.1, and a global method toprove Theorem 8.2, as we have said before. The global method is a modificationof that of [Wa3]. By Lemma 5.15, it suffices to only consider the case s ′ = s ′ ǫ when ǫ = 1.Throughout this subsection, we assume that ǫ = 1.Recall that s rs ( F ) is the subset of elements in s rs ( F ) coming from s ′ rs ( F ).Let C be the set of Cartan subspaces of s coming from those of s ′ , and | C | aset of representatives for H -conjugacy classes of Cartan subspaces in C .Let f be in C ∞ c ( s ( F )) . Then, by the Weyl integration formula, we have b I η ( X, f ) = Z s ( F ) b i η ( X, Y ) κ ( Y ) f ( Y ) | D s ( Y ) | − / d Y = X c | W c | − Z c reg ( F ) b i η ( X, Y ) κ ( Y ) I η ( Y, f ) d Y = X c ∈| C | | W c | − Z c reg ( F ) b i η ( X, Y ) κ ( Y ) I η ( Y, f ) d Y. Thus, to show b I η ( X, f ) = 0 for any X / ∈ s rs ( F ) , it suffices to show the followinglemma. Lemma 8.3. For any X / ∈ s rs ( F ) and any Y ∈ s rs ( F ) , we have b i η ( X, Y ) = 0 .Proof. First we need some preparation. Define an involution τ on C ∞ c ( s ( F )): f τ ( X ) := f ( X t ), where X = (cid:18) AB (cid:19) ∈ s ( F ) and X t is its transpose. Thefollowing two properties can be easily checked:1. τ commutes with Fourier transform, i.e., ( b f ) τ = c f τ ;2. for X = (cid:18) AB (cid:19) ∈ s rs ( F ), I η ( X, f τ ) = η (det AB ) I η ( X, f ) for any f ∈C ∞ c ( s ( F )).In particular, if Y ∈ s rs ( F ) , then I η ( Y, f τ ) = I η ( Y, f ); if an elliptic X is notin s rs ( F ) , then I η ( X, f + f τ ) = 0.Now let X / ∈ s rs ( F ) be an elliptic element. For any f ∈ C ∞ c ( s ( F )) , by theabove discussion, we see that0 = b I η ( X, f + f τ ) = 2 X c ∈| C | | W c | − Z c reg ( F ) b i η ( X, Y ) κ ( Y ) I η ( Y, f ) d Y. Y ∈ s rs ( F ) we may choose a specific f ∈ C ∞ c ( s ( F )) so that X c ∈| C | | W c | − Z c reg ( F ) b i η ( X, Y ) κ ( Y ) I η ( Y, f ) d Y = b i η ( X, Y ) . Therefore b i η ( X, Y ) = 0 for any elliptic X / ∈ s rs ( F ) and any Y ∈ s rs ( F ) .Now let X / ∈ s rs ( F ) be a non-elliptic element. It suffices to assume that X is of the form X ( A ) = (cid:18) n A (cid:19) for some A ∈ GL n, rs ( F ). Since X / ∈ s rs ( F ) is non-elliptic, we can assume that A is of the form (cid:18) A A (cid:19) where A ∈ GL n , rs ( F ) is elliptic and not in N(GL n ( E )), and A is in GL n , rs ( F ). Recallthe discussions in § r ≃ s n × s n of s such that X ∈ r .Moreover, under the natural isomorphism ι : r ∼ → s n × s n , the image of X is ( X , X ) where X i = X ( A i ) for i = 1 , 2. Write s i = s n i for i = 1 , 2. Let M = H × H where H i = H n i for i = 1 , 2. Then M acts on r naturally. For Z ∈ r rs ( F ), let b i η, r ( Z, · ) be the kernel function that represents the distribution f b I η,M ( Z, f ) for f ∈ C ∞ c ( r ( F )). It is obvious that b i η, r ( Z, Y ) = b i η, s ( Z , Y ) b i η, s ( Z , Y )where ( Z , Z ) and ( Y , Y ) are the images of Z and Y under ι in s × s respectively, and b i η, s i ( Z i , · ) is the kernel function that represents the