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THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS: THE ENDOSCOPIC CASE by RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, and MICHAŁ ZYDOR ABSTRACT In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups U × U in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain n n+1 cuspidal data, called ∗-regular, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods introduced by Jacquet-Lapid-Rogawski. The other, built upon Zeta integrals, is expressed in terms of functionals on the Whittaker model. A direct proof of the equality between the two expressions is also given. Finally several useful auxiliary results about the spectral expansion of the Jacquet-Rallis trace formula are provided. CONTENTS 1. Introduction ...................................................... 184 1.1. The endoscopic cases of the Gan-Gross-Prasad conjecture ........................ 184 1.2. The spectral expansion of the Jacquet-Rallis trace formula for the linear groups ............. 188 1.3. On the ∗-regular contribution for the Jacquet-Rallis trace formula for the linear groups ........ 190 1.4. Outline of the paper . ............................................ 195 2. Preliminaries ..................................................... 196 2.1. General notation ............................................... 196 2.2. Algebraic groups and adelic points . ..................................... 197 2.3. Haar measures ................................................ 200 2.4. Heights, weights and Harish-Chandra function ............................. 202 2.5. Spaces of functions .............................................. 209 2.6. Estimates on Fourier coefficients....................................... 221 2.7. Automorphic forms and representations .................................. 223 2.8. Relative characters .............................................. 227 2.9. Cuspidal data and coarse Langlands decomposition ............................ 229 2.10. Automorphic kernels ............................................. 239 3. The coarse spectral expansion of the Jacquet-Rallis trace formula for Schwartz functions ........... 243 3.1. Notations . .................................................. 243 3.2. The coarse spectral expansion for Schwartz functions ........................... 246 3.3. Auxiliary expressions for I .......................................... 248 3.4. Auxiliary function spaces and smoothed constant terms .......................... 252 3.5. A relative truncation operator ........................................ 258 3.6. Proof of Theorem 3.3.7.1 ........................................... 261 3.7. Proof of Proposition 3.3.8.1 ......................................... 264 4. Flicker-Rallis period of some spectral kernels .................................... 267 4.1. Flicker-Rallis intertwining periods and related distributions ........................ 267 4.2. A spectral expansion of a truncated integral ................................. 272 4.3. The case of ∗-regular cuspidal data ..................................... 273 5. The ∗-regular contribution in the Jacquet-Rallis trace formula . ......................... 277 5.1. Relative characters .............................................. 277 5.2. The ∗-regular contribution .......................................... 280 6. Spectral decomposition of the Flicker-Rallis period for ∗-regular cuspidal data ................. 282 6.1. Notation .................................................... 283 6.2. Statements of the main results ........................................ 284 6.3. Proof of Theorem 6.2.5.1.2 .......................................... 286 6.4. Proof of Theorem 6.2.6.1 ........................................... 289 6.5. Convergence of Zeta integrals ........................................ 293 © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2022 https://doi.org/10.1007/s10240-021-00129-1 184 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 7. Canonical extension of the Rankin-Selberg period for H-regular cuspidal data ................. 295 7.1. Statements of the main results ........................................ 295 7.2. Proof of Theorem 7.1.3.1 ........................................... 297 7.3. Convergence of Zeta integrals ........................................ 300 8. Contributions of ∗-regular cuspidal data to the Jacquet-Rallis trace formula: second proof ........... 301 8.1. Main result .................................................. 302 8.2. Proof of Theorem 8.1.4.1 ........................................... 305 9. Flicker-Rallis functional computation........................................ 308 9.1. Local comparison ............................................... 308 9.2. Global comparison .............................................. 317 10. Proofs of the Gan-Gross-Prasad and Ichino-Ikeda conjectures . ......................... 318 10.1. Identities among some global relative characters .............................. 318 10.2. Proof of Theorem 1.1.5.1 ........................................... 322 10.3. Proof of Theorem 1.1.6.1 ........................................... 323 Acknowledgements ..................................................... 326 Appendix A: Topological vector spaces .......................................... 326 References ......................................................... 333 1. Introduction 1.1. The endoscopic cases of the Gan-Gross-Prasad conjecture 1.1.1. One of the main motivation of the paper is the obtention of the remaining cases, the so-called “endoscopic cases”, of the Gan-Gross-Prasad and the Ichino-Ikeda conjectures for unitary groups. To begin with, we shall give the main statements we prove. 1.1.2. Let E/F be a quadratic extension of number fields and c be the non-trivial element of the Galois group Gal(E/F).Let A be the ring of adèles of F. Let η be the quadratic idele class character associated to the extension E/F. Let n 1 be an integer. Let H be the set of isomorphism classes of non-degenerate c-Hermitian spaces h over E of rank n.For any h ∈ H , we identify h with a representative and we shall denote by n n n U(h ) its automorphisms group. Let h ∈ H be the element of rank 1 given by the norm n 0 1 N . E/F We attach to any h ∈ H the following algebraic groups over F: • the unitary group U of automorphisms of h; • the product of unitary groups U = U(h) × U(h ⊕ h ) where h ⊕ h denoted the h 0 0 orthogonal sum. We have an obvious diagonal embedding U → U . 1.1.3. Arthur parameter. — Let G be the group of automorphisms of the E-vector n 1 space E .Weview G as an F-group by Weil restriction. By a Hermitian Arthur parameter of G , we mean an irreducible automorphic representation for which there exists a par- tition n +···+ n = n of n and for any 1 i r a cuspidal automorphic representation 1 r of G (A) such that i n Strictly speaking, it is a generic discrete Arthur parameter. By simplicity, we shall omit the adjectives generic discrete. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 185 n+1 (−1) 1. each is conjugate self-dual and the Asai L-function L(s, , As ) has a i i pole at s = 1; 2. the representations are mutually non-isomorphic for 1 i r; 3. the representation is isomorphic to the full induced representation Ind ( ... ) where P is a parabolic subgroup of G of Levi factor 1 r n G ×···× G . n n 1 r Remark 1.1.3.1. — It is well-known (see [Fli88]) that condition 1 above is equiva- n+1 lent to the fact that is (GL ,η )-distinguished in the sense of Section 4.1.2 below. i n ,F The integer r and the representations ( ) are unique (up to a permutation). i 1ir We set S =(Z/2Z) . Let G = G × G .Bya Hermitian Arthur parameter of G, we mean an automorphic n n+1 representation of the form = where is a Hermitian Arthur parameter n n+1 i of G for i = n, n + 1. For such a Hermitian Arthur parameter, we set S = S × S . n n+1 1.1.4. Let h ∈ H and σ be a cuspidal automorphic representation of U (A).We n h say that a Hermitian Arthur parameter of G is a weak base-change of σ if for almost all places of F that split in E, the local component is the split local base change of σ .If v v this is the case, we write = BC(σ). Remark 1.1.4.1. —By the work of Mok [Mok15] and Kaletha-Minguez-Shin- White [KMSW], we know that if σ admits a weak base-change then it admits a strong base-change that is a Hermitian Arthur parameter of G such that is the base-change n v of σ for every place v of F (where the local base-change of ramified representations is also constructed in loc. cit. and characterized by certain local character relations). Besides, a result of Ramakrishnan [Ram18] implies that a weak base-change is automatically a strong base-change. Therefore, we could have used the notion of strong base-change instead. However, we prefer to stick with the terminology of weak base-change in order to keep the statement of the next theorem independent of [Mok15]and [KMSW]. 1.1.5. Gan-Gross-Prasad conjecture. — Our first main result is the global Gan-Gross- Prasad conjecture [GGP12, Conjecture 24.1] in the case of U(n)× U(n+ 1) and can be stated as follows. In the following, for a reductive group H over F, we denote by [H] the quotient H(F)\H(A) equipped with the quotient of a Haar measure on H(A) by the counting measure on H(F). Theorem 1.1.5.1. — Let be a Hermitian Arthur parameter of G. The following two state- ments are equivalent: 186 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 1. The complete Rankin-Selberg L-function of satisfies L( ,) = 0; 2. There exists h ∈ H and an irreducible cuspidal automorphic subrepresentation σ of U such n h that is a weak base change of σ and the period integral P defined by P (ϕ) = ϕ(h) dh [U ] induces a non-zero linear form on the space of σ . Remark 1.1.5.2. — If the Arthur parameter is moreover simple (that is if is cuspidal), the theorem is proved by Beuzart-Plessis-Liu-Zhang-Zhu (cf. [BPLZZ21,The- orem 1.7]). Previous works had to assume extra local hypothesis on , which implied that was also simple (see [Zha14b], [Xue19], [BP21a]and [BP21c]) or only proved the direction 2. ⇒ 1. of the theorem ([GJR09], [IY19], [JZ20]). As observed in [Zha14b, Theorem 1.2] and [BPLZZ21, Theorem 1.8] we can deduce from Theorem 1.1.5.1 the following statement (whose proof is word for word that of [Zha14b]): Theorem 1.1.5.3. — Let be a Hermitian Arthur parameter of G . Then there exists n+1 n+1 a simple Hermitian Arthur parameter of G such that the Rankin-Selberg L-function satisfies: n n L( , × ) = 0. n n+1 1.1.6. Ichino-Ikeda conjecture. — Let σ = σ be an irreducible cuspidal auto- morphic representation of U that is tempered everywhere in the following sense: for every place v, the local representation σ is tempered. By [Mok15]and [KMSW], σ admits a weak (hence a strong) base-change to G. Set n+1 L(s,) L(s,σ) = L(s + i − 1/2,η ) L(s + 1/2,σ, Ad) i=1 i i where L(s,η ) is the completed Hecke L-function associated to η and L(s,σ, Ad) is the completed adjoint L-function of σ (defined using the local Langlands correspondence for Currently known bounds towards the Ramanujan conjecture do not exclude the possibility of certain local Rankin-Selberg L-factors of to have a pole at s = 1/2. Therefore, the non-vanishing of the central value could a priori be affected if we replace L(s,) by a partial L-function and here we have to include all places (including Archimedean ones). THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 187 Gfrom [Mok15], [KMSW]).Wedenoteby L(s,σ ) the corresponding quotient of local L-factors. For each place v of F, we define a local normalized period P : σ ×σ → C as v v h,σ follows. It depends on the choice of a Haar measure on U (F ) as well as an invariant inner product (.,.) on σ and is given by v v −1 P (ϕ ,ϕ ) = L( ,σ ) (σ (h )ϕ ,ϕ ) dh ,ϕ ,ϕ ∈σ , v v v v v v v v v h,σ v v v U (F ) where, thanks to the temperedness assumption, the integral is absolutely convergent [Har14, Proposition 2.1] and the local factor L(s,σ ) has no zero (nor pole) at s = . Moreover, by [Har14, Theorem 2.12], if ϕ =⊗ ϕ ∈ σ , then for almost all places v we have (1.1.6.1) P (ϕ ,ϕ ) = vol(U (O ))(ϕ ,ϕ ) . v v v v v v h,σ h We also recall that the global representation σ has a natural invariant inner product given by (ϕ,ϕ) = |ϕ(g)| dg,ϕ ∈σ. Pet [U ] Our second main result is the global Ichino-Ikeda conjecture for unitary groups formulated in [Har14, Conjecture 1.3] and can be stated as follows (this result can be seen as a refinement of Theorem 1.1.5.1, the precise relation requiring the local Gan- Gross-Prasad conjecture and Arthur’s multiplicity formula for unitary groups will not be discussed here). Theorem 1.1.6.1. — Assume that σ is a cuspidal automorphic representation of U that is tempered everywhere and let = be the weak (hence the strong) base-change of σ to G. n n+1 Suppose that we normalize the period integral P and the Petersson inner product (.,.) by choosing the h Pet invariant Tamagawa measures d hand d gon U (A) and U (A) respectively. Assume also that Tam Tam h the local Haar measures dh on U (F ) factorize the Tamagawa measure: d h = dh . Then, for v v Tam v every nonzero factorizable vector ϕ =⊗ ϕ ∈σ , we have P (ϕ ,ϕ ) |P (ϕ)| 1 v v h h,σ −1 v =|S | L( ,σ) (ϕ,ϕ) 2 (ϕ ,ϕ ) Pet v v v wherewerecallthat S denotes the finite group S × S . n n+1 Note that the product over all places in the theorem is well-defined by (1.1.6.1). Moreover, once again, this theorem is proved in [BPLZZ21] under the extra assumption We warn the reader that our convention is to include the global normalizing L-values in the definition of Tama- gawa measures, cf. Section 2.3 for precise definitions. 188 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR that is cuspidal (in which case |S |= 4). Previous results in that direction includes [Zha14a], [BP21a], [BP21c] where some varying local assumptions on σ entailing the cuspidality of were imposed. In a slightly different direction, the paper [GL] estab- lishes the above identity up to an unspecified algebraic number under some arithmetic assumptions on σ . 1.2. The spectral expansion of the Jacquet-Rallis trace formula for the linear groups 1.2.1. Motivations. — As in [Zha14b], [Zha14a], [Xue19], [BP21a], [BP21c]and [BPLZZ21], our proofs of Theorems 1.1.5.1 and 1.1.6.1 follow the strategy of Jacquet and Rallis [JR11] and are thus based on a comparison of relative trace formulas on unitary groups U for h ∈ H and the group G. Let’s recall that these trace formulas have two h n different expansions: one, called the geometric side, in terms of distributions indexed by geometric classes and the other, called the spectral side, in terms of distributions indexed by cuspidal data. As usual, the point is to get enough test functions to first compare the geometric sides which gives a comparison of spectral sides. For specific test functions, the trace formula boils down to a simple and quite easy equality between a sum of relative regular orbital integrals and a sum of relative charac- ters attached to cuspidal representations. This is the simple trace formula used by Zhang in [Zha14b]and [Zha14a] to prove special cases of Theorems 1.1.5.1 and 1.1.6.1.Inre- turn one has to impose restrictive local conditions on the representations one considers. In [Zyd16], [Zyd18], [Zyd20], Zydor established general Jacquet-Rallis trace for- mulas. Besides, in [CZ21], Chaudouard-Zydor proved the comparison of all the geomet- ric terms for matching test functions, that is functions with matching local orbital inte- grals. Using these results, Beuzart-Plessis-Liu-Zhang-Zhu in [BPLZZ21]proved 1.1.5.1 and 1.1.6.1 when is cuspidal. Their main innovation is a construction of Schwartz test functions only detecting certain cuspidal data. In this way, they were able to con- struct matching test functions for which the spectral expansions reduce to some relative characters attached to cuspidal representations. 1.2.2. In this paper, we also want to use the construction of Beuzart-Plessis-Liu- Zhang-Zhu. But for this, we need two extra ingredients. First we need the slight extension of Zydor’s work to the space of Schwartz test functions. For the geometric sides, this was done in [CZ21]. For the test functions we need, the spectral side of the trace formulas for unitary groups still reduces to relative characters attached to cuspidal representations and we need nothing more. But, for the group G, we shall extend the spectral side of the trace formula to the space of Schwartz functions. Second there is an even more serious question: since the representation is no longer assumed to be cuspidal, the spectral contribution associated to is much more involved. In this section, we shall explain alternative and somewhat more tractable expressions for the spectral contributions in the trace formula for G. For the specific cuspidal datum attached to , we get a precise result as we shall see in Section 1.3 below. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 189 1.2.3. The spectral expansion for the Schwartz space. — Let X(G) be the set of cuspidal data of G (see Section 2.9.1). To χ is associated a direct invariant factor L ([G]) of L ([G]) (see [MW89, Chap. II] or Section 2.9 for a review). Let f be a function in the Schwartz space S(G(A)) (cf. Section 2.5.2 for a definition). Let K , resp. K ,bethe f f ,χ 2 2 kernel associated to the action by right convolution of f on L ([G]), resp. L ([G]). Following [Zyd20] (see Section 3.2.3), we introduce the modified kernel K f ,χ depending on a parameter T in a certain real vector space. Set H = G and G = GL × GL both seen as subgroups of G (the embedding H → G being the “di- n,F n+1,F agonal” one where the inclusion G → G is induced by the identification of E with n n+1 n+1 the hyperplane of E of vanishing last coordinate). The following theorem is an exten- sion to Schwartz functions of [Zyd20, théorème 0.1]. Theorem 1.2.3.1. — (see Theorem 3.2.4.1) 1. For any T in a certain positive Weyl chamber, we have |K (h, g )| dg dh < ∞ f ,χ [H] [G ] χ∈X(G) 2. Let η be the quadratic character of G (A) defined in Section 3.1.6.For each χ ∈ X(G), the integral (1.2.3.1) K (h, g )η (g ) dg dh f ,χ [H] [G ] coincides with a polynomial-exponential function in T whose purely polynomial part is con- stant and denoted by I (f ). 3. The distributions I are continuous, left H(A)-equivariant and right (G (A),η )- χ G equivariant. Moreover the sum (1.2.3.2) I(f ) = I (f ) is absolutely convergent and defines a continuous distribution. The (coarse) spectral expansion of the trace formula for G is precisely the expres- sion (1.2.3.2). 1.2.4. The definition of I given in Theorem 1.2.3.1 is convenient to relate the spectral expansion to the geometric expansion. However, to get more explicit forms of the distributions I , we shall use the following three expressions: (1.2.4.3) ( K )(h, g )η (g ) dg dh f ,χ G [H] [G ] 190 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR n+1 (1.2.4.4) F (h, T) K (h, g )η (g ) dg dh f ,χ G [H] [G ] n+1 (1.2.4.5) F (g , T) K (h, g ) dhη (g )dg f ,χ G [G ] [H] Essentially they are given by integration of the kernel K along [H]×[G ].However, f ,χ to have a convergent expression for a general χ , one needs to use some truncation de- pending on the same parameter T as above. We introduce the Ichino-Yamana truncation operator, denoted by , whose definition is recalled in Section 3.3.2.In(1.2.4.3), we ap- ply it to the left-variable of K . But one can also use the Arthur characteristic function f ,χ n+1 F (·, T) whose definition is recalled in 3.3.4.In(1.2.4.4), this function is evaluated at h ∈ H(A) through the embedding H = G → G .In(1.2.4.5), it is evaluated at the n n+1 component g of the variable g =(g , g ) ∈ G (A) = GL (A) × GL (A). n n+1 n n n+1 The link with the distribution I is provided by the following theorem (which is a combination of Propositions 3.3.3.1 and 3.3.5.1 and Theorem 3.3.9.1). Note that we shall not need the full strength of the theorem in this paper. However it will be used in a greater generality in a subsequent paper. Theorem 1.2.4.1. — Let f ∈ S(G(A)) and χ ∈ X(G). 1. For any T in some positive Weyl chamber, the expressions (1.2.4.3), (1.2.4.4) and (1.2.4.5) are absolutely convergent. 2. Each of the three expressions is asymptotically equal (in the technical sense of assertion 2 of Theorem 3.3.9.1)) to a polynomial-exponential function of T whose purely polynomial term is constant and equal to I (f ). 1.2.5. Note that powerful estimates for modified kernels are introduced and used in the proofs of Theorems 1.2.3.1 and 1.2.4.1. We refer the reader to Theorem 3.3.7.1 for a precise statement. 1.3. On the ∗-regular contribution for the Jacquet-Rallis trace formula for the linear groups 1.3.1. From now on we assume that the cuspidal datum χ is relevant ∗-regular that is χ is the class of a pair (M,π) with the property that the normalized induction G(A) := Ind (π), where we have fixed a parabolic subgroup P with Levi component M, P(A) is a Hermitian Arthur parameter of G. To we associate, following [Zha14a, §3.4], a relative character I . The precise definition of this object is recalled in Section 8.1.3. Let us just say here that it is associated to two functionals λ and β on the Whittaker model W(,ψ ) of ,where ψ is a certain generic automorphic character of the N N standard maximal unipotent subgroup N of G, that naturally show up in integrals of Rankin-Selberg type. More precisely, λ is the value at s = of a family of Zeta integrals, studied by Jacquet-Piatetski-Shapiro-Shalika [JPSS83], representing the Rankin-Selberg THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 191 L-function L(s,) whereas β is essentially the pole at s = 1 of another family of Zeta in- tegrals, first introduced by Flicker [Fli88], representing the (product of) Asai L-functions n+1 n (−1) (−1) L(s,, As ) := L(s, , As )L(s, , As ). The relative character I is then G n n+1 given in terms of these functionals by (1.3.1.1) I (f ) = λ((f )W )β (W ), f ∈ S(G(A)), ϕ η ϕ ϕ∈ where the sum runs over an orthonormal basis of (for the Petersson inner product) and W denotes the Whittaker function associated to the Eisenstein series E(ϕ) (obtained, as −1 usual, by integrating E(ϕ) against ψ over [N]). The following is our main technical result whose proof occupies most part of the paper. Theorem 1.3.1.1. — Let χ be a cuspidal datum associated to a Hermitian Arthur parameter as above. Then, for every function f ∈ S(G(A)) we have − dim(A ) I (f ) = 2 I (f ) where A denotes the maximal central split torus of M. Remark 1.3.1.2. — It is perhaps worth emphasizing that the contribution of χ is purely discrete in the Jacquet-Rallis trace formula. Such a phenomenon happens in Jacquet relative trace formula, see [Lap06]. By contrast, the contribution of the same kind of cuspidal datum χ to the Arthur-Selberg trace formula is purely continuous (unless, of course, if is cuspidal). We shall provide two different proofs of Theorem 1.3.1.1, one based on trunca- tions, the other using integral representations of Asai and Rankin-Selberg L-functions. Let’s explain separately the main steps of each approach. 1.3.2. A journey through truncations. — We first begin with the approach based on truncations. The first step is to get a spectral decomposition of the function (1.3.2.2) K (g, g )η (g )dg , f ,χ G [G ] of the variable g ∈[G]. The kernel itself K has a well-known spectral decomposition based on the Lang- f ,χ lands decomposition. Then the problem is basically to invert an adelic integral and a complex integral. It is solved by Lapid in [Lap06] (up to some non-explicit constants) but we will use a slightly different method avoiding delicate Lapid’s contour moving. Instead 192 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR we replace the integral (1.3.2.2) by its truncated version (1.3.2.3) (K )(g, g )η (g )dg , f ,χ G [G ] where the mixed truncation operator defined by Jacquet-Lapid-Rogawski [JLR99]isap- plied to the right variable of the kernel. We can recover (1.3.2.2) by taking the limit when T→+∞. It is easy to get the spectral decomposition of (1.3.2.3) (see Proposition 4.2.3.3). Using an analog of the famous Maaß-Selberg relations due to Jacquet-Lapid-Rogawski (see [JLR99] and Lemma 4.3.6.2 below), we get in Proposition 4.3.6.1 that (1.3.2.3)is equal to a finite sum of contributions (up to an explicit constant) of the following type exp(−λ, T ) (1.3.2.4) J (g,λ, f ) dλ. Q,χ G,∗ θ (−λ) ia Q Q∈P(M) G,∗ Here it suffices to say that ia is some space of unramified unitary characters and that J (g,λ, f ) is a certain relative character built upon Flicker-Rallis intertwining peri- Q,χ ods (introduced by Jacquet-Lapid-Rogawski). The integrand is a familiar expression of Arthur’s theory of (G, M)-families with quite standard notations. It turns out that the family (J (g,λ, f )) is indeed an Arthur (G, M)-family of Schwartz functions in Q,χ Q∈P(M) the parameter λ. Let’s emphasize that this Schwartz property relies in fact on deep esti- mates introduced by Lapid in [Lap06]and [Lap13]. By a standard argument, it is then easy to get the limit of (1.3.2.4)when T→+∞ which gives the spectral decomposition of (1.3.2.2) (see Theorem 4.3.3.1). Note that the spectral decomposition we get is already discrete at this stage. From this result, one gets the equality T − dim(A ) (1.3.2.5) ( K )(h, g )η (g ) dhdg = 2 I (f ). f ,χ G P,π [H] [G ] The left-hand side has been defined in Section 1.2.4 and the relative character I is P,π defined as follows: I ((f )ϕ) · J (ϕ) RS η ϕ∈ where the sum is over an orthonormal basis, I (ϕ) is the regularized Rankin-Selberg RS period of the Eisenstein series E(ϕ) defined by Ichino-Yamana and J (ϕ) is a Flicker- Rallis intertwining period (for more detail we refer to Section 5.1.5). In particular, the left-hand side of (1.3.2.5) does not depend on T. So Theorem 1.2.4.1 implies Theorem 1.3.2.1. — (see Theorem 5.2.1.1 for a slightly more precise statement) − dim(A ) I (f ) = 2 I (f ). χ P,π THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 193 Remark 1.3.2.2. — As the reading of Section 10.2 should make it clear, this state- ment suffices to prove the Gan-Gross-Prasad conjecture namely Theorem 1.1.5.1.How- ever, to get the Ichino-Ikeda conjecture, namely Theorem 1.1.6.1,wewillwanttouse statements about comparison of local relative characters written in terms of Whittaker functions. For this purpose, Theorem 1.3.1.1 will be more convenient. The link between regularized Rankin-Selberg period of Eisenstein series and Whit- taker functionals has been investigated by Ichino-Yamana (see [IY15]). The following the- orem relates the Flicker-Rallis intertwining periods to the functional β (W ) in (1.3.1.1). η ϕ It uses a local unfolding method inspired from [FLO12, Appendix A] (see Section 9). Theorem 1.3.2.3. — For all ϕ ∈, we have J (ϕ) =β (W ) η η ϕ In this way, one proves the following theorem which implies Theorem 1.3.1.1. Theorem 1.3.2.4. I = I . P,π 1.3.3. Second proof: the use of Zeta integrals. — The spectral decomposition of (1.3.2.2) essentially boils down to a spectral expansion of the period integral P (ϕ) := ϕ(g )η (g )dg G ,η G [G ] for test functions ϕ ∈ S ([G]),where S ([G]) denotes the Schwartz space of [G] consisting χ χ of smooth functions rapidly decaying with all their derivatives that are “supported on χ ” (see Section 2.5 for a precise definition). Choose a parabolic subgroup P = MN with Levi component M. By Langlands L spectral decomposition and of the special form of χ,any ϕ ∈ S ([G]) admits a spectral decomposition (1.3.3.6) ϕ = E(ϕ )dλ ia where ia denotes the real vector space of unramified unitary characters of M(A) and G(A) ϕ belongs to the normalized induction space Ind (π ⊗λ) and E(ϕ ) is the associated λ λ P(A) Eisenstein series. Theorem 1.3.3.1. — For every ϕ ∈ S ([G]), we have − dim(A ) P (ϕ) = 2 β (W ), G ,η η ϕ where W stands for the Whittaker function of the Eisenstein series E(ϕ ). ϕ 0 0 194 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR The proof of Theorem 1.3.3.1 is close to the computation by Flicker [Fli88]ofthe Flicker-Rallis period of cusp forms in terms of an Asai L-function and local Zeta integrals. More precisely, we first realize P (ϕ) as the residue at s = 1 of the inner product of the G ,η restriction ϕ with some Eisenstein series E(s,φ) (where φ is an auxiliary Schwartz |[G ] n n+1 function on A ⊕ A ). Mimicking the unfolding of loc. cit. we connect this inner product FR with an Eulerian Zeta integral Z (s, W ,φ) involving the Whittaker function W of ϕ ϕ ϕ −1 (obtained as before by integration against ψ ). We should emphasize here that, since ϕ is not a cusp form, the unfolding gives us more terms but using the special nature of the cuspidal datum χ we are able to show that these extra terms do not contribute to FR the residue at s = 1. The formation of Z (s, W ,φ) commutes with the spectral expan- sion (1.3.3.6)when (s) 1 and, as follows from the local theory, the Zeta integrals FR ∗ Z (s, W ,φ) for λ ∈ ia are essentially Asai L-functions whose meromorphic continu- λ M ations, poles and growths in vertical strips are known. Combining this with an application of the Phragmen-Lindelöf principle, we are then able to deduce Theorem 1.3.3.1. Let us mention here that, as in the proof of (1.3.2.5), a key point is the fact (due to Lapid ([Lap13]or[Lap06])) that the spectral transform λ →ϕ is, in a suitable technical sense, “Schwartz” that is rapidly decreasing together with all its derivatives. The second step is to integrate (1.3.2.2)over g ∈[H]. To do so, we define a regular- ization of the integral over [H] that doesn’t require truncation. More precisely, denoting by T ([G]) the space of functions of uniform moderate growth on [G],wecan definethe “χ - part” T ([G]) of T ([G]) (see Section 2.5) of which S ([G]) is a dense subspace. More- χ χ over, starting with ϕ ∈ T ([G]) we can also form its Whittaker function W and consider RS the usual Rankin-Selberg integral Z (s, W ) that converges for (s) 1 and represents, when ϕ is an automorphic form, the Rankin-Selberg L-function for G × G . n n+1 Theorem 1.3.3.2. — (see Theorem 7.1.3.1) The functional ϕ ∈ S ([G]) → ϕ(h)dh [H] extends by continuity to a functional on T ([G]) denoted by ϕ ∈ T ([G]) → ϕ(h)dh. Moreover, χ χ [H] RS for every ϕ ∈ T ([G]), the Zeta function s → Z (s, W ) extends to an entire function on C and we χ ϕ have RS ϕ(h)dh = Z ( , W ). [H] The proof of this theorem is similar to that of Theorem 1.3.3.1: we first show that, for ϕ ∈ S ([G]) and (s) 1, we have s RS ϕ(h)|det h| dh = Z (s + , W ) [H] THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 195 by mimicking the usual unfolding for the Rankin-Selberg integral. Once again, as ϕ is not necessarily a cusp form, we get extra terms in the course of the unfolding but, thanks to the special nature of the cuspidal datum χ , we are able to show that they all RS vanish. At this point, we use the spectral decomposition (1.3.3.6) to express Z (s, W ) RS RS as the integral of Z (s, W ) when (s) 1. By Rankin-Selberg theory, Z (s, W ) ϕ ϕ λ λ is essentially a Rankin-Selberg L-function whose meromorphic continuation, location of the poles, control in vertical strips and functional equation are known. Combining this with another application of the Phragmen-Lindelöf principle, we are able to bound RS RS Z ( , W ) = ϕ(h)dh in terms of Z (s, W ) for (s) 1 and this readily gives the ϕ ϕ [H] theorem. One direct consequence of Theorem 1.3.3.2 is that the regularized period RS 1 E(h,(f )ϕ)dh coincides with Z ( ,(f )W ) = λ((f )W ). Thus by a combi- ϕ ϕ [H] nation of Theorems 1.3.3.1 and 1.3.3.2,weget − dim(A ) (1.3.3.7) K (h, g )η (g )dg dh = 2 I (f ). f ,χ G [H] [G ] Finally we have to show that the left-hand side is equal to I (f ).Infact, we show (see Theorem 8.1.4.1 and Section 8.2.3)thatwehave K (h, g )η (g )dg dh = K (h, g ) dhη (g )dg f ,χ G f ,χ G [H] [G ] [G ] [H] where the right-hand side is (conditionally) convergent. We can conclude that it is equal to I (f ) by applying Theorem 1.2.4.1 to the expression (1.2.4.5). 1.4. Outline of the paper We now give a quick outline of the content of the paper. Section 2 contains pre- liminary material. Notably, we fix most notation to be used in the paper, we explain our convention on normalization of measures, we introduce the various spaces of functions we need and we discuss several properties of Langlands decomposition along cuspidal data as well as kernel functions that are important for us. Section 3 contains the state- ments and proofs concerning the spectral expansion of the Jacquet-Rallis trace formula for G that were discussed in Section 1.2 above. In Section 4, we introduce the Flicker-Rallis intertwining periods and prove the spectral expansion of the Flicker-Rallis period of the kernel associated to ∗-regular cus- pidal datum. In Section 5 we deduce from it Theorem 1.3.2.1 namely the spectral ex- pansion for I (f ).Sections 6 and 7 are devoted to the proofs of Theorems 1.3.3.1 and 1.3.3.2 respectively. These two theorems are combined in Section 8 to give another proof of the spectral expansion of I (f ) (Theorem 1.3.1.1). In Section 9, we relate the Flicker- Rallis intertwining periods to the functional β . From this, we deduce Theorem 1.3.2.3 η 196 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR and Theorem 1.3.2.4. The final Section 10 explain the deduction of Theorems 1.1.5.1 and 1.1.6.1 from Theorem 1.3.1.1. Finally, we have gathered in Appendix A some useful facts on topological vector spaces and holomorphic functions valued in them. It contains in particular some variations on the theme of the Phragmen-Lindelöf principle for such functions that will be crucial for the proofs of Theorems 1.3.3.1 and 1.3.3.2. 2. Preliminaries 2.1. General notation 2.1.1. For f and g two positive functions on a set X, we way that f is essentially bounded by g and we write f (x) g(x), x ∈ X, if there exists a constant C > 0such that f (x) Cg(x) for every x ∈ X. If we want to emphasize that the constant C depends on auxiliary parameters y ,..., y , we will write 1 k f (x) g(x). We say that the functions f and g are equivalent and we write y ,...,y 1 k f (x) ∼ g(x), x ∈ X, if f (x) g(x) and g(x) f (x). 2.1.2. For every C, D ∈ R∪{−∞} with D > C, we set H ={z ∈ C| (z)> >C C} and H ={z ∈ C | C < (z)< D}.A vertical strip is a subset of C which is the ]C,D[ closure of H for some C, D ∈ R with D> C. ]C,D[ When f is a meromorphic function on some open subset U of C and s ∈ U, we denote by f (s ) the leading term in the Laurent expansion of f at s . 0 0 2.1.3. When G is a group and we have a space of functions on it invariant by right (resp. left) translation, we denote by R (resp. L) the corresponding representation of G. If G is a Lie group and the representation is differentiable, we will also denote by the same letter the induced action of the Lie algebra or of its associated enveloping algebra. If G is a topological group equipped with a bi-invariant Haar measure, we denote by ∗ the convolution product (whenever it is well-defined). 2.1.4. We refer the reader to Appendix A for reminders on relevant notions of functional analysis that will be used without further comments in the core of the paper. In particular, we will use the notation ⊗ to denote the projective completed tensor product between two locally convex topological vector spaces (over C). Moreover, most of the topological vector spaces we will consider are Banach, Hilbert, Fréchet or LF. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 197 2.2. Algebraic groups and adelic points 2.2.1. Let F be a number field and A its adele ring. We write A for the ring of finite adeles and F = F ⊗ R for the product of Archimedean completions of F ∞ Q so that A = F × A .Let V be the set of places of F and V ⊂ V be the subset of ∞ f F F,∞ F Archimedean places. For every v ∈ V ,welet F be the local field obtained by completion F v × × of F at v.Wedenoteby |·| the morphism A → R given by the product of normalized absolute values |·| on each F . For any finite subset S ⊂ V \ V ,wedenoteby O v v F F,∞ the ring of S-integers in F. 2.2.2. Let G be an algebraic group defined over F. We denote by N the unipo- tent radical of G. Let X (G) be the group of characters of G defined over F. Let ∗ ∗ ∗ a = X (G) ⊗ R and a = Hom (X (G), R). We have a canonical pairing Z G Z (2.2.2.1) ·,· : a × a → R. We have also a canonical homomorphism (2.2.2.2) H : G(A) → a G G such that χ, H (g) = log|χ(g)| for any g ∈ G(A). The kernel of H is denoted by G G 1 1 1 G(A) .Wedefine [G]= G(F)\G(A) and [G] = G(F)\G(A) . We let g be the Lie algebra of G(F ) and U(g ) be the enveloping algebra of its ∞ ∞ ∞ complexification and Z(g ) ⊂ U(g ) be its center. ∞ ∞ 2.2.3. From nowonweassume that G isalsoreductive.Wewillmainlyuse the notations of Arthur’s works. For the convenience of the reader, we recall some of them. Let P be a parabolic subgroup of G defined over F and minimal for these properties. Let M be a Levi factor of P defined over F. 0 0 We call a parabolic (resp. and semi-standard, resp. and standard) subgroup of G a parabolic subgroup of G defined over F (resp. which contains M , resp. which contains P ). For any semi-standard parabolic subgroup P, we have a Levi decomposition P = M N where M contains M and we define [G] = M (F)N (A)\G(A).WecallaLevi P P P 0 P P P subgroup of G (resp. semi-standard, resp. standard) a Levi factor defined over F of a parabolic subgroup of G defined over F (resp. semi-standard, resp. standard). 2.2.4. Let K = K ⊂ G(A) be a “good” maximal compact subgroup in v∈V good position relative to M . We write K = K K where K = K and K = K .Welet k be the Lie algebra of K ∞ v v ∞ ∞ v∈V v∈V \V F,∞ F F,∞ and U(k ) be the enveloping algebra of its complexification and Z(k ) ⊂ U(k ) be its ∞ ∞ ∞ center. 198 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 2.2.5. Let P be a semi-standard parabolic subgroup. We extend the homomor- phism H : P(A) → a (see (2.2.2.2)) into the Harish-Chandra map P P H : G(A) → a P P in such a way that for every g ∈ G(A) we have H (g) = H (p) where p ∈ P(A) is given by P P the Iwasawa decomposition namely g ∈ pK. Let H = H . 0 P 2.2.6. Let A be a split torus over F. Then, A admits an unique split model over Q (which is also the maximal Q-split subtorus of Res (A)) and by abuse of notation F/Q we denote by A(R) the group of R-points of this model. In the particular case of the × × × ∞ multiplicative group G , we get an embedding R ⊂ F ⊂ A . We also write A for m,F the neutral component of A(R).Let A be the maximal central F-split torus of G. We define [G] = A G(F)\G(A). Let P be a semi-standard parabolic subgroup of G. We define A = A ,A = P M P P ∞ ∞ ∗ ∗ A and [G] = A M (F)N (A)\G(A). The restrictions maps X (P) → X (M ) → P,0 P P P M P ∗ ∗ ∗ ∗ ∗ ∗ X (A ) induce isomorphisms a a a .Let a = a , a = a ,A = A and P 0 P 0 P 0 0 P M A 0 P P P 0 ∞ ∞ A = A . 0 P 2.2.7. For any semi-standard parabolic subgroups P ⊂ Q of G, the restriction ∗ ∗ ∗ ∗ map X (Q) → X (P) induces maps a → a and a → a . The first one is injective P Q Q P whereas the kernel of the second one is denoted by a . The restriction map X (A ) → Q,∗ ∗ ∗ ∗ X (A ) gives a surjective map a → a whose kernel is denoted by a . We get also P Q P an injective map a → a . In this way, we get dual decompositions a = a ⊕ a and Q P P Q Q,∗ Q Q,∗ ∗ ∗ ∗ a = a ⊕ a . Thus we have projections a → a and a → a which we will denote P P Q P 0 P by X → X . Q,∗ Q We denote by a and a the C-vector spaces obtained by extension of scalars P,C P,C Q,∗ Q from a and a . We still denote by ·,· the pairing (2.2.2.1) we get by extension of the P P scalars to C.Wehaveadecomposition Q,∗ Q,∗ Q,∗ a = a ⊕ ia P,C P P where i =−1. We shall denote by and the associated projections and call them real and imaginary parts. The same holds for the dual spaces a . In the obvious way, we P,C Q,∗ define the complex conjugate denoted by λ of λ ∈ a . P,C 2.2.8. Let Ad be the adjoint action of M on the Lie algebra of M ∩ N .Let P Q P Q Q,∗ ρ be the unique element of a such that for every m ∈ M (A) P P Q Q | det(Ad (m))|= exp(2ρ , H (m) ). P P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 199 For every g ∈ G(A), we set Q Q δ (g) = exp(2ρ , H (g) ) P P so that, in particular, the restriction of δ to P(A) ∩ M (A) coincides with the modular character of the latter. For Q = G, the exponent G is omitted. Finally, we set ρ =ρ . 0 P 2.2.9. Let P = M N be a minimal semi-standard parabolic subgroup such 0 P P P that P ⊂ P. Let be the set of simple roots of A in M ∩ P . We denote this set by 0 P 0 P 0 0 P ∗ if P = P .Let be the image of \ (viewed as a subset of a ) by the projection 0 P P 0 0 ∗ ∗ a → a . It does not depend on the choice of P . More generally one defines .We 0 P 0 P Q,∨ Q Q have also the set of coroots ⊂ a . By duality, we get a set of simple weights .The P P P Q Q sets and determine open cones in a whose characteristic functions are denoted P P Q Q respectively by τ and τˆ . We set P P ∞,Q+ Q A = a ∈ A |α, H (a) 0, ∀α ∈ , P P ∗,Q+ Q,∨ ∗ ∨ ∨ a = λ ∈ a |λ,α 0, ∀α ∈ . P P Q+ We define similarly a using roots instead of coroots. If Q = G, the exponent G is omitted and if P = P , we replace the subscript P by 0. 0 0 For λ,μ ∈ a , we will write λ ≺ μ to indicate that μ − λ is a nonnegative linear combination of the simple roots . 2.2.10. Weyl group. — Let W be the Weyl group of (G, A ) that is the quotient by M of the normalizer of A in G(F).For P = M N and Q = M N two standard 0 0 P P Q Q parabolic subgroups of G, we denote by W(P, Q) the set of w ∈ Wsuch that w = . 0 0 −1 For w ∈ W(P, Q),wehave wM w = M . When P = Q, the group W(P, P) is simply P Q denoted by W(P). Sometimes, we shall also denote W(P, Q) by W(M , M ) if we want P Q to emphasize the Levi components (and W(M ) = W(P)). We will also write W for the Weyl group of (M , A ). P 0 2.2.11. Let M be a standard Levi subgroup of G. We denote by P(M) the set of semi-standard parabolic subgroups P of G such that M = M. There is an unique element P ∈ P(M) which is standard and the map −1 (2.2.11.3) (Q,w) → w Qw induces a bijection from the disjoint union W(P, Q) where Q runs over the set of standard parabolic subgroups of G onto P(M). 200 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 2.2.12. Truncation parameter. — We shall denote by T a point of a such that α, T is large enough for every α ∈ . We do not want to be precise here. We just need that Arthur’s formulas about truncation functions hold for the T s we consider (see [Art78] §§5, 6). The point T plays the role of a truncation parameter. For any semi-standard parabolic subgroup P, we define a point T ∈ a such that P P −1 for any w ∈ Wsuch that wP w ⊂ P, the point T is the projection of w · Ton a (this 0 P P does not depend on the choice ofw). The reader should be warned that it is not consistent with the notation of Section 2.2.7 since there T denotes instead the projection of T onto a (of course, the two conventions coincide when P is standard). 1 1 2.2.13. Let ω ⊂ P (A) be a compact subset such that P (A) = P (F)ω .Let P 0 0 0 0 0 be a standard parabolic subgroup. By a Siegel domain of [G] we mean a subset of G(A) of the form ∞ P s =ω a ∈ A |α, H (a) α, T , ∀α ∈ K P 0 0 − 0 0 where T ∈ a and such that G(A) = M (F)N (A)s . We henceforth fix a Siegel domain − 0 P P P s of [G] for every standard parabolic subgroup P and we assume that these Siegel P P domains are all associated to the same T ∈ a . In particular, for P ⊂ Qwe have s ⊂ s . − 0 Q P Moreover, there exists a compact subset K ⊂ G(A) such that ∞,P+ s ⊂ N (A)A K, for every P ⊂ G. P P 2.3. Haar measures 2.3.1. We equip a with the Haar measure that gives a covolume 1 to the lattice ∗ ∗ Hom(X (P), Z).The space ia is then equipped with the dual Haar measure so that we have φ(H) exp(−λ, H ) d Hdλ =φ(0) ia a ∞ ∗ ∗ for all φ ∈ C (a ). Note that this implies that the covolume of iX (P) in ia is given by c P ∗ ∗ − dim(a ) (2.3.1.1) vol(ia /iX (P)) =(2π) . The group A is equipped with the Haar measure compatible with the isomor- G,∗ ∞ G ∗ ∗ phism A a induced by the map H . The groups a a /a and ia ia /ia are P P P G P P P P G provided with the quotient Haar measures. For any basis B of a we denote by Z(B) the G ∗ lattice generated by B and by vol(a /Z(B)) thecovolumeofthis lattice.Wehaveon a P 0 the polynomial function: G ∨ −1 ∨ θ (λ) = vol(a /Z( )) λ,α . P P α∈ P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 201 2.3.2. Let H be a connected linear algebraic group over F (not necessarily reduc- tive). In this paper, we will always equip H(A) with its right-invariant Tamagawa measure simply denoted dh. Let us recall how it is defined in order to fix some notation. We choose a right-invariant rational volume form ω on H as well as a non-trivial continuous ad- ditive character ψ : A/F → C . For each place v ∈ V , the local component ψ of ψ induces an additive measure on F which is the unique Haar measure autodual with respect to ψ . Then using local F-analytic charts, we associate to ω a right Haar mea- sure dh =|ω | on H(F ) as in [Wei82, §2.2]. By [Gro97], there exists an Artin-Tate v H ψ v L-function L (s) such that, denoting by L (s) the corresponding local L-factor and set- H H,v ting = L (0), for any model of H over O for some finite set S ⊆ V \ V ,we H,v H,v F F,∞ have −1 (2.3.2.2)vol(H(O )) = H,v ∗ ∗ ∗ for almost all v ∈ V . Setting = L (0), where we recall that L (0) stands for the H H H leading coefficient in the Laurent expansion of L (s) at 0, the Tamagawa measure on H(A) is defined as the product ∗ −1 (2.3.2.3) dh =( ) dh . H,v v Although the local measures dh depend on choices, the global measure dh doesn’t (by the product formula). S,∗ S,∗ 2.3.3. For S ⊆ V a finite subset, we put = L (0) where L (s) stands for H H H the corresponding partial L-function and we equip H(F ),H(A ) with the right Haar S,∗ S −1 measures dh = dh and dh =( ) dh respectively. Note that we have S v H,v v v∈S H v/∈S the decomposition (2.3.3.4) dh = dh × dh . In particular, setting S ={v}, this means that H(F ) is equipped with the right Haar measure dh for every v ∈ V . v F 2.3.4. We have L (s) = L (s) where H = H/N denotes the quotient of H H H red H red by its unipotent radical. When H = N isunipotentwehave vol([N]) = 1. For H = GL , the L-function L (s) is given by L (s) =ζ (s + 1)...ζ (s + n) H F F where ζ stands for the (completed) zeta function of the number field F. In this case, we will take −1 ω =(det h) dh H i,j 1i,jn 202 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR so that (2.3.2.2) is satisfied for every non-Archimedean place v where ψ is unrami- fied. 2.3.5. The homogeneous space [G] (resp. [G] [G] ) is equipped with the quo- tient of the Tamagawa measure on G(A) by the counting measure on G(F) (resp. by the product of the counting measure on G(F) with theHaar measurewefixedon A ). For P a standard parabolic subgroup, we equip similarly [G] with the quotient of the Tam- agawa measure on G(A) by the product of the counting measure on M (F) with the Tamagawa measure on N (A). Since the action by left translation of a ∈ A on [G] P P −1 ∞ multiplies the measure by δ (a) , taking the quotient by the Haar measure on A in- duces a “semi-invariant” measure on [G] = A \[G] that is a positive linear form on P,0 P the space of continuous functions ϕ :[G] → C satisfying ϕ(ag) = δ (a)ϕ(g) for a ∈ A P P and compactly supported modulo A . 2.4. Heights, weights and Harish-Chandra function 2.4.1. Height on G.