distribution f b I η,H i ( Z i , f ) for f ∈ C ∞ c ( s i ( F )) for i = 1 , b i η ( X, Y ) = X Y ′ b i η, r ( X, Y ′ ) , where Y ′ runs over a set of representatives for the finitely many M -conjugacyclasses of elements of r ( F ) which are H -conjugate to Y . Therefore we can anddo assume that Y ∈ s rs ( F ) is in r ( F ) and of the form Y = ( Y , Y ) under thenatural map ι where Y i ∈ s i, rs ( F ) . Then b i ι, r ( X, Y ) = b i η, s ( X , Y ) b i η, s ( X , Y ) = 0 , since X / ∈ s , rs ( F ) is elliptic and Y ∈ s , rs ( F ) . We complete the proof of thelemma. Now let k be a number field, A = A ∞ × A f its ring of adeles. Let k ′ be aquadratic field extension of k , D a quaternion algebra over k containing k ′ , and η the quadratic character of A × /k × attached to k ′ by the class field theory.We define the global symmetric pairs ( G , H ) and ( G ′ , H ′ ) over k with respectto k ′ and D similarly as the local cases. Let s , s ′ be the corresponding global“Lie algebras” associated to ( G , H ) and ( G ′ , H ′ ) respectively, which are definedover k . Denote by S ( s ( A )) (resp. S ( s ′ ( A ))) the space of Schwartz functionson s ( A ) (resp. s ′ ( A )). Denote by H ( A ) the set of ( h , h ) ∈ H ( A ) such that | det h | = | det h | = 1, and by H ′ ( A ) the set of h ∈ H ′ ( A ) such that | det h | = 1.The groups H ( A ) and H ′ ( A ) are subgroups of H ( A ) and H ′ ( A ) respectively.We have the following theorem concerning the issue about convergence.54 heorem 8.4. For each φ ∈ S ( s ( A )) , Z H ( k ) \ H ( A ) X X ∈ s ell ( k ) | φ ( X h ) | d h < ∞ . Similarly, for each φ ′ ∈ S ( s ′ ( A )) , Z H ′ ( k ) \ H ′ ( A ) X Y ∈ s ′ ell ( k ) | φ ′ ( Y h ) | d h < ∞ . Proof. (1) Now we prove the assertion for φ ∈ S ( s ( A )). Here we still write Z = ( X, Y ) ∈ s = gl n ⊕ gl n and h · Z = (Ad h ) Z where h ∈ H for convenience.Recall that Z = ( X, Y ) is in s ell ( k ) if and only if neither XY nor Y X is containedin a proper parabolic subgroup of GL n ( k ). Let P be the minimal parabolicsubgroup of GL n consisting of the upper-triangular one. Put P = P × P ⊂ H .Identify R × + with the subgroup of A ×∞ consisting of elements whose componentsat each place are the same and belong to R × + . For each real number c > A c the set of a = diag( a , ..., a n ) ∈ SL n ( R ) such that a i a i +1 ≥ c for all1 ≤ i ≤ n − a i ∈ R × + for all 1 ≤ i ≤ n , and set A c = A c × A c ⊂ H ( R ) ⊂ H ( A ∞ ) . By reduction theory, we know that there exists a maximal compact subgroup K of H ( A ), a compact subset ω ⊂ P( A ) ∩ H ( A ) and a c > G = { pak ; p ∈ ω, a ∈ A c , k ∈ K } , we have the equality H ( A ) = H ( k ) G , and thus, for each measurable function φ on H ( k ) \ H ( A ) with valued ≥ 0, the integral Z H ( k ) \ H ( A ) φ ( x ) d x is convergent if and only if the integral Z G φ ( x ) d x is such so. Fix such K , ω, c . Then the integral is convergent if there exists C ≥ p ∈ ω, k ∈ K , Z A c X Z ∈ s ell ( k ) | φ (( pak ) · Z ) | δ P ( a ) − d a ≤ C, where δ P is the modulus