— We fix an embedding ι : G→ GL for some integer N> 0. Using ι,wedefine a height on G(A) by −1 g= max (|ι(g) | ,|ι(g ) | ). i,j v i,j v 1i,jN Note that for another choice of embedding ι yielding a height . , there exists r > 0 1/r r 0 0 such that g g g for g ∈ G(A).Wehave −1 (2.4.1.1)1g, ghgh and g=g for g, h ∈ G(A).Let P ⊂ G be a standard parabolic subgroup. We set x = inf γ x∼ inf γ x,σ (x) = 1+ logx , for x∈[G] . P P P P γ∈M (F)N (A) γ∈P(F) P P Note that, for P ⊂ Q, we have (2.4.1.2) x x , for x ∈ P(F)\G(A). Q P Letting s be a Siegel domain as in Section 2.2.13,wehave(see [MW94, Eq. I.2.2(vii)]) (2.4.1.3) g ∼g, for g ∈ s . P P If H ⊂ G is a reductive subgroup, we equip H(A) with the restriction of the height . from which we deduce as above a function . on [H] for every parabolic sub- P P H H group P ⊂ H. For P ⊂ G a standard parabolic subgroup, we have (2.4.1.4) m ∼m , for m∈[M ]. M P P P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 203 More generally, if P = P ∩ H happens to be a parabolic subgroup of H with unipotent radical N = H ∩ N (so that in particular [H] ⊂[G] ), we have P P P P H H (2.4.1.5) h ∼h , for h∈[H] . P P P H H Indeed, using a Siegel domain for [H] we are reduced to show the above equivalence for h = a ∈ A where A ⊂ H is a maximal split torus. Up to conjugation, we may 0,H 0,H assume that A ⊂ A and then it readily follows from (2.4.1.3)that a ∼a∼a 0,H 0 P P for a ∈ A . 0,H We also have (2.4.1.6) aax , for (a, x) ∈ A ×[G] such that H (x) = 0. P P P Indeed, by (2.4.1.4) we may assume that P = G. Then, up to conjugation, we may assume that there exists distinct charactersλ ,...,λ ∈ X (A ) and integers N ,..., N 1such 1 k G 1 k that ⎛ ⎞ λ (a)I 1 N ⎜ ⎟ ι(a) = . , for a ∈ A . ⎝ ⎠ G λ (a)I k N Then, there exist homomorphisms ι : G → GL ,for 1 i k,suchthat ι(g) = i N ⎛ ⎞ ι (g) ⎜ ⎟ . 1 for g ∈ G. As |det ι (g)| = 1for every g ∈ G(A) ,wecan now ⎝ ⎠ i A ι (g) deduce (2.4.1.6) from the simple inequality 1/n |det(g)| max(|g | ), for g ∈ GL (A). i,j v N A i 2.4.2. Heights on vector spaces. — Let V be a vector space over F of finite dimension and let v ,...,v be a basis of V. For v = x v +···+ x v ∈ V = V ⊗ A, we set 1 d 1 1 d d A F v = max(1,|x | ,...,|x | ). V 1 v d v Note that another choice of basis would yield equivalent functions. A height on V will be for us any positive function equivalent to . . The above construction applies in particular to V = F with its standard basis e ,..., e and we will denote by . the resulting norm. Note that we have 1 n A x = max(1,|x| ), for x ∈ A. A v v 204 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 2.4.3. Weights. — Let P ⊂ G be a standard parabolic subgroup. Following [Ber88, §3.1] and [Fra98, §2.1], by a weight on [G] we mean a positive measurable function w on [G] such that for every compact subset U ⊂ G(A) we have w(xk) ∼w(x), for x∈[G] and k ∈ U. We say that two weights w , w are equivalent, and we will write w ∼w ,if w (x) ∼w (x) 1 2 1 2 1 2 for x∈[G] .By(2.4.1.1), . is a weight on [G] . P P P If w is a weight on [G] ,wedenoteby w its restriction to A . It is a weight on the P A ∞ ∞ latter group i.e. for every compact subset U ⊂ A we have w (ak) w (a) for a ∈ A A A P P and k ∈ U . Conversely, if w is a weight on A then we can view it as a weight on [G] A P exp through composition with [G] −→ a −→ A . P P Lemma 2.4.3.1. — Let w be a weight on [G] . Then, there exists N > 0 such that P 0 (2.4.3.7) w(xg) w(x)g , for (x, g)∈[G] × G(A). Proof. — First we prove the existence of N > 0such that N ∞ (2.4.3.8) w(xa) w(x)a , for (x, a)∈[G] × A . ∞ ∞ Indeed, let K ⊂ A be a compact neighborhood of 1. Then, K generates A and we A A 0 0 have (2.4.3.9)1 + loga∼ min{n 1 | a ∈ K }. Moreover, as w is a weight, there exists a constant C> 0such that w(xk ) Cw(x), for (x, k )∈[G] × K . A A P A By induction, this gives, for any n 1, n n w(xk ) C w(x), for (x, k )∈[G] × K . A A P Using (2.4.3.9), this readily gives (2.4.3.8)for some N > 0. We now prove the lemma. By the existence of Siegel domain, it suffices to show the ∞,P+ existence of N > 0 such that the estimate (2.4.3.7) holds for(x, g) ∈ A × G(A).Bythe Iwasawa decomposition G(A) = P (A)K, any g ∈ G(A) can be written g = p (g)a (g)k(g) 0 0 1 1 ∞ where p (g) ∈ P (A) , a (g) ∈ A and k(g) ∈ K. Moreover, there exists a compact K ⊂ 0 0 0 0 G(A) such that xP (A) ⊂ P (F)N (A)xK 0 0 P ∞,P+ for every x ∈ A . Therefore, by (2.4.3.9), we have 1 1 N N 0 0 w(xg) =w(xp (g)a (g)k(g)) w(xp (g))a (g) w(x)a (g) 0 0 0 0 0 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 205 ∞,P+ for every (x, g) ∈ A × G(A). To conclude, it suffices to notice that there exists N > 0 such that N N a (g) g for every g ∈ G(A). According to Franke [Fra98, Proposition 2.1], for every λ ∈ a there exists a weight d on [G] such that P,λ P d (x) ∼ exp(λ, H (x) ), for x ∈ s . P,λ 0 P These weights have the following elementary properties: for t ∈ R and λ,μ ∈ a ,wehave (2.4.3.10) d ∼(d ) and d ∼ d d . P,tλ P,λ P,λ+μ P,λ P,μ Moreover, (2.4.3.11)If λ ≺ μ (see Section 2.2.9), then d d . P P,λ P,μ ∗ P Also, if ⊂ a is a W = Norm (A )/M (F)-invariant subset then M (F) 0 0 0 P (2.4.3.12)max d (a) ∼ max exp(λ, H (a) ), for a ∈ A . P,λ 0 λ∈ λ∈ ∗,P+ Let ⊂ a is the set of maximal A -weights of the representation g → t −1 diag(ι(g), ι(g) ) for the partial order ≺ . It follows from (2.4.1.3)that (2.4.3.13) x ∼ max d (x), for x∈[G] . P P,λ P λ∈ ∗,P+ Lemma 2.4.3.2. — Let λ ∈ a . Then, we have λ,H (γ x) (2.4.3.14) d (x) ∼ sup e , for x∈[G] . P,λ P γ∈P(F) Proof. — As d (m) ∼ d (m) for m ∈[M ], up to replacing G by M ,wemay P,λ M ,λ P P assume that P = G. Moreover, we can restrict to x ∈ s in which case the left-hand side is clearly essentially bounded by the right-hand side (by definition of the weight d ). Thus, G,λ ∗,+ the lemma boils down to the inequality, where λ ∈ a , λ,H (γ x) λ,H (x) 0 0 e e , for (γ, x) ∈ G(F) × s , which is a simple reformulation of [LW13, lemme 3.5.4]. We note the immediate corollary. 206 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR ∗,P+ Corollary 2.4.3.3. — Let Q ⊂ P be standard parabolic subgroups of G and λ ∈ a . Then, we have (2.4.3.15) d (x) d (x), for x ∈ Q(F)\G(A). Q,λ P,λ 2.4.4. Neighborhoods of infinity. — Let P ⊂ Q be standard parabolic subgroups of G and be the set of roots of A in n /n . We define the following weights: 0 P Q (2.4.4.16) d (x) = min d (x), x∈[G] , P,α P α∈ and (2.4.4.17) d (x) = min d (x), x∈[G] . Q,α Q α∈ Since for every β ∈ , there exist a family of nonnegative integers (n ) Q ,not all α P α∈ \ 0 0 zero, such that n α ≺ β , we see that there exists n > 0such that P α P α∈ \ 0 0 Q Q Q 1/n (2.4.4.18) d (x) min d (x) max(d (x), d (x) ), for x∈[G] . P,α P P P P α∈ \ 0 0 For every C > 0, we set Q Q ω [> C]:={x ∈ P(F)N (A)\G(A) | d (x)> C}. P P Let π and π be the two natural projections P Q P(F)N (A)\G(A) [G] [G] P Q The next lemma summarizes some classical results from reduction theory. Lemma 2.4.4.1. 1. There exists> 0 such that π sends ω [>] onto [G] . Q P 2. Let > 0.Then, forevery λ ∈ a we have d (x) ∼ d (x) for x ∈ ω [>].In P,λ Q,λ 0 P particular, Q Q d (x) ∼ d (x) and x ∼x , for x ∈ω [>]. P Q P Q P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 207 3. For every > 0, the restriction of π to ω [>] has uniformly bounded fibers. More Q P precisely, there exist c > 0 and r > 0 such that for every x∈[G] and 1 > 0, we have P −1 −r #((π ) (x) ∩ω [>]) c Q P Q Q 4. For every > 0, we can find C > 0 such that for all (x, y) ∈ ω [>]× ω [> C], P P P P π (x) =π (y) implies x = y. Q Q 5. The map π is a local homeomorphism which locally preserves measures. The map π is proper and the pushforward of the invariant measure on P(F)N (A)\G(A) by it is the invariant measure on [G] . Proof. — 1. is just the existence of Siegel domains noting that if s and s are Siegel Q P domains as in Section 2.2.13 for [G] and [G] respectively, there exists> 0such that Q P the inverse image ofω [>] in s contains s . Similarly, for any> 0, this inverse image P Q is contained in another, perhaps bigger, Siegel domain s for [G] and this immediately gives2.(wherefor thelastequivalencewehaveused (2.4.3.13)). Properties 3. and 4. are Lemma 2.11 and Lemma 2.12 of [Lan76] respectively. Finally, 5. is clear. Note that, from points 1. and 2. of the above lemma, we immediately deduce that (2.4.4.19) d (x) d (x), for x ∈ P(F)N (A)\G(A). Lemma 2.4.4.2. — Let H ⊂ G be a reductive subgroup such that P = P ∩ H and Q = H H Q ∩ H are parabolic subgroups of H with unipotent radicals N = N ∩ H and N = N ∩ H P P Q Q H H respectively. Then, we have Q Q (2.4.4.20) d (h) d (h), for h∈[H] . P P H If moreover, G = Res H , P = Res (P ) and Q = Res (Q ) for some finite extension K/F K K/F H K K/F H K K/F, then Q Q (2.4.4.21) d (h) ∼ d (h), for h∈[H] . P P H Proof. — Let A ⊂ P be a maximal split torus. Up to conjugation by P(F),we 0,H H Q Q may assume that A ⊂ A . By the existence of Siegel domains, and since d , d are 0,H 0 P P Q Q weights, we just need to show (2.4.4.20)for x = a ∈ A . Since and are invariant 0,H P P P P by the Weyl groups W = Norm (A )/M (F) and W respectively, by (2.4.3.12), we M (F) 0 0 have d (a) ∼ min exp(α, H (a) ) and α∈ (2.4.4.22) H ∞ d (a) ∼ min exp(α, H (a) ), for a ∈ A . P 0,H α∈ H 208 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR By the assumption, we have n /n ⊂ n /n so that the restriction of to A con- P Q P Q 0,H H H P tains .With(2.4.4.22), this gives the inequality (2.4.4.20). We can deduce (2.4.4.21) similarly noting that, in the case where G = Res H ,P = Res (P ) and Q = K/F K K/F H K Q Q Res (Q ) , the restriction of to A is equal to . K/F H K 0,H P P 2.4.5. Xi function. — Let U ⊆ G(A) be a compact neighborhood of 1. We set P −1/2 (x) = vol (xU) , x∈[G] . [G] P Replacing U by another compact neighborhood of 1 would yield an equivalent function which is why we dropped the subset U from the notation. We have (2.4.5.23) (x) ∼ d (x), for x∈[G] . P,ρ P Indeed, there exists> 0 such that, with the notation of Section 2.4.4,P (F)N (A)s U ⊂ 0 P P ω [>]. Thus, by Lemma 2.4.4.1 3. and 5., we have vol (gU) ∼ vol (gU), for g ∈ s . [G] P (F)N (A)\G(A) P P 0 P On the other hand, there exists a compact neighborhood U of 1 in G(A) such that P (F)N (A)gU ⊆ P (F)N (A)gU for every g ∈ s . 0 0 0 P P By Lemma 2.4.4.1 5., this gives vol (gU) = vol (N (A)gU) ∼ vol (gU), for g ∈ s . [G] P (F)\G(A) 0 P (F)N (A)\G(A) P P 0 0 P Finally, as the left action of A multiplies the invariant measure on [G] by the inverse 0 0 −1 modular character δ ,weobtain vol (gU) ∼ vol (gU) ∼ exp(−2ρ , H (g) ) for g ∈ s [G] [G] 0 0 P P P which is another way to state (2.4.5.23). By [Lap13, §2, (9)], we also have P 2 −d (2.4.5.24) There exists d > 0such that (g) σ (g) dg < ∞; 0 P [G] From (2.4.5.23), we also deduce the existence of N 1such that P 0 (2.4.5.25) (g)g , g ∈[G] . Note that the definition of the function in loc. cit. coincides, up to equivalence, with ours by (2.4.5.23). THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 209 2.5. Spaces of functions 2.5.1. Let V be a Fréchet space. We say of a function f : G(A) → Vthat it is smooth if it is right invariant by a compact-open subgroup J of G(A ) and for every g ∈ G(A ) the function g ∈ G(F ) → f (g g ) ∈ VisC . This definition applies in f f ∞ ∞ f ∞ particular for V = C and, for P ⊂ G a parabolic subgroup, we denote by C ([G] ) the space of smooth complex-valued functions on [G] . We also denote by C ([G] ) (resp. L ([G] )) the spaces of complex-valued contin- c P P loc uous compactly supported (resp. locally integrable) functions on [G] . 2.5.2. Let C be a compact subset of G(A ) and let J ⊂ K be a compact-open subgroup. Let S(G(A), C, J) be the space of smooth functions f : G(A) → C which are biinvariant by J, supported in the subset G(F ) × C and such that the semi-norms f = sup g |(R(X)L(Y)f )(g)| r,X,Y g∈G(A) are finite for every integer r 1and X, Y ∈ U(g ). This family of semi-norms defines a topology on S(G(A), C, J) making it into a Fréchet space. The global Schwartz space S(G(A)) is the locally convex topological direct limit over all pairs (C, J) of the spaces S(G(A), C, J). It is a strict LF space. Moreover, the Schwartz space S(G(A)) is an algebra for the convolution product denoted by ∗. It contains the dense subspace C (G(A)) of smooth and compactly supported functions. For an integer r 0, we will also consider the r ∞ space C (G(A)) generated by products f f where f is a compactly supported function ∞ ∞ on G(F ⊗ R) which admits derivatives up to the order r and f is a smooth compactly supported function on G(A ). For every integer n 1, we define similarly the global Schwartz space S(A ) using the norm . (see Section 2.4.2). 2.5.3. In order to organize in some uniform way the different spaces of func- tions that we are about to define, we now introduce, mostly following the terminology of [BK14], some nice categories of complex linear representations of G(A).Recallfrom [BK14, §2.3] that a Fréchet representation V of G(F ) is said to be a F-representation if there exists a family of semi-norms (. ) defining the topology of V such that the V,n n G(F )-action is continuous with respect to each of them. Moreover, a F-representation V of G(F ) is called a SF-representation if for every vectorv ∈ V, the map g ∈ G(F ) → g·v is ∞ ∞ smooth (that is C )and forevery X ∈ U(g ), the resulting linear operator v ∈ V → X·v is continuous (cf. [BK14, §2.4.3]). In a similar fashion, we say that a representation of G(A) on a Fréchet space V is aF-representation if there exists a family of semi-norms (. ) defining the topology of V V,n n such that the G(A)-action is continuous with respect to each of them. In particular, con- tinuous representations of G(A) on Banach spaces are automatically F-representations. 210 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Let V be a F-representation of G(A).Avector v ∈ Vis smooth if the orbit map g ∈ G(A) → g · v is smooth. We denote by V the subspace of smooth vectors. It is aG(A)-invariant subspace and for every compact-open subgroup J ⊆ G(A ), we equip ∞ J the subspace (V ) of J-fixed vectors with the topology associated to the semi-norms v → Xv where . runs over a family of semi-norms defining the topology of V, V V X runs over (a basis of) U(g ) and Xv := R(X)(g → g v) . With this topology, ∞ ∞ ∞ |g =1 ∞ J (V ) becomes a Fréchet space and even a SF representation of G(F ) in the sense ∞ J ∞ J recalled above. Moreover, the inclusions (V ) ⊂ (V ) for J ⊆ J ⊆ G(A ) are closed ∞ ∞ J embeddings and so V = (V ) has a structure of strict LF space. For every v ∈ Vand f ∈ S(G(A)), the integral (2.5.3.1) f ·v = f (g)g ·vdg G(A) converges in V and this defines an action of the algebra (S(G(A)),∗) on V. Moreover, by the Dixmier-Malliavin theorem [DM78], we have (2.5.3.2)V = S(G(A)) · V where the right-hand side stands for the set of finite linear combinations f · v with i i v ∈ Vand f ∈ S(G(A)). i i 2.5.4. In this paper, by a SLF representation of G(A) we mean a C-vector space V equipped with a linear action of G(A) and, for every compact-open subgroup J ⊆ G(A ), the structure of a Fréchet space on V such that the following conditions are satisfied: • For every J ⊆ G(A ),V is a SF representation of G(F ) in the sense of [BK14, f ∞ §2.4.3]; • V = V where J runs over all compact-open subgroups of G(A ) (i.e. the action of G(A ) on V is smooth); J J • For every J ⊆ J ⊆ G(A ), the inclusion V ⊆ V is a closed embedding. By the second and third points above, a SLF representation of G(A) has a natural struc- ture of strict LF space. If V is a F-representation of G(A), the subspace of smooth vectors V has a natural structure of SLF representation of G(A) by the previous paragraph. Most of the function spaces that we are going to introduce carry natural structures of F- or SLF representations of G(A). Let V be a SLF representation of G(A). Then, (2.5.3.1) still defines an action of (S(G(A)),∗) on V and the equality (2.5.3.2) holds with V instead of V . The following lemma implies, by the open mapping theorem for LF spaces, that equivariant morphisms of SLF representations with closed image are necessarily topo- logical embeddings. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 211 Lemma 2.5.4.1. — Let V be a SLF representation of G(A) and W ⊂ V be a closed G(A)- J J invariant subspace. Then, equipping W with the topology induced from V for every compact-open subgroup J ⊆ G(A ) gives W the structure of a SLF representation of G(A) and the corresponding LF topology on W (given as the locally convex direct limit of the topologies on the W ’s) coincides with the topology induced from V. Proof. — It is immediate that W is a SLF representation of G(A) (as a closed in- variant subspace of a SF representation is itself a SF representation). On the other hand, to check that the LF topology on W is induced from V, it suffices to notice the decompo- sitions V = V , W = W ρ ρ ρ∈K ρ∈K f f where K denotes the set of isomorphism classes of smooth irreducible representations of K and for δ ∈ K ,V and W stand for the corresponding isotypical subspaces. Indeed, f f ρ ρ J J V and W have structures of Fréchet spaces (these are closed subspaces of V and W for ρ ρ some compact-open subgroup J ⊂ G(A ) respectively) and these decompositions identify the LF spaces V, W with the topological direct sums of the families (V ) , (W ) ρ ρ∈K ρ ρ∈K f f respectively: this is because for every ρ,V maps continuously into some V and con- versely for every J, V maps continuously into V ⊕···⊕ V where ρ ,...,ρ are the, ρ ρ 1 n 1 n finitely many, irreducible representations of K with a nonzero J-fixed vector (and simi- larly for W). Therefore, the end of the lemma is a consequence of the following general property of topological direct sums: (2.5.4.3)Let (V ) be a family of locally convex topological vector spaces and for each i i∈I i ∈ I, let W be a subspace of V equipped with the induced topology. Then, the i i locally convex topological direct sum V induces on its subspace W i i i∈I i∈I the locally convex direct sum topology. 2.5.5. Let P be a semi-standard parabolic subgroup of G. We denote by L ([G] ) the space of L -measurable functions on [G] . It is a Hilbert space when equipped with the scalar product ϕ ,ϕ = ϕ (g)ϕ (g)dg 1 2 P 1 2 [G] associated to the Tamagawa invariant measure on [G] .Wedenotesimilarly by L ([G] ) the Hilbert space of measurable functions ϕ on [G] satisfying ϕ(ag) = P,0 P 1/2 ∞ 2 δ (a) ϕ(g) for almost all a ∈ A and such that |ϕ(g)| dg is convergent. [G] P,0 More generally, if w is a weight on [G] , we write L ([G] ) for the Hilbert space P P of functions that are square-integrable with respect to the measure w(g)dg. This space is 212 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR equipped with a continuous (non-unitary) representation of G(A) by right-translation. Its 2 ∞ subspace L ([G] ) of smooth vectors consists of smooth functions ϕ :[G] → C such P P that R(X)ϕ ∈ L ([G] ) for every X ∈ U(g ). By the Sobolev inequality, see [Ber88, §3.4, P ∞ 2 ∞ Key Lemma], for every ϕ ∈ L ([G] ) we have P −1/2 (2.5.5.4) |ϕ(g)| (g)w(g) , g ∈[G] . Moreover, from Riesz representation theorem, we have: 2 ∞ (2.5.5.5) For every continuous linear form T ∈ (L ([G] ) ) and f ∈ S(G(A)), 2 ∞ there exists ϕ ∈ L ([G] ) such that T(R(f )ψ) =ψ,ϕ for every ψ ∈ −1 P P L ([G] ). To save some space, for N ∈ R, d 0 and any weight w on [G] , we will adopt the following notation 2 2 2 2 L ([G] ) = L ([G] ) and L ([G] ) = L ([G] ). P N P P d P N,w σ,d,w . w σ w P P Moreover, for w = 1 we will simply drop the index w. The following result is a conse- quence of [Ber88, Theorem p. 688] as the composition of two Hilbert-Schmidt operators is nuclear. Proposition 2.5.5.1. — There exists d 0 such that for every compact-open subgroup J ⊂ G(A ) and every weight w on [G] the inclusion of Fréchet spaces f P 2 ∞,J 2 ∞,J L ([G] ) ⊂ L ([G] ) P P σ,d ,w w ∞,J is nuclear. In particular, any summable family in L ([G] ) becomes absolutely summable in σ,d ,w 2 ∞,J L ([G] ) (see Lemma A.0.6.1). 2.5.6. We let S ([G] ) be the space of measurable complex-valued functions ϕ on [G] such that ϕ = sup x |ϕ(x)|< ∞ ∞,N x∈[G] for every N > 0. We equip S ([G] ) with the topology associated to the family of semi- norms (. ) .ItisaFréchetspace.Notethat S ([G] ) is not a F-representation of ∞,N N>0 P G(A) (because the action by right translation is not continuous) but the closed subspace 00 0 S ([G] ) ⊂ S ([G] ) of continuous functions is so. P P 2.5.7. The Schwartz space S([G] ) of [G] is defined as the space of smooth func- P P tions ϕ :[G] → C such that for every N > 0and X ∈ U(g ) we have P ∞ ϕ = sup x |(R(X)ϕ)(x)|< ∞. ∞,N,X x∈[G] P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 213 00 ∞ Then, S([G] ) is a SLF representation of G(A) which is equal to S ([G] ) .Moreover, P P 2 ∞ 00 for every weight w on [G] , S([G] ) is dense in L ([G] ) (as S ([G] ) is dense in P P P P L ([G] )). From (2.5.5.4) and the open mapping theorem, we have an equality of SLF representations (where the right-hand side is equipped with the locally convex projective limit topology) 2 ∞ (2.5.7.6) S([G] ) = L ([G] ) . P P N>0 2.5.8. The Harish-Chandra Schwartz space C([G] ) of [G] is defined as the space P P of smooth functions ϕ :[G] → C such that for every d > 0and X ∈ U(g ) we have P ∞ P −1 d ϕ = sup (x) σ (x) |(R(X)ϕ)(x)|< ∞. ∞,d,σ,X P x∈[G] For J a compact-open subgroup of G(A ), we equip C([G] ) with the topology induced f P from the family of semi-norms (. ) . This makes C([G] ) into a SLF represen- ∞,d,σ,X d,X P tation of G(A) when equipped with the action by right translation. The Schwartz space S([G] ) is dense in C([G] ).Moreover, by (2.5.5.4), we have the alternative description P P 2 ∞ (2.5.8.7) C([G] ) = L ([G] ) . P P σ,d d>0 2.5.9. For every weight w on [G] ,welet T ([G] ) be the space of complex P P Radon measure ϕ on [G] such that −1 ϕ −1 = w(x) |ϕ(x)|< ∞. 1,w [G] We equip T ([G] ) with the topology associated to the norm . −1 . It is a continuous P 1,w 0 0 Banach representation of G(A).For N> 0, we write T ([G] ) for T ([G] ) and we set P N P 0 0 T ([G] ) = T ([G] ). P P N>0 We equip this space with the corresponding (non-strict) LF topology. We have a sesquilin- ear pairing 0 0 (ϕ,ψ) ∈ T ([G] ) × S ([G] ) → ϕ,ψ = ψ(x)ϕ(x) P P P [G] 0 00 which identifies T ([G] ) with the topological dual of S ([G] ). P P 214 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 2.5.10. For every weight w on [G] ,wedefine T ([G] ) as the space of smooth P w P functions ϕ :[G] → C such that for every X ∈ U(g ) we have P ∞ −1 −1 ϕ = sup w(x) |(R(X)ϕ)(x)|< ∞. ∞,w ,X x∈[G] We equip T ([G] ) , for J a compact-open subgroup of G(A ), with the topology induced w P f from the family of semi-norms (. −1 ) . This makes T ([G] ) a SLF representation ∞,w ,X X w P of G(A).For N > 0, we write T ([G] ) and T ([G] ) for T N([G] ) and T N([G] ) N P w,N P P P . w. P P respectively. The space of functions of uniform moderate growth on [G] is defined as T ([G] ) = T ([G] ). P N P N>0 We equip this space with the corresponding (non-strict) LF topology. The Schwartz space S([G] ) is dense in T ([G] ) but we warn the reader that usually S([G] ) is not dense in P P P T ([G] ) (unless [G] is compact). By (2.5.5.4), we have the alternative descriptions (as a N P P LF space) 2 ∞ 0 ∞ (2.5.10.8) T ([G] ) = L ([G] ) = T ([G] ) . P P P −N N N>0 N>0 2.5.11. More generally, for a weight w on [G] and N > 0, we define S ([G] ) P w,N P as the space of smooth functions ϕ :[G] → C such that for every r 0and X ∈ U(g ) P ∞ we have −N r ϕ r = sup x w(x) |(R(X)ϕ)(x)|< ∞. ∞,−N,w ,X x∈[G] We equip S ([G] ) , for J a compact-open subgroup of G(A ), with the topology in- w,N P f duced from the family of semi-norms (. r ) and this makes S ([G] ) into ∞,−N,w ,X X,r0 w,N P a SLF representation of G(A). We set S ([G] ) = S ([G] ) w P w,N P N>0 and we equip this space with the corresponding (non-strict) LF topology. Once again, S([G] ) is dense in S ([G] ) but in general not in S ([G] ).Moreover, by (2.5.5.4), P w P w,N P we have the alternative description (as a LF space) 2 ∞ (2.5.11.9) S ([G] ) = L ([G] ) . w P r P −N,w N>0 r0 We note the following lemma. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 215 Lemma 2.5.11.1. — Let ϕ ∈ T ([G] ).If w is bounded from above on the support of ϕ, then R(f )ϕ ∈ S ([G] ) for every f ∈ S(G(A)). w P Proof. — Indeed, if w is bounded from above on the support of ϕ, then, as w is a weight, it is also bounded from above on the support of R(f )ϕ for f ∈ C (G(A)). As R(f )ϕ ∈ T ([G] ), it readily follows that R(C (G(A)))ϕ ⊂ S ([G] ).Moreover, by P w P Dixmier-Malliavin we have S(G(A)) = S(G(A)) ∗ C (G(A)) and S ([G] ) is stable by w P right convolution by S(G(A)). The lemma follows. 2.5.12. By [Ber88, end of Section 3.5] (see also [Cas89b, Corollary 2.6]) we have (2.5.12.10) For every compact-open subgroup J ⊂ G(A ), the Fréchet spaces S([G] ) f P and C([G] ) are nuclear. Assume that G = G × G where G and G are two connected reductive groups 1 2 1 2 over F. Let J ⊂ G (A ),J ⊆ G (A ) be two compact open subgroups and set J = J × J . 1 1 f 2 2 f 1 2 By (2.5.12.10), (A.0.7.8) and a reasoning similar to (the proof of) [BP20, Proposition 4.4.1 (v)] we obtain: (2.5.12.11) There are topological isomorphisms J J J J J J 1 2 1 2 S([G ]) ⊗S([G ]) S([G]) and C([G ]) ⊗C([G ]) C([G]) 1 2 1 2 sending a pure tensor ϕ ⊗ϕ to the function (g , g ) →ϕ (g )ϕ (g ). 1 2 1 2 1 1 2 2 By the above, given two continuous linear forms L ,L on C([G ]), C([G ]) respectively, 1 2 1 2 the linear form L ⊗ L on C([G ]) ⊗ C([G ]) extends by continuity to a linear form on 1 2 1 2 C([G]) that we shall denote by L ⊗L . 1 2 2.5.13. Constant terms and pseudo-Eisenstein series. — Let Q ⊃ P be another standard parabolic subgroup. We have two continuous G(A)-equivariant linear maps 0 0 S ([G] ) → S ([G] ), ϕ → E (ϕ) and P Q 0 0 T ([G] ) → T ([G] ), ϕ → ϕ Q P P defined as the following compositions of pullbacks and pushforwards Q Q∗ Q P P∗ E (ϕ) =π π (ϕ), ϕ =π π (ϕ) P Q∗ P P∗ Q where π and π areasinSection 2.4.4. More concretely, we have Q P E (ϕ, x) = ϕ(γ x), for ϕ ∈ S ([G] ) and x∈[G] P Q γ∈P(F)\Q(F) 216 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR whereas the map ϕ → ϕ sends T ([G] ) into T ([G] ) and is given on this subspace by P Q P the familiar formula ϕ (x) = ϕ(ux)du, for ϕ ∈ T ([G] ) and x∈[G] . P Q P [N ] The pseudo-Eisenstein map ϕ → E (ϕ) sends S([G] ) continuously into S([G] ) and P Q the constant term map ϕ → ϕ sends T ([G] ) continuously into T ([G] ) for every P N Q N P N> 0 (as follows from (2.4.1.2)). We also have the adjunction 0 0 (2.5.13.12) ϕ ,ψ =ϕ, E (ψ) for ϕ ∈ T ([G] ), ψ ∈ S ([G] ). P P Q Q P Lemma 2.5.13.1. — There is a constant c > 0 such that for every N 0, cN N ϕ → sup δ (x) x |ϕ (x)| P P x∈[G] is a continuous semi-norm on S ([G] ). Proof. — Indeed, this follows from Corollary 2.4.3.3 and (2.4.3.13) noting that for Q ∗,Q+ ∗,P+ every λ ∈ a there exists c > 0such that λ + 2cρ ∈ a . 0 P 0 2.5.14. Approximation by the constant term. — Recall that in Section 2.4.4,wehave introduced a weight d on [G] . The next proposition is a reformulation of the well- known approximation property of the constant term (see [MW89, Lemma I.2.10]). Proposition 2.5.14.1. 1. Let N> 0,r 0 and X ∈ U(g ). Then, there exists a continuous semi-norm . on ∞ N,X,r T ([G] ) such that N Q N −r (2.5.14.13) |R(X)ϕ(x) − R(X)ϕ (x)| x d (x) ϕ P N,X,r for ϕ ∈ T ([G] ) and x ∈ P(F)N (A)\G(A). N Q Q 2 ∞ 2. Let w be a weight on [G] and ϕ ∈ L ([G] ) . Then, there exists N > 0 such that for Q Q every r 0, we have 2 −1 N −r (2.5.14.14) |ϕ(ax) −ϕ (ax)| w(a)δ (a) da x d (x) P Q r P P for x ∈ P(F)N (A)\G(A). Proof. — 1. Since ϕ ∈ T ([G] ) → ϕ ∈ T ([G] ) is continuous (as follows from N Q P N P (2.4.1.2)), there exists at least a continuous semi-norm . on T ([G] ) such that N,X N Q |R(X)ϕ(x) − R(X)ϕ (x)| x ϕ , P N,X P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 217 for ϕ ∈ T ([G] ) and x ∈ P(F)N (A)\G(A). N Q Q By the above estimate, and with the notation of Section 2.4.4, it suffices to find a contin- uous semi-norm . such that (2.5.14.13)issatisfiedfor x ∈ ω [> C] where C > 0 N,X,r is some fixed but arbitrary constant. We can choose C > 0such that ω [> C]⊂ P(F)s and therefore, it suffices to show (2.5.14.13)for x ∈ s only. If P is a maximal parabolic subgroup of Q i.e. \ ={α} for some root α,the result is then a direct consequence of [MW89, Lemma I.2.10]. To deduce the general case, we choose a tower of parabolic subgroups P = P ⊂ P ⊂···⊂ P = Qwith \ 0 1 n P Q i−1 P ={α } for every 1 i n. Then, we have \ ={α ,...,α } and, for 1 i n, i 1 n 0 0 s is included in s . Therefore, by [MW89, Lemma I.2.10], for every 1 i n,wecan Q P find a continuous semi-norm . on T ([G] ) such that i,N,X,r N P |R(X)ϕ (x) − R(X)ϕ (x)| x exp(−rα , H (x) )ϕ , P P i 0 P i,N,X,r i i−1 P i for ϕ ∈ T ([G] ) and x ∈ s . N Q Q Then, we get |R(X)ϕ(x) − R(X)ϕ (x)| |R(X)ϕ (x) − R(X)ϕ (x)| P P P i i−1 i=1 x max exp(−rα , H (x) )ϕ i 0 N,X,r 1in N −r x d (x) ϕ N,X,r P P for ϕ ∈ T ([G] ) and x ∈ s where ϕ = ϕ is a continuous semi-norm N Q Q N,X,r P i,N,X,r on T ([G] ). N Q 2. Let N > 0 that we will assume large enough. Then, for some N > 0, we have 2 ∞ ∞ continuous inclusions L ([G] ) ⊂ L ([G] ) ⊂ T ([G] ).Thus, by 1.,for every Q Q N/2 Q w −N r 0we can find elementsX ,..., X ∈ U(g ) such that 1 k ∞ 2 N −r 2 −N |ψ(x) −ψ (x)| x d (x) |R(X )ψ(y)| y dy P i P Q [G] 1i 2 ∞ for ψ ∈ L ([G] ) and x ∈ P(F)N (A)\G(A). Applying this inequality to the function Q Q L ϕ(x) =ϕ(ax) where a ∈ A , we are reduced to show the convergence of 2 −1 −N |R(X )ϕ(ay)| w(a)δ (a) day dy i Q [G] A 1ik Q 218 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR −1 2 ∞ for N large enough. After the change of variable y → a y, and since ϕ ∈ L ([G] ) , this boils down to showing that −1 −N w(a)a y da w(y), for y∈[G] . −1 From Lemma 2.4.3.1 (applied to the weight w ), there exists N > 0such that w(a) N −N 0 ∞ −1 1 w(ax)x for(a, x) ∈ A ×[G] and there exists N > 0such that a y da 1 Q 1 Q Q for y∈[G] . Then, by (2.4.1.6), any N N + N works. Q 0 1 2.5.15. The proof of the next result is partly inspired from [Fra98, §4, p. 204]. Proposition 2.5.15.1. — Let P ⊂ Q be standard parabolic subgroups of G. For every weight − + w on [G] ,there existweights w , w on [G] such that: Q P P P 2 2 • The constant term ϕ → ϕ maps L ([G] ) continuously into L ([G] ); P Q − P • The pseudo-Eisenstein map E extends to a continuous linear map from L ([G] ) into L ([G] ); + − • For every> 0, we have w (x) ∼w(x) ∼w (x) for x ∈ ω [>]. P P P Proof. — First, we observe that it suffices to prove that for every weight w on [G] , there exists a weight w on [G] such that: 2 2 • The constant term ϕ → ϕ maps L ([G] ) continuously into L ([G] ); P Q − P • For every> 0, we have w (x) ∼w(x) for x ∈ ω [>]. P P Indeed, if it were the case, by the adjunction (2.5.13.12), for every weight w on [G] ,E would extend to a continuous linear map L ([G] ) → L ([G] ) where P Q −1 + −1 − w = (w ) P P is a weight satisfying, for any given> 0, + −1 −1 w (x) ∼(w(x) ) =w(x), for x ∈ω [>]. P P Let r > 0 and set − r w (x) = min(1, d (x)) inf w(ux), x∈[G] . u∈[N ] Clearly, w is a weight on [G] and we will check that it has the desired properties when r is sufficiently large. First, for> 0, there exists a compact K ⊂ N (A) such that N P THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 219 P(F)N (A)x ⊂ P(F)N (A)xK for every x ∈ω [>].As w is a weight, this shows that P Q N w (x) ∼w(x), for x ∈ω [>]. P P On the other hand, for ϕ ∈ L ([G] ) and by the Cauchy-Schwarz inequality we have 2 − 2 − (2.5.15.15) |ϕ (x)| w (x)dx |ϕ(x)| w (γ x)dx P P [G] [G] P Q γ∈P(F)\Q(F) 2 r |ϕ(x)| w(x) min(1, d (γ x)) dx. [G] γ∈P(F)\Q(F) By Lemma 2.4.4.1 3., if r is sufficiently large we have min(1, d (γ x)) 1, for x∈[G] . γ∈P(F)\Q(F) Forsuchachoice,(2.5.15.15)gives 2 − 2 2 |ϕ (x)| w (x)dx |ϕ(x)| w(x)dx, for ϕ ∈ L ([G] ) P Q P w [G] [G] P Q which is exactly saying that the constant term ϕ → ϕ maps L ([G] ) continuously into P Q L ([G] ). Recall that if w is a weight on [G] ,wedenoteby w its restriction to A . Q A Corollary 2.5.15.2. — Let w and w be weights on [G] such that w ∼w .Then, Q A 1. If w w, the linear map 2 ∞ 2 ∞ 2 ∞ L ([G] ) → L ([G] ) × L ([G] ) , Q P Q w w PQ (2.5.15.16) ϕ →((ϕ ) ,ϕ) P P is a closed embedding. 2. Dually, if w w, the linear map 2 ∞ 2 ∞ 2 ∞ L ([G] ) × L ([G] ) → L ([G] ) + P Q Q w w PQ (2.5.15.17) P P (ϕ ) ,ϕ → ϕ + E (ϕ ) PQ is an open surjection. 220 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Proof. — 1. By the open mapping theorem, it suffices to show that the image of 2 2 P ∞ ∞ (2.5.15.16) is the closed subspace of tuples ((ϕ ) ,ϕ) ∈ L ([G] ) × L ([G] ) P P Q PQ w such that ϕ = ϕ for every P Q. That the image is included in this subspace follows 2 ∞ 2 ∞ from Proposition 2.5.15.1. Conversely, let ϕ ∈ L ([G] ) be such that ϕ ∈ L ([G] ) Q P − P for every P Q. Then, we need to show that ϕ ∈ L ([G] ) (as, applying the same rea- 2 ∞ soning to the derivatives of ϕ, we can actually deduce ϕ ∈ L ([G] ) ). 1 1 Let G(A) ⊂ G(A) be the inverse image of 0 ∈ a by H . We equip G(A) Q Q Q Q ∞ 1 with the unique measure dx such that, through the identification G(A) = A × G(A) , Q Q the invariant measure on G(A) decomposes as dadx.For every x ∈ G(A), we set x = exp(−H (x))x ∈ G(A) .Let> 0. For P Q, we introduce the set Q Q ω ={x ∈ P(F)N (A)\G(A) | d (x)> x }. P, P First we show that (2.5.15.18) |ϕ(x)| w(x)dx < ∞. π (ω ) Q P, Q Q Indeed, as ω ⊆ ω [> C] for some C > 0, by Lemma 2.4.4.1 3. and 5. we can replace P, P the domain of integration by ω . Moreover, since ϕ ∈ L ([G] ) and w (x) ∼w(x) for P P P, P Q Q x ∈ω [> C] (cf. Proposition 2.5.15.1), the function x ∈ω → ϕ (x) is square-integrable P P, with respect to the measure w(x)dx. Thus, it only remains to show that | | ϕ(x) −ϕ (x) w(x)dx < ∞. P, Q Q But this follows from Proposition 2.5.14.1 2. as d (x) ∼ d (x ) and there exists N > 0 P P −N such that x dx converges. P(F)N (A)\G(A) From (2.5.15.18), it only remains to show that x → |ϕ(x)| w(x) is integrable over the complement [G] \ π (ω ). Q P, PQ However, by Lemma 2.4.4.1 2., there exists c > 0 such that this subset is contained in the set of x∈[G] such that d (x) cx Q,α for every α ∈ . We readily check that for sufficiently small, the resulting domain is compact modulo A .Thus, as ϕ ∈ L ([G] ) and w ∼w , the claim follows. Q A Q w A THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 221 2. Again by the open mapping theorem, it suffices to show that the map (2.5.15.17) is surjective. But this follows by duality from 1., (2.5.5.5) and the Dixmier-Malliavin the- orem. 2.6. Estimates on Fourier coefficients 2.6.1. Let P ⊂ G be a standard parabolic subgroup, ψ : A/F → C be a non- trivial character and V be the vector space of additive algebraic characters N → G . P P a Let ∈ V (F) and set ψ := ψ ◦ :[N ]→ C where denotes the homomorphism P A P A between adelic points N (A) → A.For ϕ ∈ C ([G]), we set −1 ϕ (g) = ϕ(ug)ψ (u) du, g ∈ G(A). N ,ψ [N ] The adjoint action of M on N induces one on V that we denote by Ad .Wefixa P P P height . on V (A) as in Section 2.4.2. V P Lemma 2.6.1.1. 1. There exists c > 0 such that for every N , N 0, 1 2 N N ∗ −1 1 2 cN ϕ → sup Ad (m ) m δ (m) |ϕ (m)| P N ,ψ V M P P P m∈M (A) is a continuous semi-norm on S([G]). 2. Let N> 0.Then, forevery N 0, ∗ −1 N −N ϕ → sup Ad (m ) m |ϕ (m)| N ,ψ V M P P P m∈M (A) is a continuous semi-norm on T ([G]). Proof. — Bounding brutally under the integral sign, we have |ϕ (g)| |ϕ| (g) N ,ψ P for ϕ ∈ C ([G]) and g ∈ G(A).Let N 0and J ⊆ G(A ) be a compact-open sub- group. By Lemma 2.5.13.1 and (2.4.1.4), it suffices to show the existence of elements X ,..., X ∈ U(g ) such that 1 k ∞ ∗ −1 −1 (2.6.1.1) |ϕ (m)| Ad (m ) (R(X )ϕ) (m) N ,ψ i N ,ψ P V P ∞ J for every ϕ ∈ C ([G]) and m ∈ M (A).Let u ∈ N (A). By definition of . ,wejust P P V need to show the existence of X ,..., X ∈ U(g ) such that 1 k ∞ −1 (2.6.1.2) |ϕ (m)| (Ad(m)u) |(R(X )ϕ) (m)| N ,ψ i N ,ψ P A P i 222 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR ∞ J for every ϕ ∈ C ([G]) and m ∈ M (A). This last claim is a consequence of the two following facts whose proofs are elementary and left to the reader: (2.6.1.3) For every non-Archimedean place v, there exists a constant C 1, with C = v v 1 for almost all v,suchthat |(Ad(m )u )| > C implies ϕ (m) = 0for v v v v N ,ψ ∞ J every ϕ ∈ C ([G]) and m ∈ M (A). be such that u = e . Then, we (2.6.1.4)Let v be an Archimedean place and let X ∈ g v v have (R(X)ϕ) (m) = dψ ((Ad(m)u) )ϕ (m) for all ϕ ∈ C ([G]) and N ,ψ v v N ,ψ P P m ∈ M (A) where dψ : F → iR is the differential of ψ at the origin. P v v v 2.6.2. Let n 1 be a positive integer. We let GL acts on F by right multipli- cation and we denote by e = (0,..., 0, 1) the last element of the standard basis of F . We also denote by P the mirabolic subgroup of GL , that is the stabilizer of e in GL . n n n n We identify A with G ,and thus A with R , in the usual way. The next lemma GL m >0 n GL will be used in conjunction with Lemma 2.6.1.1 to show the convergence of various Zeta integrals. Lemma 2.6.2.1. — Let C> 1. Then, for N 1 and N 1 the integral 1 C 2 C −N −N 1 2 s ag e g |det g| dadg GL A P (F)\GL (A)×R n n >0 converges for s ∈ H uniformly on every (closed) vertical strip. ]1,C[ Proof. — The integral of the lemma can be rewritten as −N −N 1 2 s (2.6.2.5) g ξ ag |det ag| dadg. GL A [GL ] R n >0 n ξ∈F \{0} 3 N n There exists N > 0such that v vg g for (v, g) ∈ A × GL (A). Therefore, 3 A n the inner integral above is essentially bounded by −N /N (s) N 2 3 n (s) |det g| g aξ |a| da >0 n ξ∈F \{0} hence, for 1< (s)< C, by −N /N N +N 2 3 n (s) 2 4 g aξ |a| da >0 n ξ∈F \{0} for some N > 0. However, since the inner integral in (2.6.2.5) is left invariant by GL (F), 4 n as a function of g, we may replace g in the estimate above by g .Asfor N 1we GL n THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 223 have −N g dg < ∞ GL [GL ] [BP21a, Proposition A.1.1 (vi)], it only remains to show that for N 1 the integral −N ns aξ |a| da >0 n ξ∈F \{0} converges for 1 < (s)< C uniformly in vertical strips. This is a consequence of the following claim: (2.6.2.6)For every k n, if N is sufficiently large we have −N −k aξ |a| , a ∈ R . >0 ξ∈F \{0} Indeed, we have |a|= max(|aξ |)aξ i A 1in for (a,ξ) ∈ R × (F \{0}). Therefore, we just need to prove (2.6.2.6) when k = n.Let >0 n n n C ⊂ A be a compact subset which surjects onto A /F .Wehave aξ + av n aξ n max(1,|a|), max(1,|a|)aξ n A A A for (a,ξ,v) ∈ R ×(F \{0}) × C. Hence, >0 −N/2 −N aξ aξ + av n dv A A n n ξ∈F \{0} ξ∈F \{0} −N/2 −N/2 −n aξ + av dv =|a| v dv n n A A n n n A /F A ξ∈F for a ∈ R . The last integral above is absolutely convergent when N 1[BP21a,Propo- >0 sition A.1.1 (vi)] and the claim (2.6.2.6) follows. 2.7. Automorphic forms and representations 2.7.1. Let P be a standard parabolic subgroup of G. We define the space A (G) of automorphic forms on [G] as the subspace of Z(g )-finite functions in T ([G] ).We P ∞ P let A (G) (resp. A (G)) be the subspace of cuspidal (resp. square-integrable) au- P,cusp P,disc tomorphic forms i.e. the space of forms ϕ ∈ A (G) such that ϕ = 0for every proper P Q parabolic subgroup Q P (resp. such that |ϕ|∈ L ([G] )). P,0 224 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR For J ⊂ Z(g ) an ideal of finite codimension, we denote by A (G) the sub- ∞ P,J space of automorphic forms ϕ ∈ A (G) such that R(z)ϕ = 0for every z ∈ J and we set A (G) = A (G) ∩ A (G), P,cusp,J P,J P,cusp A (G) = A (G) ∩ A (G). P,disc,J P,J P,disc There exists N 1such that A (G) is a closed subspace of T ([G] ) and we equip P,J N P A (G) with the induced topology from T ([G] ). This topology does not depend P,J N P on the choice of N by Lemma 2.5.4.1 and the open mapping theorem. The subspaces A (G) and A (G) of A (G) are closed and we equip them with the induced P,cusp,J P,disc,J P,J topologies. Then, by Lemma 2.5.4.1, A (G), A (G) and A (G) are all SLF P,cusp,J P,disc,J P,J representations of G(A) (in the sense of Section 2.5.4) for the action by right translation. For convenience, we also endow A (G) = A (G), A (G) = P P,J P,cusp A (G) and A (G) = A (G) with the corresponding locally con- P,cusp,J P,disc P,disc,J J J vex direct limit topologies. These spaces are not LF because the poset of ideals of finite codimension in Z(g ) does not admit a countable cofinal subset. However, for every maximal ideal m ⊂ Z(g ), the subspaces A (G) = A n(G), A (G) = A n(G) and P m P,m P,cusp m P,cusp,m n n A (G) = A n(G) P,disc m P,disc,m are strict LF spaces and we have decompositions as locally convex topological direct sums A (G) = A (G) , A (G) = A (G) , P P m P,cusp P,cusp m m m A (G) = A (G) P,disc P,disc m where m runs over all maximal ideals of Z(g ). For P = G, we simply set A(G) = A (G), A (G) = A (G) and A (G) = G disc G,disc cusp A (G). G,cusp 2.7.2. By a cuspidal (resp. discrete) automorphic representation π of M (A) we mean a topologically irreducible subrepresentation of A (M ) (resp. A (M )). Letπ be a cus- cusp P disc P pidal or discrete automorphic representation of M (A).Weendow π with the topology induced from A (M ) or A (M ). With this topology, it becomes a SLF represen- cusp P disc P tation of M (A). Moreover, for every compact-open subgroup J ⊂ G(A ), the subspace P f π of J-fixed and K -finite vectors is a Harish-Chandra (g , K )-module whose ∞ ∞ ∞ (K ) ∞ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 225 smooth Fréchet globalization of moderate growth (which is unique by the Casselman- Wallach globalization theorem [Cas89a], [Wal92, Chapter 11], [BK14]) is isomorphic to π (the subspace of J-fixed vectors). For every λ ∈ a , we define the twist π = π ⊗λ as the space of functions of the P,C form m∈[M ]→ exp(λ, H (m) )ϕ(m), for ϕ ∈π. P P If π is cuspidal (resp. discrete and λ ∈ ia ), π is again a cuspidal (resp. discrete) automor- phic representation. We denote by A (M ) (resp. A (M ))the π -isotypic component of π,cusp P π,disc P A (M ) (resp. A (M )) i.e. the sum of all cuspidal (resp. discrete) automorphic repre- cusp P disc P G(A) sentations of M (A) that are isomorphic to π.Let = Ind (π) (resp. A (G) = P P,π,cusp P(A) G(A) G(A) Ind (A (M )), A (G) = Ind (A (M )))bethe normalized smooth in- π,cusp P P,π,disc π,disc P P(A) P(A) duction of π (resp. A (M ), A (M )) that we identify with the space of forms π,cusp P π,disc P ϕ ∈ A (G) such that m∈[M ]→ exp(−ρ , H (m) )ϕ(mg) P P P belongs to π (resp. A (M ), A (M ))for every g ∈ G(A). Then, and π,cusp P π,disc P A (G) (resp. A (G)) are closed subspaces of A (G) (resp. A (G))if π is P,π,cusp P,π,disc P,cusp P,disc cuspidal (resp. discrete) and with the induced topologies these become SLF representa- tions of G(A). In particular, the algebra S(G(A)) acts on A (G) (resp. A (G)) P,π,cusp P,π,disc by right convolution. When the context is clear (that is when the automorphic represen- tation π is fixed), for every λ ∈ a ,wewilldenoteby I(λ) the action on A (G) P,π,cusp P,C we get by transport from the action of S(G(A)) on A and the identification P,π ,cusp A → A given by ϕ → exp(λ, H (.) )ϕ.Inthe same way, we getanac- P,π,cusp P,π ,cusp P tion on A (G) also denoted by I(λ). P,π,disc G(A) If the central character of π is unitary, we equip = Ind (π) and A (G) P,π,cusp P(A) (resp. A (G)) with the Petersson inner product P,π,disc 2 2 ϕ =ϕ,ϕ = |ϕ(g)| dg,ϕ ∈. Pet Pet [G] P,0 2.7.3. Eisenstein series. — Let P be a standard parabolic subgroup of G. For every ϕ ∈ A (G), g ∈ G(A) and λ ∈ a ,wedenoteby P,disc P,C E(g,ϕ,λ) = exp(λ, H (δg) ϕ(δg) δ∈P(F)\G(F) the corresponding Eisenstein series which is absolutely convergent for (λ) in a suitable cone. By [Lan76], [BL19] it admits a meromorphic continuation to a whenever ϕ is P,C K -finite and this still holds without this assumption by [Lap08]. Let π be a discrete ∞ 226 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR G(A) automorphic representation of M (A) and = Ind (π) ⊆ A (G) be the induced P P,disc P(A) representation of G(A). Then, for λ ∈ a where E(ϕ,λ) is regular for every ϕ ∈ , P,C ϕ → E(ϕ,λ) induces a continuous linear map → T ([G]) by [Lap08, Theorem 2.2] that actually factors through T ([G]) for some N > 0 giving a map → T ([G]) that is N N also continuous (by the closed graph theorem). 2.7.4. Let P and Q be standard parabolic subgroups of G. For any w ∈ W(P, Q) and λ ∈ a , we have the intertwining operator P,C M(w,λ) : A (G) → A (G) P,disc Q,disc defined by analytic continuation from the integral (M(w,λ)ϕ)(g) = exp(−wλ, H (g) ) −1 −1 × exp(λ, H (w ng) )ϕ(w ng) dn. −1 (N ∩wN w )(A)\N (A) Q P Q Once again, the K -finite case follows from [Lan76], [BL19] whereas the extension to general smooth discrete automorphic forms is proved in [Lap08]. 2.7.5. Assume that G = G × G where G and G are connected reductive 1 2 1 2 groups over F. We have corresponding decompositions P = P × P and M = M × M . 1 2 P P P 1 2 Let π be a discrete automorphic representation of M (A) and set as before = G(A) Ind (π). Then, there exist two, uniquely determined, cuspidal automorphic repre- P(A) G (A) sentations π , π of M (A) and M (A) respectively such that, setting = Ind (π ) 1 2 P P 1 1 1 2 P (A) G (A) and = Ind (π ), for every compact-open subgroups J ⊆ G (A ),J ⊆ G (A ) 2 2 1 1 f 2 2 f P (A) (resp. J ⊆ M (A ),J ⊆ M (A )), setting J = J × J , there is a topological isomorphism 1 P f 2 P f 1 2 1 2 J J J J 1 2 J 1 2 J (2.7.5.1) ⊗ (resp. π ⊗π π ) 1 2 1 2 J J J J 1 2 1 2 sending ϕ ⊗ ϕ ∈ ⊗ (resp. ϕ ⊗ ϕ ∈ π ⊗ π ) to the function (g , g ) → 1 2 1 2 1 2 1 2 1 2 ϕ (g )ϕ (g ). We will then write = and π =π π respectively. 1 1 2 2 1 2 1 2 2.7.6. Assume now that G is quasi-split. Let ψ : N (A) → C be a continuous N 0 non-degenerate character which is trivial on N (F). If the representation isψ -generic, 0 N i.e. if it admits a continuous nonzero linear form : → C such that ◦(u) =ψ (u) for every u ∈ N(A), it is (abstractly) isomorphic to its Whittaker model W(,ψ )={g ∈ G(A) →((g)ϕ) | ϕ ∈}. We equip this last space with the topology coming from (thus it is a SLF representation of G(A)). THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 227 If we are moreover in the situation of Section 2.7.5, there are decompositions N = N × N , ψ =ψ ψ and the isomorphism (2.7.5.1) induces one between Whittaker 0,1 0,2 N 1 2 models J J J 1 2 W( ,ψ ) ⊗W( ,ψ ) W(,ψ ) . 1 1 2 2 N 2.8. Relative characters 2.8.1. Let B a G(F )-invariant nondegenerate symmetric bilinear form on g . ∞ ∞ We assume that the restriction of B to k is negative and the restriction of B to the or- thogonal complement of k is positive. Let (X ) be an orthonormal basis of k relative ∞ i i∈I ∞ to −B. Let C =− X : this is a “Casimir element” of U(k ). K ∞ i∈I i ˆ ˆ 2.8.2. Let K and K be respectively the sets of isomorphism classes of irre- ducible unitary representations of K and of K. 2.8.3. Let π be a discrete automorphic representation of M . In the following, we denote by A either A or A .For any τ ∈ K, let A (G,τ) be the (finite P,π P,π,cusp P,π,disc P,π dimensional) subspace of functions in A (G) which transform under K according to τ . P,π A K-basis B of A (G) is by definition the union over of τ ∈ K of orthonormal bases P,π P,π B of A (G,τ) for the Petersson inner product. P,π,τ P,π 2.8.4. Let B : A (G) × A (G) → C P,π P,π be a continuous sesquilinear form. G,∗ Proposition 2.8.4.1. — Let ω be a compact subset of a . 