character of P . There exists a compact set Ω ⊂ H ( A ) such that for all p ∈ ω, a ∈ A c , k ∈ K , a − pak ∈ Ω. Then there exists φ ′ ∈S ( s ( A )) such that for all Z ∈ s ( A ) and h ∈ Ω, we have | φ ( h · Z ) | ≤ φ ′ ( Z ). Itsuffices to consider φ ′ of the form φ ′ = φ ′∞ ⊗ φ ′ f , where φ ′∞ ∈ S ( s ( A ∞ )) , φ ′ f ∈S ( s ( A f )) both with ≥ Z A c X Z ∈ s ell ( k ) φ ′ ( a · Z ) δ P ( a ) − d a. O k -lattice L in s ( k ) such that s ( k ) ∩ Supp( φ f ) ⊂ L . Denote L ell = L ∩ s ell ( k ). Since φ ′ f ( a · Z ) = φ ′ f ( Z ), it suffices to consider the integral Z A c X Z ∈ L ell ( k ) φ ′∞ ( a · Z ) δ P ( a ) − d a. If x v ∈ k v and v is an infinite place of k , write | x v | for the usual absolute valueof x v . For every x = ( x v ) ∈ A ∞ , put | x | = max v | x v | . For X = ( x i,j ) ∈ gl n ( A ∞ ),put | X | = max i,j | x i,j | . For Z = ( X, Y ) ∈ s ( A ∞ ), write | Z | = max {| X | , | Y |} .Then the following lemma implies the theorem. Lemma 8.5. Assume that n ≥ . There is a positive valued polynomial function P on the real vector space s ( A ∞ ) , which depends on L and c , such that P ( a · Z ) ≥ n − Y i =1 a i a i +1 · b i b i +1 ! | Z | , for all a = diag( a , ..., a n , b , ..., b n ) ∈ A c and all Z ∈ L ell .Proof. Take a positive valued polynomial function P on s ( A ∞ ) such that P ( X, Y ) ≥ max {| XY | , | Y X |} , for all ( X, Y ) ∈ s ( A ∞ ) . Take a positive number c L such that( X, Y ) ∈ L, d is a nonzero entry of XY or Y X ⇒ | d | ≥ c L . Let a = ( a , a , · · · , a n , b , b , · · · , b n ) in A c and let Z = ( X, Y ) = (( x i,j ) , ( y i,j ))in L ell . Write ( u i,j ) for XY . Fix i = 1 , , · · · , n − 1. Since XY is not containedin a proper parabolic subalgebra of gl n ( k ), there are i ≥ i + 1 and j ≤ i suchthat u i,j = 0 . Then | u i,j | ≥ c L , and we have P ( a · Z ) ≥ | a i u i,j a − j | ≥ c L a i a − j ≥ c L c i − i − c i − j a i a − i +1 ≥ c L c n − a i a − i +1 . This implies that a − n = n − Y i =1 ( a i /a i +1 ) in ≤ (cid:18) c L c n − P ( a · Z ) (cid:19) n − , and a = n − Y i =1 ( a i /a i +1 ) n − in ≤ (cid:18) c L c n − P ( a · Z ) (cid:19) n − . Similarly, P ( a · Z ) ≥ c L c n − b i b − i +1 , b , b − n ≤ (cid:18) c L c n − P ( a · Z ) (cid:19) n − . For all i, j = 1 , , · · · , n , we have | a · Z | ≥ a i | a i,j | b − j . Therefore, | a i,j | ≤ a − i b j | a · Z | ≤ c − ( n − i ) − ( j − a − n b | a · Z |≤ c − (2 n − (cid:18) c L c n − (cid:19) n − P ( a · Z ) n − | a · Z | . Similarly, | b i,j | ≤ c − (2 n − (cid:18) c L c n − (cid:19) n − P ( a · Z ) n − | a · Z | . By timing a i a i +1 and b i b i +1 on both sides of the above inequality, we get thelemma.