1. Let f ∈ S(G(A)) and B be a K-basis of A (G). The sum P,π P,π (2.8.4.1) I (λ, f )ϕ ⊗ϕ ϕ∈B P,π converges absolutely in the completed projective tensor product A (G)⊗A (G) uniformly P,π P,π G,∗ for λ ∈ a such that (λ) ∈ ω. In particular, the sum P,C (2.8.4.2) J (λ, f ) = B(I (λ, f )ϕ,ϕ) B P ϕ∈B P,π G,∗ is absolutely convergent uniformly for λ ∈ a such that (λ) ∈ ω. Moreover these sums P,C do not depend on the choice of B . P,π 228 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 2. The map f → J (λ, f ) is a continuous linear form on S(G(A)).Moreprecisely for C ⊂ G(A ) acompact subset and K ⊂ K a compact-open subgroup, there exists a continuous semi-norm G,∗ · on S(G(A), C, K ) such that for all λ ∈ a such that (λ) ∈ ω and f ∈ P,C S(G(A), C, K ) we have |J (λ, f )| f . Remark 2.8.4.2. — An examination of the proof below show that the assertion 2 also holds mutatis mutandis if f ∈ C (G(A)) with r large enough. The semi-norm is then taken among the norms · for which the sum of the degrees of X and Y is less than r,X,Y r. Proof. — By definition of the projective tensor product topology, it suffices to show the following: for every continuous semi-norm p on A (G), the series P,π p(I (λ, f )ϕ)p(ϕ) ϕ∈B P,π G,∗ is absolutely convergent uniformly for λ ∈ a such that (λ) ∈ ω.Let K ⊂ K be P,C a normal compact-open subgroup by which f is biinvariant. The series above can be rewritten as (2.8.4.3) p(I (λ, f )ϕ)p(ϕ), ˆ ϕ∈B τ∈K P,π,τ where only the representations τ admitting K -invariant vectors actually contribute to the sum. Note that, since K is normal in K , for such representation τ all the elements ϕ ∈ B are automatically K -fixed. Moreover, by [Wal92] §10.1, there exist c > 0and P,π,τ 0 an integer r such that for every ϕ ∈ A (G) we have P,π p(ϕ) cR(1 + C ) ϕ . K Pet ˆ ˆ For any τ ∈ K or K, let λ 0 be the eigenvalue of C acting on τ . Let us fix a large ∞ τ K enough N> 0. For every f ∈ S(G(A)), λ ∈ a , τ ∈ Kand ϕ ∈ B ,wehave P,π,τ P,C r r R(1 + C ) I (λ, f )ϕ =I (λ, L((1 + C ) )f )ϕ K P Pet P K Pet −N =(1 +λ ) I (λ, f )ϕ τ P r,N Pet N r where f = R((1 + C ) )L((1 + C )) f .Let C ⊂ G(A ) be a compact subset. Then, r,N K K f there exists a continuous semi-norm · on S(G(A), C, K ) (among those of Section 0 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 229 G,∗ 2.5.2)suchthatfor any f ∈ S(G(A), C, K ), ϕ ∈ A (G) and λ ∈ a such that (λ) ∈ 0 P,π P,C ω we have I (λ, f )ϕ f ϕ . P r,N P,π Pet Thereby we are reduced to prove for large enough N the convergence of r−N ∞ (2.8.4.4) (1 +λ ) dim(A (G, K ,τ)) τ 0 P,π τ∈K where A (G, K ,τ) ⊂ A (G) denotes the subspace of functions that transform under 0 P,π P,π K according to τ . However, there exist c > 0and m 1 such that dim(A (G, K , ∞ 2 0 P,π τ)) c (1 + λ ) (see e.g. the proof of [Mül00] Lemma 6.1). So the convergence of 2 τ −N (2.8.4.4) is reduced to that of (1 +λ ) which is well-known. ˆ τ τ∈K Proposition 2.8.4.3. — Let K ⊂ K be a normal open compact subgroup. For any integer ∞ m m 1 there exist Z ∈ U(g ),g ∈ C (G(A)) and g ∈ C (G(A)) such that ∞ 1 2 c c • Z,g and g are invariant under K -conjugation; 1 2 ∞ • g and g are K -biinvariant; 1 2 0 • for any f ∈ S(G(A)) that is K -biinvariant we have: f = f ∗ g +(f ∗ Z) ∗ g . 1 2 For large enough m, we have J (λ, f ) = B(I (λ, f )ϕ, I (λ, g )ϕ) B P P ϕ∈B P,π + B(I (λ, f ∗ Z)ϕ, I (λ, g )ϕ) P P ϕ∈B P,π −1 where the sums are absolutely convergent and g (x) = g (x ). Proof. — The first part of the proposition is lemma 4.1 and corollary 4.2 of [Art78]. Once we have noticed that the operators I (λ, g ) preserve the spaces A (G,τ),the P i P,π second part results from an easy computation in a finite dimensional space. 2.9. Cuspidal data and coarse Langlands decomposition 2.9.1. Cuspidal data. — Let X(G) be the set of pairs (M ,π) where • P is a standard parabolic subgroup of G; • π is the isomorphism class of a cuspidal automorphic representations of M (A) with central character trivial on A . P 230 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR The set of cuspidal data X(G) is the quotient of X(G) by the equivalence relation defined as −1 follows: (M ,π) ∼(M ,τ) if there exists w ∈ W(P, Q) such that wπw τ .Notethat P Q for every standard parabolic subgroup P of G, the inclusion X(M ) ⊂ X(G) descends to a finite-to-one map X(M ) → X(G).For χ ∈ X(G) represented by a pair (M ,π), P P ∨ ∨ ∨ we denote by χ the cuspidal datum associated to (M ,π ) where π stands for the complex conjugate of π . 2.9.2. Langlands decomposition. — For (M ,π) ∈ X(G),welet S ([G] ) be the P π P space of Schwartz functions ϕ ∈ S([G] ) such that ϕ (x) := exp(−ρ +λ, H (a) )ϕ(ax)da, x∈[G] , λ P P P belongs to A (G) for every λ ∈ a . P,π ,cusp λ P,C Let P ⊂ G a standard parabolic subgroup, χ ∈ X(G) be a cuspidal datum and {(M ,π ) | i ∈ I} Q i be the (possibly empty but finite) inverse image of χ in X(M ).Denoteby L ([G] ) the P P closure in L ([G] ) of the subspace P P O := E (S ([G] )). π Q χ Q i i i∈I ∞ 2 More generally, for w aweighton A (see Section 2.4.3), we let L ([G] ) be the clo- P w,χ P 2 2 2 sure of O in L ([G] ) and we define similarly a subspace L ([G] ) ⊂ L ([G] ).By P P,0 P,0 χ w χ Langlands (see e.g. [MW94, Proposition II.2.4]), we have decompositions in orthogonal direct sums 2 2 2 2 (2.9.2.1) L ([G] ) = L ([G] ) and L ([G] ) = L ([G] ). P P P,0 P,0 w w,χ χ χ∈X(G) χ∈X(G) For every subset X ⊆ X(G), we set 2 2 2,X 2 L ([G] ) := L ([G] ), L ([G] ) := L ([G] ) P P P P w,X w,χ w w,χ χ∈X χ∈X where X denotes the complement of X in X(G). When w = 1, we will drop the index w.Wehave ∞ 2 (2.9.2.2) For two weights w and w on A , the orthogonal projections L ([G] ) → P w 2 2 2 L ([G] ) and L ([G] ) → L ([G] ) coincide on the intersection P P P w,X w w ,X L ([G] ) ∩ L ([G] ). P P w w THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 231 2 2 2 Indeed, both of these projections coincide on L ([G] ) ∩ L ([G] ) = L ([G] ),where P P P w w w ∞ 2 w = max(w,w ) is again a weight on A , with the orthogonal projection L ([G] ) → P w L ([G] ) as follows readily by looking at their restrictions to the dense subspace w ,X O . χ∈X(G) 2 2 We will denote by ϕ → ϕ the orthogonal projection L ([G] ) → L ([G] ). X P P w w,X By (2.9.2.2), this notation shouldn’t lead to any confusion. These projections are G(A)- equivariant and so preserve the subspaces of smooth vectors. 2.9.3. Let X ⊆ X(G) be a subset. We set S ([G] ) = S([G] ) ∩ L ([G] ) and X P P P X 2,X S ([G] ) = S([G] ) ∩ L ([G] ). P P P By definition of L ([G] ), S ([G] ) is the orthogonal to P X P P P O := O X χ χ∈X in S([G] ). In particular, for every weight w on A ,wealsohave S ([G] ) = S([G] ) ∩ L ([G] ). X P P P w,X Let w be a weight on [G] (not necessarily factoring through a weight of A ). For F ∈{L , C, T , T , S , S } we define F ([G] ) to be the orthogonal of S c([G] ) in N w,N w X P X P F([G] ). We will also write F ([G] ) for F c([G] ). P P X P Lemma 2.9.3.1. — Let X ⊆ X(G) be a subset and Q ⊆ P be standard parabolic subgroups. Then, we have (2.9.3.3)E (S ([G] )) ⊆ S ([G] ) and T ([G] ) ⊆ T ([G] ). X Q X P X P Q X Q by adjunc- Proof. — The second inclusion follows from the first applied to X tion. It remains to prove the first inclusion. Since S ([G] ) is also the orthogonal of X P X X S ([G] ) in S([G] ), by adjunction again it suffices to establish that S ([G] ) is or- P P P Q thogonal to S ([G] ). From the definition, it is clear that S ([G] ) is orthogonal to X Q P Q Q Q O .Let κ ∈ C (a ). Then, we readily check that O is stable by multiplication by X c X (κ ◦ H ). Consequently, (κ ◦ H )S ([G] ) is also orthogonal to O but, by Lemma Q Q P Q 2.5.13.1,wehave (κ ◦ H )S ([G] ) ⊆ S([G] ). Therefore, by definition of L ([G] ), Q P Q Q Q (κ ◦ H )S ([G] ) is orthogonal to L ([G] ) and a fortiori to S ([G] ). Letting (κ ) Q P Q Q X Q n n X 232 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR be an increasing sequence of positive functions in C (a ) converging to 1 pointwise we get, by dominated convergence, ϕ ,ψ = lim(κ ◦ H )ϕ ,ψ = 0 Q Q n Q Q Q n→∞ X X for every ϕ ∈ S ([G] ) and ψ ∈ S ([G] ). This shows that S ([G] ) is indeed orthog- P X Q P Q onal to S ([G] ) and this ends the proof of the lemma. X Q 2.9.4. The following theorem is a variation on the well-known theme of “de- composition along the cuspidal support” (see [MW94, III, 2.6] and [FS98]for similar result on the space of automorphic forms). We refer the reader to Section A.0.2 for a reminder on summable and absolutely summable families in locally convex topological vector spaces. Theorem 2.9.4.1. — Let w be a weight on [G] . Then 1. For every subset X ⊂ X(G), the orthogonal projection S([G] ) → L ([G] ) extends by P P continuity to a projection 2 ∞ 2 ∞ L ([G] ) → L ([G] ) ,ϕ →ϕ . P P X w w,X 2 ∞ 2 ∞ Moreover, for every ϕ ∈ L ([G] ) the family (ϕ ) is summable in L ([G] ) P χ χ∈X(G) P w w with sum ϕ. 2 2 2. For every subset X ⊂ X(G), the orthogonal projection L ([G] ) → L ([G] ) restricts to P P a continuous projection S([G] ) → S ([G] ), ϕ →ϕ . P X P X Moreover, for everyϕ ∈ S([G] ),the family(ϕ ) is absolutely summable in S([G]) P χ χ∈X(G) with sum ϕ. 3. For every subset X ⊂ X(G) and F ∈{C, T , S }, the projection S([G] ) → S ([G] ) w P X P extends by continuity to a projection F([G] ) → F ([G] ), ϕ → ϕ , P X P X satisfying the adjunction (2.9.4.4) ϕ ,ψ =ϕ,ψ , for (ϕ,ψ) ∈ T ([G] ) × S([G] ). X P X P P P Moreover, there exists N > 0 such that for every N > 0 and every function ϕ ∈ C([G] ) 0 P (resp. ϕ ∈ T ([G] ),resp. ϕ ∈ S ([G] ))the family (ϕ ) is absolutely w P w,N P χ χ∈X(G) summable in C([G] ) (resp. T ([G] ),resp. S ([G] )) with sum ϕ. P w,N P w,N+N P 0 0 Proof. — First, we note that point 1. implies points 2. and 3. Indeed, that the orthogonal projection S([G] ) → L ([G] ) induces continuous linear projections P P X THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 233 F([G] ) → F ([G] ) for F ∈{S, C, T , S } follows from the alternative descriptions P X P w (2.5.7.6), (2.5.8.7), (2.5.10.8)and (2.5.11.9) in terms of weighted L spaces. Let J ⊂ G(A ) be a compact-open subgroup. Let ϕ ∈ S([G] ) . Then, the first point of the theorem im- 2 ∞,J plies that (ϕ ) is a summable family in L ([G] ) for every N > 0. On the other χ χ∈X(G) P 2 ∞,J hand, by Proposition 2.5.5.1,for every N > 0and > 0 the inclusion L ([G] ) ⊂ N+ 2 ∞,J 0 L ([G] ) is nuclear (note that σ . ). Therefore, by Lemma A.0.6.1,for every N P P 2 ∞,J N > 0, the family (ϕ ) is actually absolutely summable L ([G] ) i.e. it is ab- χ χ∈X(G) P solutely summable in S([G] ) (by the presentation (2.5.7.6)). That it sums to ϕ is clear. The statements on absolute summability in point 3. can be similarly deduced from point 1. noting that, by (2.5.5.4), there exists N > 0such that for everyN> 0 2 ∞ 2 ∞ L ([G] ) ⊂ T ([G] ) ⊂ L ([G] ) −1 P w,N P −1 P w ,−N+N w ,−N−N 1 1 2 ∞ 2 ∞ (resp. L ([G] ) ⊂ S ([G] ) ⊂ L ([G] ) ) r r P w,N P P −N+N ,w −N−N ,w 1 1 r0 r0 J J J so that for N > 2N , the inclusion T ([G] ) ⊂ T ([G] ) (resp. S ([G] ) ⊂ 0 1 w,N P w,N+N P w,N P J ∞,J S ([G] ) ) factors through the nuclear inclusion L ([G] ) ⊂ w,N+N P −1 P w ,−N−N 2 ∞,J 2 ∞,J 2 ∞,J L ([G] ) (resp. L r([G] ) ⊂ L r([G] ) )for −1 P P P r0 −N−N ,w r0 −N−N −,w w ,−N−N − 1 1 some > 0. Finally, by density of S([G] ) in T ([G] ), the adjunction (2.9.4.4)can be P P deduced from a similar adjunction for Schwartz functions. We prove 1. by induction on a − a .For P = P ,wehave w ∼ w and the result 0 P 0 A follows from (2.9.2.2). Assume now that 1. holds for every parabolic subgroup Q P. First we assume that w w (recall that w stands for the restriction of w to A ). A A By Corollary 2.5.15.2, in this case we have a closed embedding 2 ∞ 2 ∞ 2 ∞ (2.9.4.5)L ([G] ) → L ([G] ) × L ([G] ) ,ϕ → (ϕ ) ,ϕ . P − Q P Q Q w w w A QP 2 ∞ Moreover, by the induction hypothesis, for Q P, we have projections L ([G] ) → − Q ∞ Q L ([G] ) ,ϕ → ϕ satisfying 1. To prove the existence of the continuous projection w ,X 2 ∞ 2 ∞ ϕ ∈ L ([G] ) → ϕ ∈ L ([G] ) , it suffices to check that the continuous projection P X P w w,X Q 2 ∞ 2 ∞ ((ϕ ) ,ϕ) → ((ϕ ) ,ϕ ) of L ([G] ) × L ([G] ) preserves the image of Q Q X − Q P X QP w w A (2.9.4.5). This readily follows from the identity 2 ∞ (2.9.4.6) (ϕ ) =(ϕ ) , for every ϕ ∈ L ([G] ) and Q P. X Q Q X P We emphasize that in the above equation, ϕ is defined through the orthogonal projec- 2 ∞ 2 ∞ tion L ([G] ) → L ([G] ) whereas (ϕ ) is given by the projection T ([G] ) → P P Q X Q w w ,X A A T ([G] ) from the third part of the theorem (and which exists by the induction hypoth- X Q esis). 234 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR To show (2.9.4.6), we check that (ϕ ) ,ψ =(ϕ ) ,ψ for every ψ ∈ X Q Q Q X Q S([G] ). By the induction hypothesis again, we have ψ = ψ + ψ where ψ ∈ Q X X X X P S ([G] ) and ψ ∈ S ([G] ). Moreover, by Lemma 2.9.3.1,wehave ϕ , E (ψ ) = X Q Q X X P P P X ϕ, E (ψ ) and ϕ , E (ψ ) = 0. Therefore, X P X P Q Q P P (ϕ ) ,ψ =ϕ ,ψ =ϕ, E (ψ ) =ϕ , E (ψ) Q X Q Q X Q X P X P Q Q =(ϕ ) ,ψ X Q Q and this proves (2.9.4.6). 2 ∞ Let ϕ ∈ L ([G] ) .Byinduction,for every Q P, ((ϕ ) ) is a summable P Q χ χ∈X(G) 2 ∞ ∞ family in L ([G] ) with sum ϕ .Moreover, as w is a weight on A , (ϕ ) is − Q Q A χ χ∈X(G) 2 ∞ a summable family in L ([G] ) with sum ϕ. Since (2.9.4.5) is a closed embedding, by 2 ∞ (2.9.4.6) we deduce that (ϕ ) is a summable family in L ([G] ) with sum ϕ. This χ χ∈X(G) P ends the proof of 1. whenever w w . Note that the weight w =. satisfies the condition w w (see (2.4.1.6)). 2 ∞ Therefore, we have already establish 1. for the spaces L ([G] ) (N > 0) and thus, by (2.5.7.6) and the reasoning from the beginning, we can already deduce statement 2. for P. We now deal with the case of a general weight w.Set w = max(w,w ). Then, 2 ∞ w w = w and therefore the existence of the projections ϕ ∈ L ([G] ) → ϕ ∈ A P X L ([G] ) has already been established. By Corollary 2.5.15.2 again, we have an open w ,X surjection 2 ∞ 2 ∞ 2 ∞ L ([G] ) × L ([G] ) → L ([G] ) , Q P P w w QP (2.9.4.7) Q P Q (ψ ) ,ψ → E (ψ ) +ψ. QP We now check that the projection (ψ ) ,ψ → (ψ ) ,ψ descends to this quotient Q Q X i.e. that it preserves the kernel of (2.9.4.7). For this, it suffices to show that for every Q P, we have P P Q Q 2 ∞ (2.9.4.8) E (ψ ),ϕ =E (ψ ),ϕ , for (ψ ,ϕ) ∈ L ([G] ) × S([G] ). P X P Q P Q X Q 2 ∞ By the density of S([G] ) in L ([G] ) , we can restrict ourself to prove (2.9.4.8)when Q + Q ψ ∈ S([G] ) in which case it readily follows from (2.9.4.6). 2 ∞ 2 ∞ Let ϕ ∈ L ([G] ) → ϕ ∈ L ([G] ) be the continuous projection descended P X P w w from (ψ ) ,ψ → (ψ ) ,ψ via the surjection (2.9.4.7). By (2.9.4.8), its image Q Q X lands in L ([G] ) and it extends the projection ϕ ∈ S([G] ) → ϕ ∈ S ([G] ).Fi- P P X X P w,X 2 ∞ P Q Q nally, let ϕ ∈ L ([G] ) that we write ϕ = E (ψ ) + ψ where (ψ ) ,ψ ∈ P Q w QP Q THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 235 2 ∞ 2 ∞ Q L ([G] ) × L ([G] ) . Then, for every Q P, (ψ ) is a summable + Q P χ∈X(G) QP w χ 2 ∞ Q family in L ([G] ) with sum ψ by the induction hypothesis whereas (ψ ) is a + Q χ χ∈X(G) 2 ∞ summable family in L ([G] ) with sum ψ by the case already treated. Thus, from the continuity of the map (2.9.4.7), we deduce that P Q χ ∈ X(G) → ϕ = E (ψ ) +ψ χ χ Q χ QP 2 ∞ is a summable family in L ([G] ) with sum ϕ. We have now completed the proof by induction of 1. and hence of the theorem. 2.9.5. Let X ⊆ X(G) be a subset. By the previous proposition, we have com- patible continuous projections ϕ → ϕ from S([G] ), C([G] ) and L ([G] ) onto X P P P −N S ([G] ), C ([G] ) and L ([G] ) respectively. As S([G] ) is dense in both C([G] ) X P X P P P P −N,X 2 ∞ and L ([G] ) this entails that −N 2 ∞ (2.9.5.9) S ([G] ) is dense in C ([G] ) and L ([G] ) . X P X P P −N,X 2.9.6. Assume that G = G × G where G and G are connected reductive 1 2 1 2 groups over F and write P = P × P accordingly. Then, we have a natural identifi- 1 2 cation X(G) = X(G ) × X(G ). For subsets X ⊆ X(G ) and compact-open subgroups 1 2 i i J ⊆ G (A ), i = 1, 2, setting X = X × X and J = J × J , the projection S([G] ) → i i f 1 2 1 2 P J J J S ([G] ) (resp. C([G] ) → C ([G] ) ) corresponds via the isomorphism (2.5.12.11) to X P P X P J J i i the (completed) tensor product of the projections space S([G ] ) → S ([G ] ) (resp. i P X i P i i i J J i i C([G ] ) → C ([G ] ) )for i = 1, 2 as can readily be seen by looking at pure tensors. i P X i P i i i It follows that J J J 1 2 S ([G ]) ⊗S ([G ]) S ([G]) and X 1 X 2 X 1 2 (2.9.6.10) J J J 1 2 C ([G ]) ⊗C ([G ]) C ([G]) X 1 X 2 X 1 2 by restriction of the isomorphisms (2.5.12.11). 2.9.7. Regular cuspidal data. — We say that a cuspidal datum χ ∈ X(G) is regular if it is represented by a pair (M ,π) such that the only element w ∈ W(P) satisfying wπ π is w = 1. The next result can be deduced from Langlands spectral decomposition [Lan76] but we prefer to give a direct proof. Proposition 2.9.7.1. — Letχ ∈ X(G) be a regular cuspidal datum and P ⊂ G be a parabolic 2 2 subgroup. Then, for every ϕ ∈ L ([G]) we have ϕ ∈ L ([G] ). P P χ χ Proof. — By duality, it suffices to show that the pseudo-Eisenstein map E extends 2 2 to a continuous application from L ([G] ) into L ([G]).Let {χ | i ∈ I} be the inverse P i χ 236 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR image of χ in X(M ). Then, we have an orthogonal decomposition 2 2 L ([G] ) = L ([G] ) P P χ χ i∈I 2 2 −1/2 where L ([G] ) denotes the subspace of ϕ ∈ L ([G] ) such that m → δ (m) ϕ(mg) P P P 2 G belongs to L ([M ]) for almost all g ∈ G(A). Thus, it suffices to show that E extends χ P 2 2 to a continuous application from L ([G] ) into L ([G]) for each i ∈ I. Fix i ∈ Iand let (M ,π) be a pair representing χ (where Q ⊂ P). Then, the inverse image of χ in Q i i −1 M −1 X(M ) consists of the pairs (wM w ,wπ) where w ∈ W is such that wM w is the P Q Q Levi component of some standard parabolic subgroup wQ. Therefore, by definition of 2 2 L ([M ]),L ([G] ) is the closure of P P χ χ i i (2.9.7.11) E (S ([G] )) wπ wQ wQ 2 M in L ([G] ) where w runs over elements w ∈ W as before. Thus, we just need to check that for every such w ∈ W , ϕ ∈ S ([G] ) and ϕ ∈ S ([G] ),wehave π Q wπ wQ G G P P (2.9.7.12) E (ϕ), E (ϕ ) =E (ϕ), E (ϕ ) , G P Q wQ Q wQ since it will implies that the restriction of E to the subspace (2.9.7.11)isanisometry. By the calculation of the scalar product of two pseudo-Eisenstein series [MW94, Proposition II.2.1], we have G G E (ϕ), E (ϕ ) = M(ww )ϕ ,ϕ dλ G 0 λ Pet Q wQ −ww λ ia +λ w ∈W(Q) where λ ∈ a belongs to the range of absolute convergence of the intertwining operators M(ww ) : A (G) → A (G). 0 Q,π ,cusp wQ,(ww π) ,cusp λ 0 ww λ As χ is regular, for every w ∈ W(Q) different from 1, we have ww π wπ and there- 0 0 fore M(ww )ϕ is orthogonal to ϕ for every λ ∈ ia +λ . It follows that the above 0 λ 0 −ww λ expression reduces to G G E (ϕ), E (ϕ ) = M(w)ϕ ,ϕ dλ. G λ Pet Q wQ −wλ ia +λ P P A similar argument shows that E (ϕ), E (ϕ ) is also equal to the right-hand side −1 P wQw above. This shows (2.9.7.12) hence the proposition. Strictly speaking loc. cit. only applies to K -finite pseudo-Eisenstein series. However, the proof, which ultimately rests upon the computation of the constant terms of cuspidal Eisenstein series in their range of convergence, extends verbatim to general smooth pseudo-Eisenstein series. The skeptical reader can also assume that both ϕ and ϕ are K - finite since, by density of the respective subspaces of K -finite vectors, it suffices to check (2.9.7.12) for such functions. ∞ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 237 Corollary 2.9.7.2. — Let χ ∈ X(G) be a regular cuspidal datum, P be a standard parabolic subgroup of G and χ be the inverse image of χ in X(M ).Then, forevery ϕ ∈ S ([G]) and M P χ s ∈ H , the function >0 s−1/2 ϕ : m∈[M ]→ δ (m) ϕ (m) P,s P P P belongs to C ([M ]). Moreover, the family of linear maps χ P S ([G]) → C ([M ]),ϕ →ϕ χ χ P P,s for s ∈ H is holomorphic. >0 Proof. — Let ϕ ∈ S ([G]). By Lemma 2.9.3.1, ϕ is orthogonal to S ([M ]) for χ P,s P all s ∈ C. Hence, we just need to show that the map s → ϕ induces a holomorphic P,s function H → C([M ]). Note that by Lemma 2.5.13.1, ϕ ∈ C([M ]) for (s) 1. >0 P P,s P Thus, by the previous proposition, it suffices to show: 2 ∞ s (2.9.7.13)Ifψ ∈ L ([M ]) is such thatψ :=δ ψ ∈ C([M ]) for (s) 1then s →ψ P s P s induces a holomorphic function H → C([M ]). >0 P For X ∈ m ,wehave R(X)ψ =(2s − 1)ρ , X ψ +(R(X)ψ) s P s s (where we consider ρ as an element of the dual space m ) and it follows, by the equality (2.5.8.7), that it suffices to check that for every d > 0the map s → ψ induces a holomor- phic function H → L ([M ]). By Hölder inequality, for d > 0, (s)> 0and t 1 >0 P σ,d we have 1− (s)/t (s)/t ψ ψ ψ s 2 t 2 L L σ,d σ,td/ (s) and this implies ψ ∈ L ([M ]). The holomorphy of the map s ∈ H → ψ ∈ s P >0 s σ,d L ([M ]) is equivalent to the holomorphy of s ∈ H → ψ ,φ for every φ ∈ P >0 s M σ,d L ([M ]) but this follows from the usual criterion of analyticity for parameter inte- σ,−d gral and the domination |ψ | |ψ |+|ψ | s t t 1 2 for every s ∈ C and t > (s)> t . 2 1 Let n 1. For G = GL , regular cuspidal data admit the following explicit descrip- tion. Let χ ∈ X(GL ) be represented by a pair (M ,π) where n P M = GL ×···× GL P n n 1 k 238 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR is a standard Levi subgroup of GL and π =π ... π 1 k is a cuspidal automorphic representation of M (A) (with a central character trivial on A ). Then, χ is regular if and only if π =π for each 1 i < j k. i j 2.9.8. We now assume that G is a product of the form Res GL ×··· × K /F n 1 1 Res GL ,where K ,..., K are finite extensions of F. Let χ ∈ X(G) be a regular cus- K /F n 1 r r r G(A) pidal datum represented by a pair (M ,π) ∈ X(G).Set = Ind (π) = A (G) P P,π,cusp P(A) for the normalized smooth induction of π.Let B be a K-basis of as in Section 2.8.3. P,π For ϕ ∈ S([G]) and λ ∈ ia the series (2.9.8.14) ϕ = ϕ, E(ψ,λ) E(ψ,λ) ψ∈B P,π converges absolutely in T ([G]) for some N (that may a priori depend on λ). Indeed, this follows from the continuity of the linear map ψ ∈ A (G) → E(ψ,λ) ∈ T ([G]) P,π N for some N > 0 (see Section 2.7.3) together with Proposition 2.8.4.1 and the Dixmier- Malliavin theorem. The next theorem is a slight restatement of (part of) the main result of [Lap13]. We refer the reader to Section A.0.9 for the notion of Schwartz function valued in a TVS. Theorem 2.9.8.1 (Lapid).— There exists N > 0 such that for ϕ ∈ C([G]),the series (2.9.8.14) still makes sense (that is the scalar products ϕ, E(ψ,λ) are convergent) and converges ∗ ∗ in T ([G]) for every λ ∈ ia . Moreover, the function λ ∈ ia → ϕ ∈ T ([G]) is Schwartz, in N N P P λ particular absolutely integrable, and if ϕ ∈ C ([G]) we have the equality ϕ = ϕ dλ. ia Proof. — Note that G satisfies condition (HP) of [Lap13]: it is proven in loc. cit. that general linear groups satisfy (HP) and it is straightforward to check that prod- ucts of groups satisfying (HP) again satisfy (HP). The first part of the theorem is then a consequence of [Lap13, Proposition 5.1]. Indeed, by Dixmier-Malliavin we may as- sume that ϕ = R(f )ϕ where ϕ ∈ C([G]) and f ∈ C (G(A)).By loc. cit. the scalar product ϕ, E(ψ,λ) converges for every ψ ∈ B and there exists N > 0such that G P,π ψ → E(ψ,λ) factors through a continuous linear mapping → T ([G]) for every λ ∈ ia .As ϕ, E(ψ,λ) =ϕ , E(R(f )ψ,λ) ,ϕ ∈ B , G G P,π Note that in loc. cit. the Harish-Chandra Schwartz space C([G]) is denoted by S(G(F)\G(A)) THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 239 we deduce by Proposition 2.8.4.1 that the series (2.9.8.14) converges absolutely in ∗ ∗ T ([G]) for every λ ∈ ia . That the function λ ∈ ia → ϕ ∈ T ([G]) is Schwartz fol- N N P P λ lows similarly from [Lap13, Corollary 5.7]. The last part of the theorem is a consequence of [Lap13, Theorem 4.5] since χ regular implies that L ([G]) is included in the “induced 2 2 from cuspidal part” L ([G]) of L ([G]), with the notation of loc. cit. 2.10. Automorphic kernels 2.10.1. Let P ⊂ G be a standard parabolic subgroup. The right convolution by f ∈ S(G(A)) on each space of the decompositions (2.9.2.1) gives integral operators whose 0 0 kernels are respectively denoted by K (x, y),K (x, y),K (x, y) and K (x, y) f ,P f ,P,χ f ,P f ,P,χ where x, y ∈ G(A). If the context is clear, we shall omit the subscript f in the notation as well as the subscript P when P = G. The kernels are related by the following equality for all x, y ∈ G(A) 0 −1/2 K (x, y) = K (x, ay)δ (a) da. f ,P,χ P f ,P,χ Recall that we write [G] for the preimage of 0 by the map H :[G] → a . P P P Lemma 2.10.1.1. — There exists N > 0 such that for every weight w on [G] (see Section 0 P 2.4.3) and every continuous semi-norm . on T ([G] ), there exists a continuous semi-norm w,N w,N P 0 0 . on S(G(A)) such that for f ∈ S(G(A)) 0 −1 (2.10.1.1) |K (x, y)| f x w(x)w(y) , x, y∈[G] , f ,P,χ S P χ∈X(G) 0 N −1 1 (2.10.1.2) |K (x, y)| f x w(x)w(y) , x, y∈[G] , f ,P,χ P P χ∈X(G) and −1 (2.10.1.3) K (., y) f w(y) , y∈[G] . f ,P,χ w,N S P χ∈X(G) Proof. — Obviously, (2.10.1.3) implies (2.10.1.1)and (2.10.1.2) thus we will only prove this last estimate. First, we note that there exists N > 0 such that, for every weight w and every f ∈ S(G(A)) the operator R(f ) of right convolution by f induces a continuous map T ([G] ) → T ([G] ).Let f ∈ S(G(A)) and for χ ∈ X(G),let R (f ) be the compo- P w,N P χ sition of R(f ) with the “χ -projection” defined by Theorem 2.9.4.1. Then, from the third point of this theorem, we deduce the existence of N > 0 such that, for every weight w, 0 0 R (f ) sends T ([G] ) continuously into T ([G] ) and for every ϕ ∈ T ([G] ),the χ P w,N P P w 0 w 240 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR family (R (f )ϕ) is absolutely summable in T ([G] ). By the uniform bounded- χ χ∈X(G) w,N P ness principle, this implies the existence of a constant C> 0such that (2.10.1.4) R (f )ϕ Cϕ −1, for every ϕ ∈ T ([G] ) χ w,N 1,w P 0 w χ∈X(G) −1 where we recall that . is the norm defining the Banach space T ([G] ) see Section 1,w P −1 2.5.9 (we emphasize that the peculiar appearance of w as an index in the right hand side is purely the effect of a slight inconsistency in our notation). Using the fact that f → R (f )ϕ is continuous, we get, once again by the uniform boundedness principle, the existence of a continuous semi-norm . on S(G(A)) such that for f ∈ S(G(A)), (2.10.1.5) R (f )ϕ f ϕ −1, for every ϕ ∈ T ([G] ). χ w,N S 1,w P 0 w χ∈X(G) Applying (2.10.1.5)to ϕ = δ the Dirac measure at y ∈[G] gives the inequality y P (2.10.1.3) (note that R (f )δ = K (., y)). χ y f ,P,χ 2.10.2. Let P be a standard parabolic subgroup of G and let M = M .Let χ ∈ 0 ∞ X(G) and A (G) be the closed subspace of A (G) generated by left A -invariant P,disc P,χ,disc M functions whose class belongs to L ([G] ). We have a isotypical decomposition P,0 A (G) = ⊕A (G) P,π,disc P,χ,disc indexed by a set of discrete automorphic representations π of M(A).Let B be a K- P,χ basis of A (G) that is the union ∪ B over π as above of K-bases of A (G) π P,π P,π,disc P,χ,disc (see Section 2.8.3). In thesameway we define B =∪ B for any τ ∈ K. P,χ,τ π P,π,τ In the following, we add a subscript x or y to R(X) to indicate that this operator is applied to the variable x or y. The next lemma is an extension to Schwartz functions of results of Arthur, see [Art78,§4]. Lemma 2.10.2.1. — There exists a continuous semi-norm · on S(G(A)) and an integer N such that for all X, Y ∈ U(g ),all x, y ∈ G(A) and all f ∈ S(G(A)) we have −1 |P(M )| χ∈X(G) P ⊂P × | R (X)E(x, I (λ, f )ϕ,λ)R (Y)E(y,ϕ,λ)| dλ x P y G,∗ ia P ϕ∈B τ∈K P,χ,τ N N L(X)R(Y)f x y . G G THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 241 Moreover for all x, y ∈ G(A) and all χ ∈ X(G) we have 0 −1 K (x, y) = |P(M )| E(x, I (λ, f )ϕ,λ)E(y,ϕ,λ) dλ. P P f ,χ G,∗ ia P ⊂P ϕ∈B 0 P,χ Proof. — Let χ ∈ X(G) and τ ∈ K. Let P be a standard parabolic subgroup. For G,∗ f , g ∈ S(G(A)), λ ∈ ia and x, y ∈ G(A) we define: B (λ, x, y, f , g) = E(x, I (λ, f )ϕ,λ)E(y, I (λ, g)ϕ,λ) P,χ,τ P P ϕ∈B P,χ,τ and L (λ, x, y, f ) = E(x, I (λ, f )ϕ,λ)E(y,ϕ,λ). P,χ,τ P ϕ∈B P,χ,τ We denote by e the measure supported on K given by deg(τ) trace(τ(k))dk where dk is the Haar measure on K giving the total volume 1. We have e ∗ e = e .Let’s define τ τ τ ∨ ∨ −1 f by f (x) = f (x ) and let f = e ∗ f ∗ e . We shall use the following properties one can τ τ τ readily check: (2.10.2.6) R (X)R (Y)L (λ, x, y, f ) = L (λ, x, y, L(X)R(Y)f ), X, Y ∈ U(g ) x y P,χ,τ P,χ,τ ∞ (2.10.2.7) B (λ, x, y, f , g) = L (λ, x, y, f ∗ g ) P,χ,τ P,χ,τ (2.10.2.8) L (λ, x, x, f ∗ f ) = B (λ, x, x, f , f ) 0 P,χ,τ P,χ,τ 1 1 ∨ ∨ ∨ 2 2 (2.10.2.9) |L (λ, x, y, f ∗ g )| L (λ, x, x, f ∗ f ) L (λ, y, y, g ∗ g ) P,χ,τ P,χ,τ P,χ,τ (2.10.2.10) L (λ, x, y, f ) = L (λ, x, y, f ) P,χ,τ P,χ,τ τ (2.10.2.11) L (λ, x, y, f ) = 0,τ ∈ K,τ =τ. P,χ,τ τ From properties (2.10.2.6), we see that we are reduced to the case X = Y = 1. It suffices to show that there exists an integer N and that, for any normal open compact subgroup ∞ K K ⊂ K , there exists a continuous semi-norm · on the subspace S(G(A)) of K - 0 0 1 K bi-invariant Schwartz functions such that for all x, y ∈ G(A) and all f ∈ S(G(A)) we have N N T (x, y, f ) f x y G G where we introduce −1 T (x, y, f ) = |P(M )| |L (λ, x, y, f )| dλ, τ P P,χ,τ τ G,∗ ia χ∈X(G) P ⊂P P 0 242 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR and T (x, y, f ) = T (x, y, f ). τ∈K Indeed by the uniform boundedness principle and by (2.10.2.10)and (2.10.2.11)itiseasy T (x, y, f ) = T (x, y, f ).Let m 1 large enough and let K ⊂ K to conclude. Note that τ τ τ 0 be a normal open compact subgroup. By a slight variant of Proposition 2.8.4.3,wecan ∞ m find Z ∈ U(g ), g ∈ C (G(A)) and g ∈ C (G(A)) such that ∞ 1 2 c c • Z is invariant under K -conjugation; • g and g are invariant under K-conjugation; 1 2 0 ˆ • for any f ∈ S(G(A)) and any τ ∈ Kwe have: f = f ∗ g +(f ∗ Z) ∗ g 1 2 f = f ∗ g +(f ∗ Z) ∗ g . τ τ 1,τ τ 2,τ Thus the expression T (x, y, f ) is bounded by (the sums below are over τ ∈ K) 1 1 ∨ ∨ 2 2 (2.10.2.12) ( T (x, x, f ∗ f )) ( T (y, y, g ∗ g ) τ τ τ 1,τ τ 1,τ τ τ 1 1 ∨ ∨ 2 2 +( T (x, x,(f ∗ Z) ∗(f ∗ Z) )) ( T (y, y, g ∗ g ) . τ τ τ 2,τ τ 2,τ τ τ Arthur shows in [Art78, p. 931 and corollary 4.6] that for every N > 0 large enough there exists C> 0such that ∨ 2N (2.10.2.13) T (y, y, g ∗ g ) Cy τ i,τ i,τ G 1 ∨ for i = 1, 2and all y ∈ G(A) . At this point we are reduced to bound T (x, x, f ∗ f ) τ τ τ τ for any f ∈ S(G(A)). Following [Art78, p. 931] (the compactness of the support of f plays no essential role there), we get for any K-finite functions f ∈ S(G(A)): ∨ 0 (2.10.2.14) T (x, x, f ∗ f ) K (x, x). f ∗f Let N > 0 large enough. Using the weight w=· , we deduce from Lemma 2.10.1.1 that there exist N > N and a semi-norm · on S(G(A)) such that for all f ∈ S(G(A)) and all x, y ∈ G(A) we have 0 N −N K (x, y) f x y . f G G In particular, we deduce that for any N > 0 large enough there exists a continuous semi- norm · on S(G(A)) such that for all f ∈ S(G(A)) and all x ∈ G(A) 0 0 2 2 2N (2.10.2.15) K (x, x) = |K (x, y)| dy f x . f ∗f f G [G] 0 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 243 We fix N, C > 0 and a continuous semi-norm · on S(G(A)) such that (2.10.2.13)and (2.10.2.15) hold. Let’s define f = f and ˆ τ 0 τ∈K f =f ∗ Z +f 1 0 0 for f ∈ S(G(A)). These are again continuous semi-norms on S(G(A)). Using the majorization (2.10.2.14) for the K-finite function f and the majorizations (2.10.2.12), (2.10.2.13)and (2.10.2.15)weget T (x, y, f ) C (x y ) f G G 1 1 K for all x, y ∈ G(A) and f ∈ S(G(A)) . Then we can deduce the first majorization. To get the last equality, it is enough to observe that both members are defined and continuous on S(G(A)) and that the equality holds on the dense subset of K-finite and compactly supported functions (see [Art78, lemma 4.8]). 3. The coarse spectral expansion of the Jacquet-Rallis trace formula for Schwartz functions This section has two goals. The first, accomplished in Theorem 3.2.4.1,istoex- tend the coarse spectral expansion I = I of the Jacquet-Rallis trace formula for χ∈X(G) linear groups G (as proved in [Zyd20]) to the Schwartz space. The second, given in The- orem 3.3.9.1, is to provide spectral expressions more suitable for explicit calculations. An asymptotic estimate of modified automorphic kernels (stated in Theorem 3.3.7.1) plays a central role. 3.1. Notations 3.1.1. Let E/F be a quadratic extension of number fields. Let η be the quadratic character of A attached to E/F. Let n 1 be an integer. Let G = GL be the alge- n,F braic group of F-linear automorphisms of F .Let G = Res (G × E) be the F-group n E/F F obtained by restriction of scalars from the algebraic group GL of E-linear automor- n,E phisms of E .Wedenoteby c the Galois involution. We have a natural inclusion G ⊂ G which induces an inclusion A ⊂ A which is in fact an equality. The restriction map G G ∗ ∗ ∗ X (G ) → X (G ) gives an isomorphism a a . n G G 3.1.2. Let (B , T ) be a pair where B is the Borel subgroup of G of upper n n n n triangular matrices and T is the maximal torus of G of diagonal matrices. Let (B , T ) n n n n be the pair deduced from (B , T ) by extension of scalars to E and restriction to F: it is a n n pair of a minimal parabolic subgroup of G and its Levi factor. Let K ⊂ G (A) and K = K ∩ G (A) ⊂ G (A) be the “standard” maximal com- n n n n n n pact subgroups. Notice that we have K ⊂ K . n 244 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 3.1.3. The map P → P = Res (P × E) induces a bijection between the sets E/F F of standard parabolic subgroups of G and G whose inverse bijection is given by P → P = P ∩ G . ∗ ∗ Let P be a standard parabolic subgroup of G . The restriction map X (P) → X (P ) ∗ dim(a ) identifies X (P) with a subgroup of X(P ) of index 2 . It also induces an isomorphism a → a which fits into the commutative diagram: P P G (A) a G (A) a n P For any standard parabolic subgroups P ⊂ Q, the restriction of the function τ to a coincides with the function τ .However we have forall x ∈ G (A) Q Q ρ , H (x) = 2ρ , H (x) . P P P P Remark 3.1.3.1. —The map a → a does not preserve Haar measures. In fact, P P dim(a ) the pull-back on a of the Haar measure on a is 2 times the Haar measure on a . P P P ∞ ∞ In particular, although the groups A and A can be canonically identified, the Haar P P ∞ dim(a ) measure on A is 2 times the Haar measure on A . P P 3.1.4. We shall use the natural embeddings G ⊂ G and G ⊂ G where n n+1 n n+1 the smaller group is identified with the subgroup of the bigger one that fixes e and n+1 preserves the space generated by (e ,..., e ) where (e ,..., e ) denotes the canonical 1 n 1 n+1 n+1 basis of F . 3.1.5. Let G = G × G and G = G × G .Thus G is an F-subgroup of G. n n+1 n n+1 Let ι : G → G × G n n n+1 be the diagonal embedding. Let H be the image of ι (so H is isomorphic to G ). For an element g ∈ G(A) we will always write g and g for its components in n n+1 G (A) and G (A) respectively. n n+1 3.1.6. Let det (resp. det ) be the morphism G → G given by the determi- n n+1 m,F nant on the first (resp. second) component. Let η be the character G (A)→{±1} given by n+1 n η (g ) =η(det (g )) η(det (g )) g ∈[G ]. G n n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 245 3.1.7. Let K = K × K : it is a maximal compact subgroup of G(A).Wede- n n+1 fine pairs (P , M ) = (B × B , T × T ) and (P , M ) = (B × B , T × T ) of 0 0 n n+1 n n+1 0 0 n n+1 n n+1 minimal parabolic F-subgroups of G and G with their Levi components. As in Section 3.1.3, we have a bijection given by P → P = P ∩ G between the sets of standard parabolic subgroups of G and G . 3.1.8. Any parabolic subgroup P of G admits a decomposition P = P × P . n n+1 We introduce the “Rankin-Selberg set” F as the set of F-parabolic subgroups of G of RS the form P = P × P where P is a standard parabolic subgroup of G and P is n n+1 n n n+1 a semi-standard parabolic subgroup of G such that P ∩ G = P (here we use the n+1 n+1 n n embedding G → G ). n n+1 For P ∈ F , we set RS P = P ∩ H, P = P ∩ G , P = P ∩ G and P = P ∩ G . H n n+1 n n n+1 n+1 Let P, Q ∈ F be such that P ⊂ Q (from now on, when we write P ⊂ Q ∈ F we RS RS n+1 n+1 always implicitly assume that both P and Q are in F ). Then, we have a = a and RS P n+1 n+1 n+1 n+1 the characteristic functions τ , τ , σ coincide with τ , τ , σ respectively. P P P P n+1 n+1 P n+1 n+1 P n+1 n+1 We will only use the latter set of functions for convenience. We let n+1 Q dim(a ) n+1 =(−1) . + + We set a = a and a = a (see Section 2.2.9). n+1 B n+1 n+1 B n+1 3.1.9. For P, Q ∈ F , we define two weights on [G] by RS P −1 (g) = inf g γ g and P n+1 γ∈M (F)N (A) P P n+1 n+1 Q, Q Q n+1 n+1 d (g) = min(d (g ), d (g )) n n+1 P P P n+1 n+1 for g ∈[G] .Notethat (h) ∼ 1for h∈[H] . Moreover, by Lemma 2.4.4.2,wehave P P P Q, Q (3.1.9.1) d (g) d (g), for g ∈[G] . P P From Lemma 2.4.3.1, we also deduce: (3.1.9.2) For every weight w on [G ] , there exists N > 0such that n+1 P 0 n+1 w(g ) w(g ) (g) , for g =(g , g )∈[G] . n+1 n P n n+1 P 246 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 3.1.10. We will throughout consider a parameter T ∈ a that we assume most n+1 of the time to be “sufficiently positive”. More precisely, set d(T) = inf α(T) α∈ n+1 where we write for . Then, when we write “for T sufficiently positive”, we n+1 B n+1 mean “for T such that d(T) max(εT, C)”where . is an arbitrary norm on the real vector space a ,C > 0 is a large enough constant andε> 0 is an arbitrary (but in n+1 practice small enough) constant. 3.2. The coarse spectral expansion for Schwartz functions 3.2.1. Let f ∈ S(G(A)) be a Schwartz test function (see Section 2.5.2). 3.2.2. Let P be a parabolic subgroup of G. The right convolution by f on L ([G] ) gives an integral operator whose kernel is denoted by K .Let χ ∈ X(G). P P,f 2 2 Replacing L ([G] ) by its closed subspace L ([G] ) (see (2.9.2.1)), we get a kernel de- P P noted by K .Wehave K = K .If P = G, we omit the subscript P. If the P,χ,f P,f P,χ,f χ∈X(G) context is clear, we will also omit the subscript f . 3.2.3. A modified kernel. — For h ∈ H(A), g ∈ G (A), χ ∈ X(G) and T ∈ a we n+1 set (3.2.3.1) K (h, g ) = f ,χ τˆ (H (δ g ) − T )K (γ h,δg ), P P n P f ,P,χ P n+1 n+1 n n+1 P∈F γ∈P (F)\H(F) δ∈P (F)\G (F) RS H where • we recall that δ = (δ ,δ ) and g = (g , g ) according to the decomposition n n+1 n n+1 G = G × G ; n n+1 • in the notation H (δ g ),weconsider δ g as an element of G (A) (via the P n n n+1 n n n+1 embedding G → G ); n n+1 • T is defined as in Section 2.2.12. n+1 Remark 3.2.3.1. — This is the kernel used in [Zyd20] for compactly supported functions. Since we are considering a Schwartz function f , the sums over γ and δ are not necessarily finite. However, the component δ may be taken in a finite set depending on g (see [Art78] Lemma 5.1) and we can check that the sum defining K is also absolutely f ,χ convergent. To see this, we can use the following majorization: for every N > 0there exists N > 0such that N −N −N (3.2.3.2) K (h, g ) g h g , for (h, g )∈[H] ×[G ] . f ,P,χ N P P n P P n+1 P n+1 χ∈X(G) THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 247 This inequality is a simple consequence of Lemma 2.10.1.1 applied to the weight w = 2N −N . since g (g )g (recall that gg gg ). P P P P P n+1 n n n+1 3.2.4. Theorem 3.2.4.1. — Let T ∈ a . n+1 1. The map f ∈ S(G(A)) → |K (h, g )| dg dh f ,χ [H] [G ] χ∈X(G) is given by a convergent integral and defines a continuous semi-norm on S(G(A)). 2. As a function of T, the integral T T (3.2.4.3) I (f ) = K (h, g )η (g ) dg dh χ f ,χ [H] [G ] coincides with an exponential-polynomial function in T whose purely polynomial part is con- stant and denoted by I (f ). 3. The distributions I are continuous, left H(A)-invariant and right(G (A),η )-equivariant. χ G 4. The sum (3.2.4.4) I(f ) = I (f ) is absolutely convergent and defines a continuous distribution I. Remark 3.2.4.2. — The last statement is the “coarse spectral expansion” of the Jacquet-Rallis trace formula for G as introduced by Zydor in [Zyd20]. Proof. — All the statements but the continuity and the extension to Schwartz func- tions are proved in [Zyd20, Theorems 3.1 and 3.9] for compactly supported functions. The assertion 1 follows from the combination of majorization (3.3.7.10)ofThe- orem 3.3.7.1 below and Proposition 3.3.5.1 for the map (3.3.5.6). Note that assertion 1 implies the continuity of I . The assertion 2 can be proved as in [Zyd20, proof of Theo- rems 3.7]. One only needs the slight extension of assertion 1 to modified kernels defined in (3.7.4.2) associated to Levi subgroups (see comment above (3.7.4.2)). Continuity and assertion 4 are then the result of the explicit formula of [Zyd20, Theorem 3.7 ] which also holds for Schwartz functions. Finally one proves assertion 3 as in [Zyd20,Theorems 3.9]. 248 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 3.3. Auxiliary expressions for I 3.3.1. The goal of this section is to provide new expressions for the distribution I defined in Theorem 3.2.4.1. In this paper, we will use these expressions to explicitly compute I . The main results are subsumed in Theorem 3.3.9.1. Before giving the state- ments, we have to explain the main objects. Note that the proof of Theorem 3.3.9.1 relies on two other results, namely Theorem 3.3.7.1 and Proposition 3.3.8.1 whose proofs will be given in subsequent sections. On the other hand Theorem 3.3.7.1 was also used in the proof of Theorem 3.2.4.1. 3.3.2. The Ichino-Yamana truncation operator. — Let T ∈ a .In[IY15], Ichino- n+1 Yamana defined a truncation operator which transforms functions of uniform moderate growth on [G ] into rapidly decreasing functions on [G ]. By applying it to the right n+1 n component of [G]=[G ]×[G ], we get a truncation operator which we denote by n n+1 (the subscript r is for right). It associates to any function ϕ on [G] the function on [H] defined by the following formula: for any h∈[H]: T G (3.3.2.1) ( ϕ)(h) = τˆ (H (δh) − T )ϕ (δh) P P P G ×P r P n+1 n+1 n+1 n n+1 P∈F δ∈P (F)\H(F) RS H where we follow notations of Section 3.2.3. Note that in the expression H (δh),weview n+1 δh as an element of G (A) by the composition H→ G → G where the second map n+1 n+1 is the second projection. We denote by ϕ the constant term of ϕ along G × P . G ×P n n+1 n n+1 For properties of we shall refer to [IY15]. However for our purposes it is con- venient to state the following proposition. Proposition 3.3.2.1. — For any positive integers N and N , any open compact subgroup K ⊂ G(A ), there is an integer r > 0 and a finite family (X ) of elements of U(g ) of degree rsuch f i i∈I C that for any ϕ ∈ C (G(F)\G(A)/K ) we have for all h∈[H] T −N −N ϕ)(h) h sup x |(R(X )ϕ)(x)| r H G x∈G(A) i∈I Proof. — The result, a variant of Arthur’s Lemma 1.4 of [Art80], is proven in [IY15], Lemma 2.4. Let g ∈[G]. We shall apply the truncation operator to the map x ∈[G]→ K (x, g). After evaluating at h ∈[H] we get an expression we shall simply denote by f ,χ K (h, g). f ,χ Proposition 3.3.2.2. — For every N 0, there exists a continuous semi-norm . on S(G(A)) such that for f ∈ S(G(A)) T −N −N (3.3.2.2) | K (h, g)| f h g f ,χ S r H G χ∈X(G) THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 249 for h∈[H] and g ∈[G]. Proof. — Using the weight w =· on G(A) we deduce from Lemma 2.10.1.1 that N > 0and for all N 0 a continuous semi-norm · on S(G(A)) such that: N +N 0 −N (3.3.2.3) |K (x, y)| f x y , f ,χ χ∈X(G) for all x, y ∈ G(A). The right derivatives in the first variable of the kernel K (x, y) can f ,χ be expressed in terms of the kernel K associated to left derivatives of f .Let K be f ,χ 0 a compact subgroup of G(A ). We deduce from (3.3.2.3) and Proposition 3.3.2.1 that there is a continuous semi-norm · on S(G(A)) such that for any f ∈ S(G(A)) that is left-invariant under K we have T −N −N K (h, g)| f h g f ,χ r H G χ∈X(G) for all h∈[H] and g ∈[G]. This gives the proposition (a semi-norm on S(G(A)) is con- tinuous if and only if its restriction to S(G(A)) is continuous for every compact-open subgroup K ⊂ G(A )). 0 f 3.3.3. Convergence of a first integral. — It is given by the following proposition. Proposition 3.3.3.1. — The map (3.3.3.4) f ∈ S(G(A)) → | K (h, g )|dhdg , f ,χ [H]×[G ] χ∈X(G) is given by a convergent integral and defines a continuous semi-norm on S(G(A)). Proof. — It is a straightforward consequence of Proposition 3.3.2.2. n+1 3.3.4. Arthur function F (·, T).— For T ∈ a sufficiently positive, we shall use n+1 n+1 Arthur function F (·, T) (see [Art78] §6). It is the characteristic function of the set of x ∈ G (A) for which there exists a δ ∈ G (F) such that δx ∈ s (see Section n+1 n+1 G n+1 n+1 2.2.13)and , H (δx) − T ≤ 0for all ∈ .Recallalsothat F (·, T) descends 0 B n+1 to characteristic function of a compact subset of Z (A)G (F)\G (A).Wewillalso n+1 n+1 n+1 n+1 use the function F (·, T) defined relatively to G . n+1 3.3.5. Two other convergent integrals. Proposition 3.3.5.1. — The maps n+1 (3.3.5.5) f ∈ S(G(A)) → F (h, T)|K (h, g )| dhdg f ,χ [H]×[G ] χ∈X(G) 250 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR n+1 (3.3.5.6) f ∈ S(G(A)) → F (g , T)|K (h, g )| dhdg f ,χ [H]×[G ] χ∈X(G) are given by convergent integrals and define continuous semi-norms on S(G(A)), where as usual g = (g , g ) ∈ G (A) × G (A). n n+1 n n+1 n+1 Proof. — Observe that the restriction of F (·, T) to [H] is compactly supported. Then the convergence and the continuity of the integral (3.3.5.5)followfromthe ma- jorization (3.3.2.3). Using Lemma 2.10.1.1, (see also comments on inequality (3.2.3.2)), we see that for every N> 0 there exists N > 0 and a continuous semi-norm . on S(G(A)) such that −N N −N (3.3.5.7) K (h, g ) f g h g , f ,P,χ n G H n+1 n+1 χ∈X(G) for (h, g )∈[H]×[G ] and f ∈ S(G(A)). The convergence and continuity of the integral (3.3.5.6) result from the above in- n+1 equality and the fact that the restriction of F (·, T) to [G ] is compactly supported. 3.3.6. A second modified kernel. — Let f ∈ S(G(A)).For T ∈ a , χ ∈ X(G) and n+1 (h, g )∈[H]×[G ], we set T G (3.3.6.8) κ (h, g ) = τ (H (γ h) − T )K (γ h,δg ). P P P f ,P,χ f ,χ P n+1 n+1 n+1 P∈F γ∈P (F)\H(F) RS H δ∈P (F)\G (F) Remark 3.3.6.1. — Here the expression H (h) is understood as the value at n+1 h ∈ H(A) = G (A) ⊂ G (A) of the map H . The expression definingκ is absolutely n n+1 P n+1 f ,χ convergent as the sum over γ ∈ P (F)\H(F) is finite (see [Art78, Lemma 5.1]) and K H f ,P,χ is rapidly decaying in the second variable, see (3.3.2.3). 3.3.7. Asymptotics of the modified kernels. — We find that the modified kernels (3.2.3.1)and (3.3.6.8) are asymptotic for large parameters T to the kernels truncated by Arthur’s characteristic function. More precisely we have: Theorem 3.3.7.1. 1. For every N> 0, there exists a continuous semi-norm . on S(G(A)) such that S,N T G −NT −N −N n+1 (3.3.7.9) K (h, g ) − F (g , T)K (h, g ) e h g f f ,χ S,N f ,χ n H G χ∈X(G) THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 251 for f ∈ S(G(A)), (h, g )∈[H]×[G ] and T ∈ a sufficiently positive. In particular, n+1 for every N > 0, there exists a continuous semi-norm . on S(G(A)) such that S,N T G n+1 (3.3.7.10) K (h, g ) − F (g , T)K (h, g ) dhdg f ,χ f ,χ n [H]×[G ] χ∈X(G) −NT e f S,N for f ∈ S(G(A)). 2. For every N> 0, there exists a continuous semi-norm . on S(G(A)) such that S,N T G −NT −N −N n+1 (3.3.7.11) κ (h, g ) − F (h, T)K (h, g ) e h g f f ,χ S,N f ,χ H G χ∈X(G) for f ∈ S(G(A)), (h, g )∈[H]×[G ] and T ∈ a sufficiently positive. In particular, n+1 for every N > 0, there exists a continuous semi-norm . on S(G(A)) such that S,N T G n+1 (3.3.7.12) κ (h, g ) − F (h, T)K (h, g ) dhdg f ,χ f ,χ [H]×[G ] χ∈X(G) −NT e f S,N for f ∈ S(G(A)). Proof. — The proof of 3.3.7.1 will be given in Section 3.6 after some preparation provided by Sections 3.4 and 3.5. Note that the asymptotics (3.3.7.10), resp. (3.3.7.12), is an obvious consequence of (3.3.7.9), resp. (3.3.7.11). 3.3.8. Recall that we have built distributions I in Theorem 3.2.4.1 from the kernel K . The following Proposition shows that one could have defined I using the kernel κ . Proposition 3.3.8.1. — Let χ ∈ X(G) 1. The integral T T i (f ) = κ (h, g )η (g )dg dh, χ f ,χ [H]×[G ] is absolutely convergent for T ∈ a sufficiently positive. n+1 2. The map T → i (f ),when T is sufficiently positive, coincides with an exponential- polynomial whose purely polynomial part is constant and equal to I (f ). Proof. — The first assertion follows from Proposition 3.3.5.1 and the asymptotics (3.3.7.11)ofTheorem 3.3.7.1. The second assertion will be proved in Section 3.7. 252 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 3.3.9. We can now state the final theorem of the section. Theorem 3.3.9.1. — Let χ ∈ X(G) and f ∈ S(G(A)).For T ∈ a sufficiently positive n+1 T T let P (f ) and I (f ) be one of the following pairs of expressions: χ χ T G n+1 (3.3.9.13) I (f ) and F (g , T)K (h, g )η (g ) dhdg ; χ G χ n [H]×[G ] T G n+1 (3.3.9.14) i (f ) and F (h, T)K (h, g )η (g ) dhdg ; χ G [H]×[G ] T T (3.3.9.15) i (f ) and K (h, g )η (g )dhdg . χ G χ r [H]×[G ] T T 1. The integral defining I (f ) is absolutely convergent and the map T → P (f ) coincides χ χ for T ∈ a sufficiently positive with an exponential-polynomial whose constant term equals n+1 I (f ). 