(2) The convergence of the second integral (for φ ′ ∈ S ′ ( s ( A ))) can be deducedeasily from Lemma 10.8 of [Wa3], since the twisted conjugation by A c is theusual conjugation.By the above theorem, we have a well-defined distribution I η on s ( A ), definedby I η ( φ ) = Z H ( k ) \ H ( A ) X X ∈ s ell ( k ) φ ( X h ) η ( h ) d h, φ ∈ S ( s ( A )) , and a well-defined distribution I on s ′ ( A ), defined by I ( φ ′ ) = Z H ′ ( k ) \ H ′ ( A ) X Y ∈ s ′ ell ( k ) φ ′ ( Y h ) d h, φ ′ ∈ S ( s ′ ( A )) . If φ = Q v φ v , φ ′ = Q v φ ′ v , it is routine to see that I η ( φ ) = X X ∈ [ s ell ( k )] τ ( H X ) Y v κ v ( X ) I η ( X, φ v ) ,I ( φ ′ ) = X Y ∈ [ s ′ ell ( k )] τ ( H ′ Y ) Y v I ( Y, φ ′ v ) , where τ ( H X ) = vol( H X ( k ) \ ( H X ∩ H ( A ) )) , τ ( H ′ Y ) = vol( H ′ Y ( k ) \ ( H ′ Y ∩ H ′ ( A ) ) , [ s ell ( k )] denotes the set of H ( k )-orbits in s ell ( k ), and [ s ′ ell ( k )] denotes the setof H ′ ( k )-orbits in s ′ ell ( k ). If X ∈ s rs ( k ) and Y ∈ s ′ rs ( k ) so that X ↔ Y , then H X ≃ H ′ Y (same reason as the local case). We choose Haar measures on H X ( A )and H ′ Y ( A ) so that they are compatible. Thus, if X ∈ s ell ( k ) , Y ∈ s ′ ell ( k ) suchthat X ↔ Y , we have τ ( H X ) = τ ( H ′ Y ) . .3 Proof of Theorem 8.2 Now, we fix f ∈ C ∞ c ( s ( F )) and f ′ ∈ C ∞ c ( s ′ ( F )) so that they are smooth transferof each other. Here we allow that f may not lie in C ∞ c ( s ( F )) , as we havementioned in the proof of Proposition 5.17. We also refer the reader to theproof of Proposition 5.17 to see the definition of smooth transfer in this moregeneral situation.Fix X ∈ s rs ( F ) , Y ∈ s ′ rs ( F ) such that X ↔ Y . Our aim is to search for anonzero constant c which is independent of f, f ′ , X and Y such that κ ( X ) b I η ( X , f ) = c b I ( Y , f ′ ) . In the following, we choose some global data. Fields We choose a number field k , a quadratic field extension k ′ of k , and aquaternion algebra D over k containing k ′ so that:1. k is totally imaginary;2. there exists a finite place w of k such that k w ≃ F, k ′ w ≃ E and D ( k w ) ≃ D;3. there exists another finite place u of k such that u is inert in k ′ .Such a number field k and a quaternion algebra D do exist (cf. [Wa3, Propositionin § k w with F , k ′ w with E and D ( k w ) with D.Denote by A the ring of adeles of k , by O k the ring of integers of k , and O k ′ thering of integers of k ′ . Fix a continuous character A /k whose local componentat w is our fixed character ψ of k w . Denote by ψ this global character, whenthere is no confusion. Groups We define the global symmetric pairs ( G , H ) and ( G ′ , H ′ ) over k with respect to k ′ and D similarly as the local case. We still use h and h ′ todenote the Lie algebras of H and H ′ respectively, use s and s ′ to denote theglobal Lie algebras corresponding to ( G , H ) and ( G ′ , H ′ ) respectively, if thereis no confusion. Thus X ∈ s rs ( k w ) and Y ∈ s ′ rs ( k w ). Places Denote by V (resp. V ∞ , V f ) the set of all (resp. archimedean, non-archimedean) places of k . Fix two O k -lattices: L = gl n ( O k ) ⊕ gl n ( O k ) ⊂ s ( k )and L ′ = gl n ( O k ′ ) ⊂ s ′ ( k ). For each v ∈ V f , put L v = L ⊗ O k O k,v , L ′ v = L ′ ⊗ O k O k,v . We fix a finite set S ⊂ V such that:1. S contains u, w and V ∞ ;2. for each v ∈ V − S , everything is unramified, i.e. G and G ′ are unramifiedover k v , L v and L ′ v are self-dual with respect to ψ v and h , i .We denote by S ′ the subset S − V ∞ − { w } of S . Orbits For each v ∈ V f , we choose an open compact subset Ω v ⊂ s ′ ( k v ) suchthat:1. if v = w , we require that: Y ∈ Ω w ⊂ s ′ rs ( k w ), b I ( · , f ′ ) is constant on Ω w ,and κ ( · ) b I η ( · , f ) is constant and hence equal to κ ( X ) b I η ( X , f ) on the setof X ∈ s rs ( k w ) which matches an element Y in Ω w ,58. if v = u , we require Ω u ⊂ s ′ ell ( k u );3. if v ∈ S but v = w, u , choose Ω v to be any open compact subset;4. if v ∈ V f − S , let Ω v = L ′ v .Then by the strong approximation theorem, there exists Y ∈ s ′ ( k ) such that Y ∈ Ω v for each v ∈ V f . Furthermore, by the condition (ii) above, Y ∈ s ′ ell ( k ).Take an element X ∈ s ell ( k ) such that X ↔ Y . Functions For each v ∈ V , we choose functions φ v ∈ S ( s ( k v )) and φ ′ v ∈S ( s ′ ( k v )) as follows:1. if v = w , let φ v = f and φ ′ v = f ′ ;2. if v ∈ S ′ , by Proposition 7.6, we require that: • if X v ∈ Supp( φ v ), there exists Y v ∈ c ′ Y ( k v ) such that X v ↔ Y v ,where we denote by c ′ Y the Cartan subspace in s ′ containing Y ; • if Y v ∈ Supp( φ ′ v ), there exists Y ′ v ∈ c ′ Y ( k v ) such that Y v and Y ′ v are H ′ ( k v )-conjugate; • φ v is a transfer of φ ′ v ; • κ v ( X ) b I η ( X , φ v ) = c v b I ( Y , φ ′ v ) = 0, where c v = γ ψ ( h ( k v )) γ ψ ( h ′ ( k v )) − ;3. for v ∈ V − S , set φ v = L v , φ ′ v = L ′ v ; then φ v = b φ v , φ ′ v = c φ ′ v , and byLemma 5.18 we have κ v ( X ) b I η ( X , φ v ) = κ v ( X ) I η ( X , φ v ) = I ( Y , φ ′ v ) = b I ( Y , φ ′ v );4. for v ∈ V ∞ , identifying ( H ( k v ) , s ( k v )) with ( H ′ ( k v ) , s ′ ( k v )), we choose φ v = φ ′ v ∈ S ( s ( k v )) such that: • b I η ( X , φ v ) = b I ( Y , φ ′ v ) = 0; • if X ∈ s ( k ) is H ( k v )-conjugate to an element in the support of c φ v ateach place v ∈ V , then X is H ( k )-conjugate to X ; • if Y ∈ s ′ ( k ) is H ′ ( k v )-conjugate to an element in the support of c φ ′ v at each place v ∈ V , then Y is H ′ ( k )-conjugate to Y .This is possible. The key point is that, by invariant theory, we have naturalmaps (cf. Remark 5.3) s ′ / H ′ ֒ → s / H −→ A nk , where A nk is the n -dimensional affine space over k so that A nk = Spec( O ( s ) H ).We refer the reader to [Wa3, Lemme in § φ ∈ S ( s ( A )) and φ ′ ∈ S ( s ′ ( A )) to be: φ = Y v ∈ V φ v , φ ′ = Y v ∈ V φ ′ v . inal proof According to the conditions on φ u (resp. φ ′ u ), we know thatif X ∈ s ( k ) (resp. Y ∈ s ′ ( k )) is such that X ∈ Supp( φ ) H ( A ) (resp. Y ∈ Supp( φ ′ ) H ′ ( A ) ), then X ∈ s ell ( k ) (resp. Y ∈ s ′ ell ( k )). Here we use Supp( φ ) H ( A ) to denote the union of H ( A )-orbits intersecting Supp( φ ), and Supp( φ ′ ) H ′ ( A ) to denote the union of H ′ ( A )-orbits intersecting Supp( φ ′ ). Suppose that X ∈ s ell ( k ) is such that I η ( X, φ ) = Y v ∈ V I η ( X, φ v ) = 0 . Then, by the conditions on φ v , X comes from s ′ ( k v ) at each place v not equalto w . We claim that X must