2. For every N> 0, there exists a continuous semi-norm . on S(G(A)) such that S,N T T −NT |I (f ) − P (f )| f e S,N χ χ for T ∈ a sufficiently positive and f ∈ S(G(A)). n+1 Proof. — For assertion 1, the statement about P (f ) is just the statement about T T I (f ) and i (f ) that has been given in Theorem 3.2.4.1 and Proposition 3.3.8.1.The χ χ three integrals are absolutely convergent by Propositions 3.3.3.1 and 3.3.5.1.Let’s prove assertion 2. The asymptotics for the first pair, resp. second, is a direct consequence of the asymptotics (3.3.7.10), resp. (3.3.7.12), of Theorem 3.3.7.1. So it remains to prove the third asymptotics. In fact, for every N > 0, there exists a continuous semi-norm . S,N on S(G(A)) such that T G −NT n+1 K (h, g ) − F (h, T)K (h, g ) dhdg f e χ χ S,N [H] [G ] for f ∈ S(G(A)) and T sufficiently positive. This can be proved as in [IY15, proof of Proposition 3.8] and is left to the reader. So the third asymptotics follows from the second one. 3.4. Auxiliary function spaces and smoothed constant terms 3.4.1. For G∈{G , G , G , G , H, G},welet T (G) be the space of tuples n n+1 F n n+1 RS ( ϕ) ∈ T ([G] ) P P∈F P RS P∈F RS THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 253 such that for every P ⊂ Q ∈ F ,wehave RS ϕ −( ϕ) ∈ S Q([G] ) P Q P P where we have set P = P ∩ G, Q = Q ∩ G, the weight θ is defined by n+1 d if G∈{G , G }, n n+1 n+1 n+1 θ = P d if G∈{G , G }, n n+1 ⎪ n+1 d if G∈{H, G}, and we refer the reader to Section 2.5.11 for the definition of the function spaces Q Q S Q([G] ). By Lemma 2.4.4.2, in every case we have θ d and therefore, by Propo- P P sition 2.5.14.1, the above condition is equivalent to the following: there exists N > 0 such that for every r 0and X ∈ U(G ) (where we denote by G the Lie algebra of ∞ ∞ G(F ⊗ R)), we have 0 −r (3.4.1.1) (R(X) ϕ)(g) −(R(X) ϕ)(g) g θ (g) , for g ∈ P(F)\G(A). P Q r,X P P Q Q n+1 Note that, by Lemma 2.4.4.2 again, for every P ⊂ Q ∈ F ,wehave d (h) ∼ d (h) for RS P P n+1 h∈[H] so that under the identification H = G ,wehave P n (3.4.1.2) T ([H]) = T ([G ]). F F n RS RS We define similarly T ([G]) as the space of tuples RS ( ϕ) ∈ S ([G] ) P P∈F P RS P P∈F RS such that for every P ⊂ Q ∈ F ,wehave RS ϕ −( ϕ) ∈ S Q,([G] ) P Q P P Q, where the weights and d have been defined in Section 3.1.9. Similarly, using (3.1.9.1) and Proposition 2.5.14.1, the condition above is equivalent to: there exists N > 0such that for every r 0and X ∈ U(g ),wehave 0 ∞ N Q, 0 −r (3.4.1.3) (R(X) ϕ)(g) −(R(X) ϕ)(g) g d (g) , for g ∈ P(F)\G(A). P Q r,X P P 3.4.2. Obviously, T ([G]) (resp. T ([G])) embeds as a closed subspace of RS F RS T ([G] ) × S Q([G] ) P P P∈F P⊂Q∈F RS RS 254 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR (resp. S ([G] ) × S Q,([G] )) P P P∈F P⊂Q∈F RS RS and as such it inherits a LF topology. More precisely, for every N > 0, the inverse im- age T ([G]) (resp. T ([G]))of T ([G] )× S Q ([G] ) (resp. of F ,N N P P RS F ,N P∈F P⊂Q∈F d ,N RS RS RS S ([G] ) × S Q, ([G] )))in T ([G]) (resp. in T ([G])) inher- ,N P P F P∈F P P⊂Q∈F RS F RS RS d ,N RS its a strict LF topology (by Lemma 2.5.4.1)and we endow T ([G]) (resp. in T ([G])) RS F RS with the locally convex direct limit of these topologies. By Theorem 2.9.4.1,wehave (3.4.2.4)For every ϕ = ( ϕ) ∈ T ([G]) (resp. ϕ = ( ϕ) ∈ T ([G])), the P P∈F F P P∈F RS RS RS RS family χ ∈ X(G) →ϕ :=( ϕ ) P χ P∈F RS is absolutely summable in T ([G]) (resp. in T ([G]))withsum ϕ. RS F RS For the purpose of the next proposition, we recall that T ([G]) denotes the space of Radon measures of moderate growth on [G] (see Section 2.5.9). Proposition 3.4.2.1. 1. For every ϕ ∈ T ([H]) and f ∈ S(G(A)),the family P ∈ F → R(f )ϕ RS P belongs to T ([G]). RS 2. Let G∈{G , G}.Then, forevery ϕ ∈ T ([G ]) and f ∈ S(G(A)),the family n+1 P ∈ F → R(f )ϕ , RS P wherewehaveset P = P ∩ G , belongs to T ([G]). RS 3. For every (G , G )∈{(G , G ),(G, H)} and every ( ϕ) ∈ T ([G ]),the fam- 1 2 n P P∈F F 1 n RS RS ily of restrictions P ∈ F → ϕ | , RS P [G ] wherewehaveset P = P ∩ G , belongs to T ([G ]). 2 2 F 2 RS 4. For every( ϕ) ∈ T ([G]) and( ψ) ∈ T ([G ]), the family of products P P∈F P P∈F F n+1 RS RS RS RS P ∈ F → (g , g )∈[G] → ϕ(g , g ) ψ(g ) RS n n+1 P P n n+1 P n+1 belongs to T ([G]). RS Proof. — 1. By Lemma 2.5.11.1, it suffices to prove that for every P ⊂ Q ∈ F , RS we have: THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 255 • is bounded on the support of ϕ ; P P Q, • d is bounded on the support of ϕ −(ϕ ) . P Q P P H H The first requirement is clear as ϕ is supported on [H] ⊂[G] and is bounded P P P P H H on [H] . For the second one, we need to show the existence of C > 0such that ϕ and P P H H Q, (ϕ ) coincide on {g ∈[G] | d (g)> C}. By adjunction, for every ψ ∈ S ([G] ),we Q P P P H P have Q Q ϕ ,ψ =ϕ , E (ψ) and (ϕ ) ,ψ =ϕ , E (ψ) . P P Q Q Q P P Q Q H H P H H H P H Thus, it suffices to show that for C> 0 sufficiently large, for every function ψ ∈ S ([G] ) Q, supported in {g ∈[G] | d (g)> C} we have Q Q E (ψ) | = E (ψ). [H] P Q P With the notation of Section 2.4.4, this is in turn equivalent to: Q, (3.4.2.5) There exists C > 0such that for every g ∈ P(F)N (A)\G(A) with d (g)> C, π (g)∈[H] implies g ∈ P (F)N (A)\H(A). Q H Q Q H H Q, Q n+1 Writing g = (g , g ), the condition d (g)> C is equivalent to (g , g ) ∈ω [> C] n n+1 n n+1 P P n+1 P n+1 whereas π (g) ∈[H] (resp. g ∈ P (F)N (A)\H(A)) is equivalent to π (g ) = Q H Q n Q H H Q n+1 n+1 π (g ) (resp. g = g ). Thus, the claim follows directly from Lemma 2.4.4.1.4. n+1 n n+1 n+1 2. Let P ⊂ Q ∈ F and set P = P ∩ G, Q = Q ∩ G, P = P ∩ G , Q = Q ∩ G . RS Q Q Since G∈{G , G},wehave θ = d . Thus, by a similar argument, we are reduced to n+1 P P show: (3.4.2.6) There exists C > 0such that for every g ∈ P(F)N (A)\G(A) with d (g)> C, π (g)∈[G ] implies g ∈ P (F)N (A)\G (A). Q Q By Lemma 2.4.4.1 1. and Lemma 2.4.4.2, there exists> 0 such that the set {g ∈ P (F)N (A)\G (A) | d (g )>} surjects onto [G ] . Moreover, by Lemma 2.4.4.1 4., there exists C > 0such that for Q Q P P g, g ∈ P(F)N (A)\G(A) with d (g)> Cand d (g )> , π (g) = π (g ) implies g = g . P P Q Q The claim follows. 3. follows from the characterization (3.4.1.1) since, by Lemma 2.4.4.2,for P ⊂ Q ∈ Q Q 1 2 F we have θ ∼ θ . Similarly, 4. follows readily from the characterizations (3.4.1.1), RS P P 1 2 Q, Q, Q n+1 (3.4.1.3) as, by definition of d ,wehave d (g) d (g ) for g ∈[G] . n+1 P P P P n+1 256 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 3.4.3. For P ∈ F , ϕ ∈ S ([G] ) and ψ ∈ T ([G ] ), we define a function RS P n+1 P P n+1 ϕ,ψ ∈ T ([G ] ) by n P ϕ,ψ (g ) = ϕ(g , g )ψ(g ), g ∈[G ] . n n n+1 n+1 n n P [G ] n+1 P n+1 Note that the integral is absolutely convergent and the resulting function of uniform moderate growth by (3.1.9.2) applied to the weight w=. for N sufficiently large. n+1 Proposition 3.4.3.1. — Let ( ϕ) ∈ T ([G]) andψ ∈ T ([G ]). Then, the fam- P P∈F RS F n+1 RS ily P ∈ F → ϕ,ψ RS P P n+1 belongs to T ([G ]). F n RS Proof. — By Dixmier-Malliavin, we may assume that ( ϕ) = (R(f ) ϕ ) P P∈F P P∈F RS RS ∞ 7 for some ( ϕ ) ∈ T ([G]) and f ∈ C (G (A)). Then, we have ϕ,ψ = P P∈F n+1 P P RS F c RS n+1 ∨ ∨ −1 ϕ , R(f )ψ where f (g) = f (g ). Therefore, by Proposition 3.4.2.1 2. and 4., up P P n+1 to replacing ϕ by the product g ∈[G] → ϕ (g) R(f )ψ (g ), P P P n+1 n+1 it suffices to prove that: (3.4.3.7) The family of functions P ∈ F → p ( ϕ) : g ∈[G ] → ϕ(g , g )dg RS n∗ P n n P P n n+1 n+1 [G ] n+1 P n+1 belongs to T ([G ]). F n RS Let P ⊂ Q ∈ F . By the characterization (3.4.1.1), we need to show the existence of RS N > 0such that for everyX ∈ U(g ) and r 0, we have 0 n,∞ R(X) ϕ(g , g )dg − R(X) ϕ(g , g )dg P n n+1 n+1 Q n n+1 n+1 [G ] [G ] n+1 P n+1 Q n+1 n+1 N Q 0 n+1 −r g d (g ) r,X n n P P n n+1 for g ∈ P (F)N (A)\G (A). For notational simplicity, we will prove this for X = 1but n n Q n it will be clear from the argument that we can choose the same exponent N for every X ∈ U(g ). n,∞ More precisely, this follows from applying the Dixmier-Malliavin theorem to the continuous smooth Fréchet representation T ([G]) for suitable N > 0 and compact-open subgroup J ⊂ G(A ). F ,N RS THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 257 n+1 Let C > 0 and set ω = ω [> C] (see Section 2.4.4 for the notation). Since ϕ ∈ n+1 n+1 −1 S ([G] ),by (3.1.9.2) applied to the weights w = (d ) and . , there exists P P P P n+1 n+1 N > 0such that for everyN> 0and r 0, we have Q Q N +N n+1 n+1 1 −N r −r (3.4.3.8) ϕ(g) g g d (g ) d (g ) , for g ∈[G] . P N,r n n+1 n+1 n P P P P P n n+1 n+1 n+1 Q P n+1 n+1 Similarly, up to enlarging N and since . . , d d on P (F)N (A)\G (A) 1 Q P n Q n n n P Q n n+1 n+1 (see (2.4.1.2), (2.4.4.19)), for every N> 0and r 0, we have N +N P Q −N n+1 r n+1 −r (3.4.3.9) ϕ(g) g g d (g ) d (g ) , Q N,r n n+1 n+1 n P Q Q P n n+1 n+1 n+1 for g ∈ P(F)N (A)\G(A). P Q P n+1 n+1 n+1 By Lemma 2.4.4.1 1. and 2., d and d are bounded from above outside π (ω) Q P Q n+1 n+1 n+1 n+1 and π (ω) respectively. Thus, from (3.4.3.8)and (3.4.3.9) we deduce that there exists n+1 N > 0such that for every r 0, the two functions g → ϕ(g , g )dg and n P n n+1 n+1 n+1 [G ] \π (ω) n+1 P n+1 n+1 g → ϕ(g , g )dg n Q n n+1 n+1 n+1 [G ] \π (ω) n+1 Q n+1 Q n+1 N Q n+1 2 −r are essentially bounded by g d (g ) for g ∈ P (F)N (A)\G (A). Thus, it only n n n n Q n P P n n n+1 remains to estimate the difference ϕ(g , g )dg − ϕ(g , g )dg P n n+1 n+1 Q n n+1 n+1 Q P n+1 n+1 π (ω) π (ω) P Q n+1 n+1 which, by Lemma 2.4.4.1 and provided C is sufficiently large, is equal to (3.4.3.10) ϕ(g , g ) − ϕ(g , g )dg . P n n+1 Q n n+1 n+1 P Q n+1 n+1 As d ∼ d and . ∼. on ω (see Lemma 2.4.4.1 2.), by (3.4.3.8), (3.4.3.9), Q P Q P n+1 n+1 n+1 n+1 for every r 0and N > 0, we have Q Q N +N n+1 n+1 1 −N r −r ϕ(g) − ϕ(g) g g d (g ) d (g ) P Q N,r n n+1 n+1 n P P P n P n+1 n+1 n+1 for g ∈ P (F)N (A)\G (A) ×ω. Combining this with the characterization (3.4.1.3)and n Q n the equality Q, Q Q Q n+1 n+1 −1 n+1 d (g) max(1, d (g )d (g ) ) = d (g ) for g ∈[G] , n n+1 n P P P P P n+1 n+1 n+1 258 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR it readily follows that there exists N > 0such that for everyN > 0and r 0, we have N +N Q −N n+1 −r ϕ(g) − ϕ(g) g g d (g ) . P Q N,r n n+1 n P P P n n+1 n+1 This in turn implies that for some N > 0 the integral (3.4.3.10) is essentially bounded 4 n+1 −r by g d (g ) for every r 0 and this ends the proof of (3.4.3.7) hence of the n n P P n n+1 proposition. 3.5. A relative truncation operator 3.5.1. Let G∈{H, G , G }.For ϕ = ( ϕ) ∈ T ([G]) and T ∈ a , we set n P P∈F F n+1 n RS RS T,G G ϕ(g) = τ (H (δg) − T ) ϕ(δg) P P P P n+1 n+1 n+1 P∈F δ∈P(F)\G(F) RS for g ∈[G]. We also set n+1 ⎨ F (g, T) ϕ(g) if G = G , T,G ϕ(g) = n+1 F (g, T) ϕ(g) if G∈{H, G }. G n Recall that S ([G]) stands for the space of functions of rapid decay on [G] (see Section 2.5.6). T,G 0 Theorem 3.5.1.1. — For T sufficiently positive, we have ϕ ∈ S ([G]).Moreprecisely, for every c > 0 and N> 0, there exists a continuous semi-norm . on T ([G]) such that F ,c,N F RS RS T,G T,G −cT (3.5.1.1) ϕ − ϕ e ϕ ∞,N F ,c,N RS for ϕ ∈ T ([G]) and T ∈ a sufficiently positive (where the semi-norms (. ) are as in F n+1 ∞,N N>0 RS Section 2.5.6). Proof. — Note that, by the identification (3.4.1.2), the statement of the theorem for G = H is exactly equivalent to the statement for G = G which itself can be obtained from the case G = G by a change of base-field from F to E. Therefore, we shall content ourself to give the proof when G = G . G T,G n+1 n As the restriction of F (., T) to [G ] is compactly supported, we have ϕ ∈ S ([G ]) and the first assertion of the theorem follows from the second. Now, as in the beginning of the proof of [Zyd20, proposition 2.8], using [Zyd20, lemme 2.1], for T sufficiently positive we obtain T,G P n n+1 ϕ(g) = F (δg, T) P⊂Q∈F δ∈P (F)\G (F) RS n n n+1 ×σ (H (δg) − T ) ϕ(δg) P P P,Q P n+1 n+1 n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 259 where we have set ϕ(g) = ϕ(g), for g ∈ P (F)\G (A), P,Q R n n P⊂R⊂Q n+1 the sum running over parabolic subgroups R ∈ F such that P ⊂ R ⊂ Q. Since σ = RS n+1 n+1 1and σ = 0for P G (see [Art78, §6, p. 940]), it follows that n+1 T,G T,G n n ϕ(g) − ϕ(g) P n+1 n+1 = F (δg, T)σ (H (δg) − T ) ϕ(δg) P P P,Q P n+1 n+1 n+1 δ∈P (F)\G (F) PQ∈F RS n n for g ∈[G ]. n 0 0 Let us fix P Q ∈ F . Since E sends S ([G ] ) into S ([G ]) continuously, RS n P n P n it suffices to show that for every c > 0and N > 0, there exists a continuous semi-norm . on T ([G ]) such that F ,c,N F n RS RS −cT −N (3.5.1.2) ϕ(g) e g ϕ P,Q F ,c,N P RS for ϕ ∈ T ([G ]),T ∈ a sufficiently positive and g ∈ P (F)\G (A) such that F n n+1 n n RS P n+1 n+1 F (g, T)σ (H (g) − T ) = 0. P P P n+1 n+1 n+1 Up to conjugacy, we may assume that P and Q are standard. We will need the following lemma which summarizes part of the analysis performed in [Art78,§6, §7]. Lemma 3.5.1.2. — There exists r > 0 such that % & e min d (g) and P ,α n+1 Q P n+1 n+1 α∈ \ 0 0 % & g max d (g) P P ,α n n+1 Q P n+1 n+1 α∈ \ 0 0 for every T ∈ a sufficiently positive and g ∈ P (F)\G (A) satisfying n+1 n n n+1 n+1 F (g, T)σ (H (g) − T ) = 0. P P P n+1 n+1 n+1 1 ∞ 1 1 Proof. — Writing g as zg where z ∈ A and g ∈ G (A) we have d (g) ∼ n+1 P ,α G n+1 n+1 1 1 d (g ) and there exists r > 0such that g g . Thus, it suffices to prove P ,α 0 P n+1 n P n+1 that the same statement holds for g ∈ P (F)\G (A) . Moreover, we may assume that n+1 n+1 g belongs to some Siegel domain s for [G ] . Then, by [Art78, Eq. (7.7)], P n+1 P n+1 n+1 P n+1 n+1 F (g, T)σ (H (g) − T ) = 0 P P P n+1 n+1 n+1 260 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Q P n+1 n+1 implies α, H (g) > α, T for every α ∈ \ and therefore, as T is sufficiently 0 0 positive, also (3.5.1.3)min α, H (g) >T Q P n+1 n+1 α∈ \ 0 0 for some constant> 0. The first inequality of the lemma follows for r > 0 large enough. n+1 For the second inequality, by [Art78, Corollary 6.2], the condition σ (H (g) − P n+1 n+1 T ) = 0 implies n+1 ⎛ ⎞ ⎝ ⎠ (3.5.1.4) H (g) − T c 1 + max α, H (g) − T P P 0 0 n+1 n+1 n+1 α∈ n+1 n+1 whereas the condition F (g, T) = 0 implies n+1 (3.5.1.5) H (g) c T n+1 for suitable constants c , c > 0and whereH (g) denotes the projection of H (g) to 0 1 0 P Q Q P n+1 n+1 n+1 n+1 a .As is the restriction of \ to a ,from(3.5.1.4)and (3.5.1.5)we 0 P 0 0 n+1 n+1 get % & H (g) c 1+T+ max α, H (g) 0 3 0 Q P n+1 n+1 α∈ \ 0 0 for some constant c > 0. Combining this with (3.5.1.3), we finally obtain % & H (g) c 1 + max α, H (g) 0 4 0 Q P n+1 n+1 α∈ \ 0 0 for some c > 0. Exponentiating then gives the second inequality of the lemma. n+1 n+1 We are now in position to prove (3.5.1.2). Let α ∈ \ .For every 0 0 n+1 parabolic subgroup P ⊂ R ⊂ Qsuch thatR ∈ F and α ∈ , there exists a RS n+1 n+1 α α unique parabolic subgroup P ⊂ R ⊂ QwithR ∈ F and = \{α}.As RS 0 0 ( ϕ) ∈ T ([G ]), by the inequality (3.4.1.1), there exists N > 0such that for every P P∈F F n 0 RS RS r 0 we can find a continuous semi-norm . on T ([G ]) with F ,r,α F n RS RS 0 −r | ϕ(g)| | ϕ(g) − ϕ(g)| g ϕ d (g) P,Q R R F ,r,α R ,α P RS n+1 P⊂R⊂Q P⊂R⊂Q R R n+1 n+1 α∈ α∈ 0 0 n+1 n+1 for ϕ ∈ T ([G ]) and g ∈ P (F)\G (A).Now,if F (g, T)σ (H (g) − T ) = 0 F n n n P P RS P n+1 n+1 n+1 for a T ∈ a that is sufficiently positive, the first inequality of the above lemma implies n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 261 n+1 that g ∈ ω [> C] for some fixed constant C > 0. In particular, by Lemma 2.4.4.1.2, n+1 we have d (g) ∼ d (g) for every P ⊂ R ⊂ Q and the above estimate becomes R ,α P ,α n+1 n+1 N −r | ϕ(g)| g d (g) ϕ . P,Q r P ,α F ,r,α P n+1 RS Q P n+1 n+1 Taking the minimum of the right hand side over α ∈ \ and using Lemma 0 0 3.5.1.2 once again, we deduce the estimates (3.5.1.2) and this ends the proof of the theo- rem. 3.6. Proof of Theorem 3.3.7.1 3.6.1. We prove 1., the proof of 2. being similar and left to the reader. Set f (g) = −1 f (g ) for g ∈ G(A).For T ∈ a , we consider the two operators n+1 T T 0 0 0 L , P : T ([H]) ⊗ T ([G ]) → S ([G ]) f f n+1 n defined as the following compositions R (f )⊗Id T 0 0 0 L : T ([H]) ⊗ T ([G ]) −−−−−→ T ([G]) ⊗ T ([G ]) f n+1 F n+1 RS .,. Res −→ T ([G ]) −→ T ([G ]) F n F RS RS n T,G −−−→ S ([G ]) and R (f )⊗Id T 0 0 0 P : T ([H]) ⊗ T ([G ]) −−−−−→ T ([G]) ⊗ T ([G ]) f n+1 F n+1 RS .,. Res −→ T ([G ]) −→ T ([G ]) F n F RS RS T,G −−−→ S ([G ]) ∨ ∨ respectively. Here, R (f ), .,. and Res denote the operators ϕ →(R(f )ϕ ) (see P P∈F H RS Proposition 3.4.2.1 1.), ( ϕ) ⊗ψ →( ϕ,ψ ) (see Proposition 3.4.3.1)and P P∈F P P P∈F RS RS n+1 T,G ( ϕ) → ( ϕ | ) (see Proposition 3.4.2.1 3.) respectively whereas and P P∈F P [G ] P∈F RS n RS T,G are the truncation operators defined in the previous section. By the closed graph theorem (which is valid for linear maps between LF spaces ∨ T,G T,G n n see [Gro55, théorème B p. 17]), each of the operators R (f ),Res, and is readily seen to be continuous: indeed, it suffices to check that the compositions of these operators with the linear maps corresponding to the “pointwise evaluations” are Q R 8 n+1 n+1 Note that ω [> C]⊂ω [> C] P P n+1 n+1 262 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR continuous which, in each case, is straightforward to check. Similarly, the operator .,. is T T separately continuous. In particular, L and P are separately continuous bilinear maps. f f T G n+1 We claim that the functions K and F (., T)K are the kernels of the operators T T 0 0 L and P respectively that is: for every ϕ ⊗ ψ ∈ T ([H]) ⊗ T ([G ]) and g ∈[G ], f f n+1 n n we have T T (3.6.1.1)L (ϕ ⊗ψ)(g ) = K (h; g , g )ϕ(h)ψ(g ) f n f n n+1 n+1 [H]×[G ] n+1 and T G n+1 (3.6.1.2)P (ϕ ⊗ψ)(g ) = F (g , T)K (h; g , g )ϕ(h)ψ(g ). f n n n n+1 n+1 [H]×[G ] n+1 Let us show (3.6.1.1), the proof of (3.6.1.2) being similar (and actually easier). Un- 0 0 folding the definitions, for ϕ ⊗ψ ∈ T ([H]) ⊗ T ([G ]) and g ∈[G ],wehave n+1 n n T G (3.6.1.3) L (ϕ ⊗ψ)(g ) = τ (H (δ g ) − T ) P P n P f n P n+1 n+1 n n+1 P∈F δ ∈P (F)\G (F) RS n n n ×R(f )ϕ ,ψ (δ g ). P P n H n n+1 Moreover, R(f )ϕ ,ψ (δ g ) P P n n+1 = R(f )ϕ (δ g , g )ψ (g ) P n P H n n+1 n+1 n+1 [G ] n+1 P n+1 = K ∨ (δ g , g ; h)ϕ (h)ψ (g ) f ,P n P P n n+1 H n+1 n+1 [G ] [H] n+1 P n+1 = K (h;δ g , g )ϕ (h)ψ (g ) f ,P n P P n n+1 n+1 n+1 [G ] ×[H] n+1 P n+1 = K (γ h;δ g ,δ g )ϕ(h)ψ(g ) f ,P n n+1 n n+1 n+1 [G ]×[H] n+1 γ∈P (F)\H(F) δ ∈P (F)\G (F) n+1 n+1 n+1 and plugging this back into (3.6.1.3)gives (3.6.1.1). T T 3.6.2. For χ ∈ X(G), we introduce variants L ,P of the previous op- f ,χ f ,χ ∨ ∨ erators by replacing in their definitions the operator R (f ) by R (f ) : ϕ → ∨ ∨ ∨ (R ∨(f )ϕ ) where R ∨(f ) denotes the composition of R(f ) with the “projec- χ P P∈F χ H RS ∨ T T tion to the χ -component” defined in Theorem 2.9.4.1. We show similarly that L ,P f ,χ f ,χ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 263 0 0 are separately continuous and that for ϕ ⊗ψ ∈ T ([H])⊗ T ([G ]) and g ∈[G ],we n+1 n n have T T (3.6.2.4)L (ϕ ⊗ψ)(g ) = K (h; g , g )ϕ(h)ψ(g ) f ,χ n f ,χ n n+1 n+1 [H]×[G ] n+1 and T G n+1 (3.6.2.5)P (ϕ ⊗ψ)(g ) = F (g , T)K (h; g , g )ϕ(h)ψ(g ). f ,χ f ,χ n n n n+1 n+1 [H]×[G ] n+1 Let N> 0. Note that, by (3.4.2.4),Theorem 3.5.1.1 and the continuity of .,. and 0 0 Res, for every ϕ ⊗ψ ∈ T ([H]) ⊗ T ([G ]),wehave n+1 T T −NT L (ϕ ⊗ψ) − P (ϕ ⊗ψ) e ∞,N N,ϕ,ψ f ,χ f ,χ χ∈X(G) T T for T ∈ a sufficiently positive. Moreover, by continuity of L and P ,eachofthe n+1 f ,χ f ,χ T T 0 semi-norms ϕ ⊗ ψ → L (ϕ ⊗ ψ) − P (ϕ ⊗ ψ) is bounded on T ([H]) ⊗ ∞,N f ,χ f ,χ N T ([G ]) by a constant multiple of ϕ ψ where we recall that . is the 1,−N 1,−N 1,−N N n+1 0 0 norm on the Banach spaces T ([H]), T ([G ]) (and the peculiar transition from the N N n+1 index N to −N is again due to a slight inconsistency of notation). Thus, by the uniform boundedness principle, we have T T −NT L (ϕ ⊗ψ) − P (ϕ ⊗ψ) e ϕ ψ ∞,N N 1,−N 1,−N f ,χ f ,χ χ∈X(G) 0 0 for ϕ ⊗ ψ ∈ T ([H]) ⊗ T ([G ]) and T ∈ a sufficiently positive. By (3.6.2.4)and n+1 N N n+1 (3.6.2.5), applying the above inequality to the Dirac measures ϕ = δ and ψ = δ , h g n+1 where h∈[H] and g ∈[G ],gives n+1 n+1 T G n+1 |K (h; g , g ) − F (g , T)K (h; g , g )| f ,χ f ,χ n n+1 n n n+1 χ∈X(G) −NT −N −N −N e h g g H n+1 n G G n+1 n+1 for (h, g , g )∈[H]×[G ]×[G ]. This inequality is precisely the content of Theorem n n+1 n n+1 3.3.7.1 1. except that we still have to argue that the implicit constant can be taken to be a continuous semi-norm on S(G(A)). Using the uniform boundedness principle once again, it suffices to check that, for every (h, g )∈[H]×[G ], the functional T G n+1 f ∈ S(G(A)) → K (h, g ) − F (g , T)K (h, g ) f ,χ f ,χ n T T =(L − P )(δ ⊗δ )(g ) h g f ,χ f ,χ n n+1 264 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR is continuous. However, this just follows from the continuity of f → R (f )δ ∈ T ([G]) which can be readily inferred from the continuity of right convolution as well RS as of the “χ -projection” of Theorem 2.9.4.1. 3.7. Proof of Proposition 3.3.8.1 3.7.1. Set ρ =ρ −ρ ∈ a P P n+1 n+1 n for P ∈ F and note that ρ = 0 if and only if P = G. RS G G n+1 n+1 3.7.2. By [Art81, §2], there exist functions ! on a × a ,for P ∈ F , RS P P P n+1 n+1 n+1 that are compactly supported in the first variable when the second variable stays in a compact and such that G n+1 (3.7.2.1) τ (H − X) = τ (H)! (H, X) n+1 Q P Q n+1 n+1 P⊂Q∈F RS for every P ∈ F and H, X ∈ a . Then, by [Zyd20, lemme 3.5] for every Q ∈ F , RS P RS n+1 the function ρ (H) X ∈ a → p (X) := e ! (H, X)d H Q Q n+1 Q n+1 n+1 n+1 is an exponential-polynomial. 3.7.3. We set −ρ ,H (m ) Q n Q n+1 f (m) = e p (H (m ))f (m) Q Q Q n Q n+1 for Q ∈ F and m∈[M ] where RS Q ρ ,H (m) Q Q f (m) = e f (k muk )η (k )dudk dk . Q H G H K ×K N (A) H Q We note that f ∈ S(M (A)). Q Q 3.7.4. Distributions attached to Levi subgroups. — Let Q ∈ F .Wenow recall the RS variant of [Zyd20, Théorème 3.8] for the Levi M .For T ∈ a and f ∈ S(M (A)),we Q n+1 Q set M ,T Q Q Q n+1 K (m , m ) = τ (H (δ m ) H P n f ,χ P P n+1 n n+1 Q⊃P∈F γ∈(P (F)∩M (F))\M (F) RS H Q Q H H δ∈(P (F)∩M (F))\M (F) Q Q − T )K (γ m ,δm ) P f ,P∩M ,χ H n+1 Q THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 265 for (m , m )∈[M ]×[M ]. Then, it can be shown in the same way as part 1. of Theorem H H 3.3.7.1 that, for T sufficiently positive, the following expression converges M ,T M ,T Q Q (3.7.4.2) I (f ) = K (m , m )η (m )dm dm , H G H f ,χ A \[M ]×[M ] Q,H ∞ ∞ ∞ where A = A ∩ A is embedded diagonally in M (A) × M (A).Moreover, by the Q,H Q Q same proof as [Zyd20, Théorème 3.7] (see also [Zyd20, Remarque 3.8], [Zyd18,Remar- M ,T que 4.9]), the function T → I (f ) is an exponential-polynomial with exponents in the set {ρ − ρ | Q ⊃ P ∈ F } (but not necessarily with a constant purely polynomial RS P Q part). Note that the composition of the embedding a = a ∩ a → a with the Q,H Q Q Q H n+1 G G n+1 n+1 projection a a yields an isomorphism a a whose Jacobian we denote Q Q,H n+1 Q Q n+1 n+1 by c . 3.7.5. Proof of Proposition 3.3.8.1.— It follows from the next lemma, the fact that M ,T T T ' ' I (f ) = I (f ) and the shape of the exponents of I (f ) recalled below. G Q χ χ Lemma 3.7.5.1. — For T ∈ a sufficiently positive, we have n+1 ρ (T) M ,T Q ' (3.7.5.3) i (f ) = c e I (f ). Q χ Q Q∈F RS Proof. — From the definition (3.3.6.8)of κ together with the identity (3.7.2.1) f ,χ applied to H = H (δ g ) − T and X = H (δ g ) − H (γ h),weobtain P n P P n P n+1 n n+1 n+1 n n+1 κ (h, g ) = ! (H (δ g ) − T , H (δ g ) Q n Q Q n f ,χ Q n+1 n n+1 n+1 n n+1 γ∈Q (F)\H(F) Q∈F RS H δ∈Q (F)\G (F) Q,T − H (γ h))K (γ h,δg ) n+1 f ,χ for (h, g )∈[H]×[G ], where we have set Q,T Q Q n+1 K (x, y) = τ (H (δ y ) P n n f ,χ P P n+1 n+1 Q⊃P∈F γ∈P (F)\Q (F) RS H H δ∈P (F)\Q (F) − T )K (γ x,δy) P f ,P,χ n+1 for (x, y)∈[H] ×[G ] . It follows that Q Q Q,T i (f ) = K (h, g )! (H (g ) − T , Q Q χ f ,χ Q n+1 n n+1 n+1 [H] ×[G ] Q∈F H RS H (g ) − H (h))η (g )dg dh. Q Q G n+1 n n+1 266 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR By the Iwasawa decompositions H(A) = Q (A)K and G (A) = Q (A)K , the summand H H corresponding to Q can be rewritten as Q,T K (m k , m k )! (H (m ) − T , H H Q Q f ,χ Q n+1 n n+1 n+1 [M ]×[M ] K ×K Q H H (m ) − H (m )) Q Q H n+1 n+1 −2ρ ,H (m ) −2ρ ,H (m ) Q Q Q H H H η (m k )e e dk dk dm dm . G H H We readily check that, for all (m , m )∈[M ]×[M ],wehave H H M ,T Q,T Q ρ ,H (m )+H (m ) Q Q Q H K (m k , m k )η (k )dk dk = e K (m , m ). H H G H H f ,χ f ,χ K ×K Using that ρ = 2ρ and ρ , H (m ) − 2ρ , H (m ) =ρ , H (m ) ,wefi- Q Q Q Q H Q Q H Q H H H n+1 nally obtain M ,T ρ ,H (m ) T Q H n+1 i (f ) = K (m , m )η (m )e H G f ,χ [M ]×[M ] Q∈F H RS ×! (H (m ) − T , H (m ) − H (m ))dm dm Q Q Q Q H H Q n+1 n n+1 n+1 n n+1 n+1 M ,T = K (m , m )η (m ) H G f ,χ A \[M ]×[M ] Q,H H Q Q∈F RS ρ ,H (m )−H (m ) Q H Q Q n+1 n+1 × e ρ ,H (am ) Q n+1 × e ! (H (am ) − T , Q Q Q n+1 n n+1 n+1 Q,H H (m ) − H (m ))dadm dm Q Q H H n+1 n n+1 M ,T ρ (T) = c e K (m , m )η (m ) Q H G f ,χ A \[M ]×[M ] Q,H Q Q∈F H RS ρ ,H (m )−H (m ) Q H Q n+1 n+1 × e × p (H (m ) − H (m ))dm dm . Q Q Q H H n+1 n n+1 To conclude, it suffices to remark that M ,T Q ρ ,H (m )−H (m ) Q H Q Q n+1 n+1 K (m , m )e p (H (m ) − H (m )) H Q Q Q H f ,χ n+1 n n+1 M ,T = K (m , m ) f ,χ for (m , m )∈[M ]×[M ]. H H THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 267 4. Flicker-Rallis period of some spectral kernels The goal of this section is to get the spectral expansion of the Flicker-Rallis integral of the automorphic kernel attached to a linear group and a specific cuspidal datum (called in Section 4.3.2 ∗-regular). This is achieved in Theorem 4.3.3.1. It turns out that the decomposition is discrete and is expressed in terms of some relative characters. 