come from s ′ ( k ). If not, there exists at least twoplaces v and v such that X does not come from s ′ ( k v ), which is a contradiction.Therefore we have I η ( φ ) = I ( φ ′ ) , since φ v is a transfer of φ ′ v at each place v not equal to w and is a partial transferof φ ′ v at the place v = w by the requirements we have imposed.On the other hand, according to the conditions on c φ v and c φ ′ v , we knowthat if X ∈ s ( k ) (resp. Y ∈ s ′ ( k )) is such that X ∈ Supp( b φ ) H ( A ) (resp. Y ∈ Supp( b φ ′ ) H ′ ( A ) ) then X is H ( k )-conjugate to X (resp. Y is H ′ ( k )-conjugate to Y ).By Poisson summation formula, we have X X ∈ s ( k ) φ ( X h ) = X X ∈ s ( k ) b φ ( X h ) , ∀ h ∈ H ( A ) , and X Y ∈ s ′ ( k ) φ ′ ( Y h ) = X Y ∈ s ′ ( k ) b φ ′ ( Y h ) , ∀ h ∈ H ′ ( A ) . Therefore, by the conditions on φ and φ ′ , we have I η ( φ ) = I η ( b φ ) , I ( φ ′ ) = I ( b φ ′ ) . Hence we obtain I η ( b φ ) = I ( b φ ′ ) , or equivalently, τ ( H X ) Y v ∈ V κ v ( X ) b I η ( X , φ v ) = τ ( H ′ Y ) Y v ∈ V b I ( Y , φ ′ v ) . Note that for v ∈ V − S , we have κ v ( X ) b I η ( X , φ v ) = b I ( Y , φ ′ v ) = 0 , and for almost all v ∈ V − S , κ v ( X ) b I η ( X , φ v ) = b I ( Y , φ ′ v ) = 1 . For v ∈ S ′ and v ∈ V ∞ , we have κ v ( X ) b I η ( X , φ v ) = c v b I ( Y , φ ′ v ) = 0 . κ w ( X ) b I η ( X , f ) = c b I ( Y , f ′ ) , where c = ( Y v ∈ S ′ c v ) − = Y v ∈ S ′ γ ψ ( h ( k v )) − γ ψ ( h ′ ( k v )) . Notice that if v ∈ V ∞ or v ∈ V − S , γ ψ ( h ( k v )) = γ ψ ( h ′ ( k v )) = 1 . Also notice that Y v ∈ V γ ( h ( k v )) = Y v ∈ V γ ψ ( h ′ ( k v )) = 1 . Therefore c = γ ψ ( h ( k w )) γ ψ ( h ′ ( k w )) − . . Since κ w ( X ) b I η ( X , f ) = κ w ( X ) b I η ( X , f ) , b I ( Y , f ′ ) = b I ( Y , f ′ ) , we complete the proof of the theorem. Acknowledgements This work was supported by the National Key Basic Re-search Program of China (No. 2013CB834202). Needless to say, the remarkablework of Jean-Loup Waldspurger on endoscopic transfer has a huge influence onthis article. The author is very grateful to Wei Zhang for suggesting this problemand sharing the idea that Waldspurger’s method might apply to this situation,to Binyong Sun for proving Theorem 8.4 which is crucial for our method. Healso thanks Kimball Martin for communicating their work [FMW] and sendingtheir preprint, and thanks Wen-Wei Li for helpful discussions. He expressesgratitude to Ye Tian and Linsheng Yin for their constant encouragement andsupport. He would like to thank the anonymous referee for explaining how toprove Theorem 8.1, which improves the main result of the article greatly, andmany other useful comments. References [AG] A. Aizenbud and D. 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(2014), no. 3, 971–1049.[Zhw2] W. Zhang, Automorphic period and the central value of Rankin-SelbergL-function , J. Amer. Math. Soc. (2014), 541–612. Chong ZhangSchool of Mathematical Sciences, Beijing Normal University,Beijing 100875, P. R. China.E-mail address: [email protected]@bnu.edu.cn