4.1. Flicker-Rallis intertwining periods and related distributions 4.1.1. Notations. — In all this section, we will fix an integer n 1 and we will use notations of Sections 3.1.1 to 3.1.3. Since n will be fixed, we will drop the subscript n from the notation: G = G ,B = B etc. So we do not follow notations of Section 3.1.5: n n we hope that it will cause no confusion. 4.1.2. Flicker-Rallis periods. — Let π be a cuspidal automorphic representation of ∞ ∗ G(A) with central character trivial on A . We shall denote by π the conjugate-dual representation of G(A). We shall say that π is self conjugate-dual if π π and that π is G -distinguished, resp. (G,η)-distinguished, if the linear form (called the Flicker-Rallis period) (4.1.2.1) ϕ → ϕ(h) dh, resp. ϕ(h)η(det(h)) dh [G ] [G ] 0 0 does not vanish identically on A (G).Then π is self conjugate-dual if and only if π is either G -distinguished or (G,η)-distinguished. However it cannot be both. This is related to the well-known factorisation of the Rankin-Selberg factorisation L(s,π ×π ◦ c) where c is the Galois involution of G(A) in terms of Asai L-functions and to the fact that the residue at s = 1 of the Asai L-functions is expressed in terms of Flicker-Rallis periods (see [Fli88]). 4.1.3. In this section, we will focus on the period in (4.1.2.1) related to distinction. However it is clear that all the results hold mutatis mutandis for the period related to η- distinction. 4.1.4. Let P = MN be a standard parabolic subgroup (with its standard decom- position). Let π be an irreducible cuspidal automorphic representation of M with central character trivial on A . It will be convenient to write M = G ×···× G with n +···+ n = n. Accord- n n 1 r 1 r ingly we have π = σ ... σ where σ is an irreducible cuspidal representation of 1 r i G . i 268 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 4.1.5. Let ϕ ∈ A (G). The parabolic subgroup P = P ∩ G of G has the fol- P,π lowing Levi decomposition M N where M = M ∩ G . We then define the following integral which is a specific example of a Flicker-Rallis intertwining period introduced by Jacquet-Lapid-Rogawski (see [JLR99] section VII, note that our definition of A (G) is P,π slightly different from theirs), J(ϕ) = ϕ(g) dg A M (F)N (A)\G (A) Clearly we get a G (A)-invariant continuous linear form on A (G).Notethat J does P,π not vanish identically if and only if each component σ is G (F)-distinguished. In this case, we have π =π . 4.1.6. Let Q ∈ P(M). As recalled in Section 2.2.11, there is a unique pair (Q,w) such that the conditions are satisfied: −1 • Q =wQw is the standard parabolic subgroup in the G-conjugacy class of Q; • w ∈ W(P; Q ). G,∗ Let λ ∈ a .Wehave M(w,λ)ϕ ∈ A (G) if λ is outside the singular hyperplanes of M ,wπ P,C the intertwining operator. We shall define (4.1.6.2) J (ϕ,λ) = J(M(w,λ)ϕ) as a meromorphic function of λ. 4.1.7. Let g ∈ G(A).Let’s definefor ϕ,ψ ∈ A (G) P,π (4.1.7.3) B (g,ϕ,ψ,λ) = E(g,ϕ,λ) · J (ψ,−λ), Q Q G,∗ as a meromorphic function of λ ∈ a . In fact, by the basic properties of Eisenstein P,C G,∗ series and intertwining operators, there exists an open subset ω ⊂ a which is the P,C G,∗ complement of a union of hyperplanes of a such that: P,C G,∗ • ω contains ia . • for all ϕ,ψ ∈ A (G),the map λ → B (g,ϕ,ψ,λ) is holomorphic on ω and P,π Q π gives for each λ ∈ω a continuous sesquilinear form in ϕ and ψ . 4.1.8. Let f ∈ S(G(A)), g ∈ G(A) and Q ∈ P(M). Let’s introduce the distribu- tion (4.1.8.4) J (g,λ, f ) = B (g, I (λ, f)ϕ,ϕ,λ) Q,π Q P ϕ∈B P,π THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 269 where B is a K-basis of A (G) (see Section 2.8.3 and λ ∈ ω (see Section 4.1.7 P,π P,π π for the notation ω ). It follows from Proposition 2.8.4.1 that J (g,λ) is a continuous π Q,π distribution on S(G(A)). 4.1.9. A (G, M)-family. Proposition 4.1.9.1. — The family (J (g,λ, f )) is a (G, M)-family in the sense Q,π Q∈P(M) of Arthur (see [Art81]): namely each map G,∗ λ ∈ a → J (g,λ, f ) Q,π G,∗ is smooth on ia (and even holomorphic on ω ) and for adjacent elements Q , Q ∈ P(M) we have π 1 2 (4.1.9.5) J (g,λ, f ) = J (g,λ, f ) Q ,π Q ,π 1 2 G,∗ on the hyperplane of ia defined by λ,α = 0 where α is the unique element in ∩(− ). Q Q M 1 2 Proof. — Let’s prove the holomorphy of J (g,λ, f ) on ω .Let C ⊂ G(A ) be Q,π π f a compact subset and let K ⊂ K be a compact-open normal subgroup such that f ∈ S(G(A), C, K ).Let ω ⊂ ω be a compact subset with a non-empty interior de- 0 π noted by ω . According to Proposition 2.8.4.1, there is a continuous semi-norm on S(G(A), C, K ) such that for λ ∈ω and any φ ∈ S(G(A), C, K ) we have: 0 0 (4.1.9.6) |J (g,λ,φ)| φ. Q,π On the other hand, we have J (g,λ, f ) = J (g,λ, f ) where f = e ∗ f ∗ e as in Q,π ˆ Q,π τ τ τ τ τ∈K the proof of Lemma 2.10.2.1. Indeed we have J (g,λ, f ) = B (g, I (λ, f)ϕ,ϕ,λ). Q,π τ Q P ϕ∈B P,π,τ Since the sum on the right-hand side is finite, the map λ → J (g,λ, f ) is holomorphic Q,π τ on ω . Using (4.1.9.6), we get for all λ ∈ω |J (g,λ, f )| f < ∞. Q,π τ τ ˆ ˆ τ∈K τ∈K Thus on ω , we observe that J (g,λ, f ) is a normally convergent series whose general Q,π term, namely J (g,λ, f ), is holomorphic. In this way λ → J (g,λ, f ) itself is holo- Q,π τ Q,π morphic on ω and on ω . Then let’s prove the second condition. Let Q , Q ∈ P(M) be such that ∩ 1 2 Q G,∗ (− ) is a singleton {α}.Letλ ∈ ia such λ,α = 0. For i = 1, 2let Q be a standard 2 P i −1 parabolic subgroup and w ∈ W(M, Q ) be such that Q =w Q w .Let β =w α ∈ i i i 1 Q i i i 1 270 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR and let s the “elementary symmetry” associated to β.Thenwehave w = s w .Let β 2 β 1 ϕ,ψ ∈ A (G). Clearly it suffices to check the equality: P,π E(g,ϕ,λ) ·(J(M(w ,λ)ψ) = E(g,ϕ,λ) · J(M(w ,λ)ψ). 1 2 Using the functional equations of intertwining operators and Eisentein series, we have M(w ,λ) = M(s ,w λ)M(w ,λ) and E(g,ϕ,λ) = E(g, M(w ,λ)ϕ,w λ).Thusuptoa 2 β 1 1 1 1 change of notations (replace P by Q ), we may assume that Q = Pand thus w = 1and 1 1 α =β . We are reduced to prove (4.1.9.7) E(g,ϕ,λ) · J(ψ) = E(g,ϕ,λ) · J(M(s ,λ)ψ) on the hyperplane λ,α = 0. The symmetry s acts on M as a transposition of two consecutive blocks of M say G and G .Notethat M(s ,λ) = M(s , 0).Thenwehave n n α α i i+1 even a stronger property: J(ψ) = J(M(s , 0)ψ) if n = n or if n = n but σ σ (see lemma 8.1 case 1 of [Lap06]). Assume that i i+1 i i+1 i i+1 ∗ ∗ n = n and σ σ . The case where σ σ is trivial (J is zero) so we shall also assume i i+1 i i i+1 i that σ σ .Then M(s , 0)ψ =−ψ ([KS88] proposition 6.3) and since s (λ) = λ we i α α have E(g,ϕ,λ) = 0so (4.1.9.7)isclear. 4.1.10. Majorization. — We will use the following proposition which results from Lapid’s majorization of Eisenstein series (see [Lap06] proposition 6.1 and section 7). For the convenience of the reader, we sketch a proof. Proposition 4.1.10.1. — Let f ∈ S(G(A)). The map λ → J (g,λ, f ) belongs to the Q,π G,∗ Schwartz space S(ia ).Moreover f → J (g,·, f ) Q,π G,∗ is a continuous map from S(G(A)) to S(ia ) equipped with its usual topology. Proof. — Let C ∈ G(A ) be a compact subset and K ⊂ K be an open-compact f 0 subgroup such that f ∈ S(G(A), C, K ).For any α,β > 0, we define an open subset ω 0 α,β G,∗ G,∗ of a which contains ia by P,C P,R G,∗ −β ω ={λ ∈ a | (λ)<α(1+(λ)) }. α,β P,C By the arguments in the proof of proposition 6.1 of [Lap06], one sees that there exist α,β > 0such that ω is included in the open set ω of Section 4.1.7. In particular, α,β π λ → J (g,λ, f ) is holomorphic on ω . Using Cauchy formula to control derivatives, Q,π α,β it suffices to prove the following majorization: there exists a continuous semi-norm · THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 271 on S(G(A), C, K ) andanopensubset ω ⊂ ω such that for any integer N 1there 0 α,β π exists c > 0so that for all f ∈ S(G(A), C, K ) and all λ ∈ω 0 α,β (4.1.10.8) |J (g,λ, f )| c . Q,π (1+λ) Let m 1 be a large enough integer. Following the notations of Proposition 2.8.4.3,we can write f = f ∗ g +(f ∗ Z) ∗ g ;weget 1 2 J (g,λ, f ) = E(g, I (λ, f )ϕ,λ)J (I (−λ, g )ϕ,λ) Q,π P Q P ϕ∈B P,π + E(g, I (λ, f ∗ Z)ϕ,λ)J (I (−λ, g )ϕ,λ) P Q P ϕ∈B P,π By a slight extension to Schwartz functions of Lapid’s majorization (see [FLO12] remark C.2 about [Lap06] proposition 6.1), the expression 2 1/2 ( |E(g, I (λ, f )ϕ,λ)| ) ϕ∈B P,π and the same expression where f is replaced by f ∗ Z satisfy a bound like (4.1.10.8). Using Cauchy-Schwartz inequality, we are reduced to bound in λ (recall that g is independent of f ) ∨ 2 1/2 (4.1.10.9) ( |J (I (−λ, g )ϕ,λ)| ) . Q P ϕ∈B P,π −1 −1 Let w be such that wQw is standard and w ∈ W(P,wQw ). At this point we will use the notations of the proof of Proposition 2.8.4.1. There exists c > 0 and an integer r such that for all ϕ ∈ A (G) we have P,π ∨ ∨ |J (I (−λ, g )ϕ,λ)|=|J(M(w,λ)I (−λ, g )ϕ)| Q P P i i cM(w,λ)I (−λ, g )ϕ P r where ϕ =R(1+ C ) ϕ . Then we need to bound the operator norm of the inter- r K Pet twining operator M(w,λ). Using the normalization of intertwining operators, the bounds of normalizing factors [Lap06] lemma 5.1 and Müller-Speh’s bound on the norm of normalized intertwining operators (see [MS04] proposition 4.2 and the proof of propo- sition 0.2), we get c > 0, N ∈ N and α,β > 0such that for all τ ∈ K , λ ∈ ω and 1 ∞ α,β ϕ ∈ A (G, K ,τ) we have P,π 0 ∨ N ∨ M(w,λ)I (−λ, g )ϕ c (1 +λ ) I (−λ, g )ϕ . P K ,r 1 τ P r i 0 i 272 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Using the same kind of arguments as in the proof of Proposition 2.8.4.1 (see also remark (2.8.4.2)), one shows that there exist α,β > 0such that (4.1.10.9)isbounded independently of λ ∈ ω . α,β 4.2. A spectral expansion of a truncated integral 4.2.1. Let χ ∈ X(G) be a cuspidal datum. We shall use the notation of Section 2.10.2. In particular, f ∈ S(G(A)) and K is the attached kernel. 4.2.2. Let’s consider a parameter T as in Section 2.2.12. Following Jacquet- Lapid-Rogawski (see [JLR99]), we introduce the truncation operator that associates to a function ϕ on [G] the following function of the variable h∈[G ]: T dim(a ) (4.2.2.1) ( ϕ)(h) = (−1) τˆ (H (δh) − T )ϕ (δh) P P P P δ∈P (F)\G (F) where the sum is over standard parabolic subgroup of G (those containing B) and ϕ is the constant term along P. Recall that P = G ∩ P. 0 T 4.2.3. We shall define the mixed truncated kernel K : the notation means χ m that the mixed truncation is applied to the second variable. This is a function on G(A)× G (A). To begin with we have: Lemma 4.2.3.1. — For (x, y) ∈ G(A) × G (A),wehave: 0 T (K )(x, y) χ m −1 = |P(M )| E(x, I (λ, f )ϕ,λ) E(y,ϕ,λ) dλ. P P G,∗ ia B⊂P P ϕ∈B P,χ Proof. — As y ∈ G (A), the mixed truncation is defined by a finite sum of constant terms of K (x,·) (in the second variable). The only point is to permute the sum over ϕ and the operator . In fact using the continuity properties of Eisenstein series (see [Lap08, theorem 2.2]) and properties of mixed truncation operator (in particular a vari- antoflemma 1.4of[Art80]), we can conclude as in the proof of Proposition 2.8.4.1. Lemma 4.2.3.2. — For any integer N, there exists a continuous semi-norm · on S(G(A)) 1 1 and an integer N such that for all X ∈ U(g ),all x ∈ G(A) and y ∈ G (A) ,all f ∈ S(G(A)) ∞ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 273 we have −1 (4.2.3.2) |P(M )| χ∈X(G) B⊂P × | (R(X)E)(x, I (λ, f )ϕ,λ) E(y,ϕ,λ)| dλ G,∗ ia ˆ ϕ∈B P,χ,τ τ∈K N −N L(X)f x y . [G] [G] Proof. — By the basic properties of the mixed truncation operator (see lemma 1.4 of [Art80]and also [LR03] proof of lemma 8.2.1), for any N, N > 0 there exists a finite family (Y ) of elements of U(g ) such the expression (4.2.3.2)ismajorized by thesum i i∈I ∞ −N over i ∈ Iof y times the supremum over g ∈ G (A) of [G] −N −1 g |P(M )| | (R(X)E)(x, I (λ, f )ϕ,λ) P P [G] G,∗ ia χ∈X B⊂P ϕ∈B ˆ P,χ,τ τ∈K × R(Y )E(g,ϕ,λ)| dλ. Then the lemma is a straightforward consequence of Lemma 2.10.2.1. Proposition 4.2.3.3. — For all x ∈ G(A) and χ ∈ X(G), we have 0 T −1 (K )(x, y) dy = |P(M )| E(x, I (λ, f )ϕ,λ) P P χ m G,∗ [G ] ia B⊂P ϕ∈B P,χ E(y,ϕ,λ) dy dλ. [G ] Proof. — First one decomposes the sum over B as a sum over τ ∈ K of finite P,χ sums over B . Then, by the majorization of Lemma 4.2.3.2 we can permute the in- P,χ,τ tegration over [G ] (which amounts to integrating over [G ] ) and the other sums or integrations in the expression we get in Lemma 4.2.3.1. 4.3. The case of ∗-regular cuspidal data 4.3.1. We shall use the notations of Section 4.2. 4.3.2. ∗-Regular cuspidal datum. — We shall say that a cuspidal datum χ ∈ X(G) is ∗-regular if for any representative (M,π) of χ and w ∈ W(M) such that wπ is isomor- ∗ ∗ phic to π or π we have w = 1. Let’s denote by X (G) the subset of ∗-regular cuspidal data. 274 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR With the notations of 4.1.4, we see that (M,π) is ∗-regular if and only if for all 1 i, j r such that n = n one of the equalities σ =σ or σ =σ implies that i = j . i j i j i 4.3.3. The next theorem is the main result of the section. Theorem 4.3.3.1. — Let f ∈ S(G(A)),let χ ∈ X(G) and let K be the associated kernel. For any g ∈ G(A), one has: 1. We have (4.3.3.1) K (g, h) dh = K (g, h) dh [G ] [G ] where both integrals are absolutely convergent. 2. If moreover χ ∈ X (G), we have, for any representative (M ,π) of χ (where P is a standard parabolic subgroup of G), − dim(a ) K (g, h) dh = 2 J (g, f ) χ P,π [G ] where one defines (see (4.1.8.4)) J (g, f ) = J (g, 0, f ). P,π P,π In particular, the integral vanishes unless π is self conjugate dual and M -distinguished where P = G ∩ P. The assertion 1 follows readily from Lemma 2.10.1.1, Fubini’s theorem and the ∞ ∞ fact that the Haar measure on A is twice the Haar measure on A (see Remark 3.1.3.1). G G The rest of the section is devoted to the proof of assertion 2 of Theorem 4.3.3.1. The main steps are Propositions 4.3.4.1 and 4.3.7.1. 4.3.4. A limit formula. — We shall use the notation lim f (T) to denote the T→+∞ limit of f (T) when α, T →+∞ for all α ∈ . Proposition 4.3.4.1. — Under the assumptions of Theorem 4.3.3.1 (but with no regularity condition on χ ), we have 0 T lim (K )(g, h) dh = 2 K (g, h) dh. χ m T→+∞ [G ] [G ] Proof. — Let’s denote F (·, T) the function defined by Arthur relative to G and its maximal compact subgroup K (see [Art78,§6] and[Art85, lemma 2.1]). It is the char- acteristic function of a compact of [G ] . Using the fact that h → K (g, h) is of uniform χ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 275 moderate growth (see Lemma 2.10.1.1), we can conclude by a variant of [Art85]theorem 3.1 (see also in the same spirit [IY15] proposition 3.8) that G 0 0 T lim F (h, T)K (g, h) −(K )(g, h) dh = 0. χ χ m T→+∞ [G ] We have lim F (h, T) = 1. Thus we deduce by Lebesgue’s theorem and the abso- T→+∞ lute convergence of the right-hand side of (4.3.3.1). G 0 0 lim F (h, T)K (g, h) dh = K (g, h) dh χ χ T→+∞ [G ] [G ] 0 0 The proposition follows by (4.3.3.1). 4.3.5. Let χ ∈ X (G).Let P be the set of standard parabolic subgroups such that there exists a cuspidal automorphic representation π of M such that (M ,π) in P P the equivalence class defined by χ.InSection 2.10.2, we defined the space A (G). P,χ,disc Since χ ∈ X (G),itisnon-zeroonlyif P ∈ P . Let P be a standard parabolic subgroup and let (M ,π) be a pair in χ.For any P ∈ P , by multiplicity-one theorem, we have P 1 χ A (G) = A (G). P ,wπ P ,χ,disc 1 w∈W(P,P ) In the following we set M = M . 1 P Let P ∈ P and g ∈ G(A). With the notations of Section 4.1 (see eq. (4.1.8.4)), for 1 χ G,∗ all Q ∈ P(M ),all λ ∈ ia we define P ,R J (g,λ, f ) = J (g,λ, f ). Q,χ Q,wπ w∈W(P,P ) It’s a continuous linear form on S(G(A)). 4.3.6. Proposition 4.3.6.1. — For all χ ∈ X (G) and all g ∈ G(A), we have 0 T (4.3.6.2) (K )(g, h) dh χ m [G ] − dim(a ) 2 exp(−λ, T ) = J (g,λ, f ) dλ. Q,χ G,∗ |P(M )| θ (−λ) P ia Q P ∈P Q∈P(M ) 1 χ 1 1 Proof. — This is an obvious consequence of the definitions, Proposition 4.2.3.3 and Lemma 4.3.6.2 below. 276 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR ∗ G Lemma 4.3.6.2. — Let χ ∈ X (G).Let P ∈ P and ϕ ∈ A . We have for all λ ∈ ia 1 χ P ,χ 1 P G exp(λ, T ) − dim(a ) E(y,ϕ,λ) dy = 2 J (ϕ,λ) . θ (λ) [G ] Q Q∈P(M ) Proof. — This is simply a rephrasing in our particular situation of a key result of Jacquet-Lapid-Rogawski (see [JLR99] theorem 40). Indeed, because χ is ∗-regular, The- orem 40 of ibid. can be stated as: exp((wλ) , T ) − dim(a ) E(y,ϕ,λ) dy = 2 J(M(w,λ)ϕ) θ (wλ) [G ] Q (Q,w) where the sum is over pair (Q,w) where Q is a standard parabolic subgroup and w ∈ W(P , Q). 4.3.7. Proposition 4.3.7.1. — Let χ ∈ X (G) and let (M ,π) be a representative where P is a standard parabolic subgroup of G.Wehave: 0 T − dim(a ) lim (K )(g, h) dh = 2 J (g, f ) P,π χ m T→+∞ [G ] where one defines (4.3.7.3) J (g, f ) = J (g, 0, f ). P,π P,π Proof. — We start from the expansion (4.3.6.2)ofProposition 4.3.6.1.For each P ∈ P ,let M = M .The family (J (g,λ, f )) is a (G, M )-family of Schwartz 1 χ 1 P Q,χ Q∈P(M ) 1 1 1 G,∗ functions on ia : this is a straightforward consequence of Propositions 4.1.9.1 and 4.1.10.1.By[Lap11] Lemma 8, we have: exp(−λ, T ) lim J (g,λ, f ) dλ = J (g, 0, f ) Q,χ P ,χ G,∗ T→+∞ θ (−λ) ia Q∈P(M ) By definition and Lemma 4.3.7.2 below, one has: J (g, 0, f ) = J (g, 0, f ) P ,χ P ,wπ 1 1 w∈W(P,P ) =|W(P, P )|J (g, 0, f ). 1 P,π Since |P(M )|= |W(P, P )| we get the expected limit. P 1 P ∈P 1 χ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 277 Lemma 4.3.7.2 (Lapid).— For any w ∈ W(P, P ), we have J (g, 0, f ) = J (g, 0, f ). P ,wπ P,π Proof. — By definition, we have J (g, 0, f ) = E(g, I (0, f )ϕ, 0) · J (ϕ) P ,wπ P 1 1 1 ϕ∈B P ,wπ where J is the linear form on A defined in Section 4.1.5 and B is any K-basis 1 P ,wπ P ,wπ 1 1 of A . Now, the intertwining operator M(w, 0) induces a unitary isomorphism from P ,wπ A to A , which sends K-bases to K-bases. Thus one has P,π P ,wπ J (g, 0, f ) = E(g, M(w, 0)I (0, f )ϕ, 0) · J (M(w, 0)ϕ) P ,wπ P 1 ϕ∈B P,π = J (g, 0, f ). P,π The last equality results from the two equalities: • E(g, M(w, 0)I (0, f )ϕ, 0) = E(g, I (0, f )ϕ, 0); P P • J (M(w, 0)ϕ) = J(ϕ) where J is the linear form on A defined in Section 4.1.5. 1 P,π The first one is the functional equation of Eisenstein series and the second one is a con- sequence of case 1 of lemma 8.1 of [Lap06]. 5. The ∗-regular contribution in the Jacquet-Rallis trace formula The goal of this section is to compute the contribution I of the Jacquet-Rallis trace formula for ∗-regular cuspidal data χ . This is achieved in Theorem 5.2.1.1 below. It turns out that for such χ the contribution I is discrete and equal (up to an explicit constant) to a relative character define in Section 5.1 built upon Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. 5.1. Relative characters 5.1.1. We will use the notations of Section 3.1. We emphasize that unlike Section 4 the group G denotes G × G and so on. n n+1 5.1.2. Let χ ∈ X(G) be a cuspidal datum and (M,π) be a representative where M is the standard Levi factor of the standard parabolic subgroup P of G. Recall that we 278 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR have introduced a character η of G (A) (see Section 3.1.6). On A (G), we introduce G P,π the linear form J defined by (5.1.2.1) J (ϕ) = ϕ(g)η (g) dg, ∀ϕ ∈ A (G) η G P,π A M (F)N (A)\G(A) where M = M ∩ G and P = P ∩ G . This is a slight variation of that defined in Section 4.1.5. We shall say that π is (M,η )-distinguished if J does not vanish identically. G η 5.1.3. Relevant and regular cuspidal data. — We shall say that χ is relevant if π is (M,η )-distinguished. ∗ ∗ ∗ Let X (G) = X (G )×X (G ) (cf. Section 4.3.2). We shall say thatχ is ∗-regular n n+1 if it belongs to the subset X (G). In particular, if χ is both relevant and regular (see Section 2.9.7)thenitis ∗-regular. 5.1.4. Rankin-Selberg period of certain Eisenstein series. — Let T ∈ a .Recallthatwe n+1 have introduced in Section 3.3.2 the truncation operator Proposition 5.1.4.1. — Let Q be a parabolic subgroup of G and Q = Q ∩ G .Let π be an irreducible cuspidal representation of M which is (M ,η )-distinguished. Let ϕ ∈ A (G). Q Q G Q,π G,∗ Then for a regular point λ ∈ a of the Eisenstein series E(g,ϕ,λ) (see Section 2.7.3), the integral (5.1.4.2) I(ϕ,λ) = E(h,ϕ,λ) dh [H] is convergent and does not depend on T. Remark 5.1.4.2. — The expression I(ϕ,λ) is nothing else but the regularized Rankin-Selberg period of E(ϕ,λ) as defined by Ichino-Yamana in [IY15]. Proof. — The convergence follows from Proposition 3.3.2.1 and the fact that Ei- seinstein series are of moderate growth. It remains to prove that the integral does not depend on T. Recall that ι induces an isomorphism from G onto H. In the proof, it will be more convenient to work with G instead of H. However, by abuse of notations, for any g ∈ G (A) and any function ϕ on G(A) we shall write ϕ(g) instead of ϕ(ι(g)). Let T ∈ a . By lemma 2.2 of [IY15], we have n+1 T+T T,P E(g,ϕ,λ) = E (δg,ϕ,λ) G ×P r r n n+1 P∈F δ∈(P∩H)(F)\H(F) RS ×! (H (δg) − T , T ) P P P n+1 n+1 n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 279 where the notations are those of Sections 3.2.3 and 3.3.2. The other notations are bor- T,P T rowed from [Zyd18, eq. (4.4)]; the operator is the obvious variant of and ! is r r P an Arthur function whose precise definition is irrelevant here. We denote by E the G ×P n n+1 constant term of E along G × P . Thus, we have n n+1 T+T E(g,ϕ,λ) dg = [H] T,P E P)(g,ϕ,λ)! (H (g) − T , T ) dg. G ×P P P n n+1 n+1 n+1 r P n+1 (P∩H)(F)\H(A) P∈F RS Let P ∈ F be such that P G. It suffices to show that the terms corresponding to P RS vanish. We identify H with G .Then P ∩ H is identified with P .Let M = M .For an n n n P appropriate choice of a Haar measure on K , such a term can be written as T,P exp(−2ρ , H (m) )( E )(mk,ϕ,λ) P P P n n [M ] K n n ×! (H (m) − T , T ) dkdm, P P P n+1 n+1 n+1 where E denotes the constant term of E along P = P × P . At this point, we may P n n+1 and shall assume that P is standard (if not, we may change B by a conjugate for the arguments). We have the usual formula for the constant term E (m,ϕ,λ) = E (m, M(w,λ)ϕ,wλ), w∈W(Q;P) where W(Q; P) is the set of elements w ∈ W that are of minimal length in double cosets P Q −1 W wW .Let w ∈ W(Q; P). Notice that the representation wπ is also (wM w ,η )- Q G distinguished. For the argument, we may and shall assume w = 1 (that is we assume that Q ⊂ P). Thus it suffices to show for all k ∈ K the integral T,P P (5.1.4.3) E (mk,ϕ,λ) dm [M ] vanishes. The group M = M has a decomposition G ×···× G with d +···+ d = n+1 P d d 1 r n+1 1 r n + 1. Each factor corresponds to a subset of the canonical basis (e ,..., e ).Wemay 1 n+1 assume that the factor G corresponds to a subset which does not contain e .Asa d n+1 consequence G is also a factor of M .Weview G × G as a subgroup of M × M . d n d d n n+1 1 1 1 Let Q × Q =(G × G ) ∩ Q ⊂ G × G . The representation π restricts to M (A) 1 2 d d n n+1 Q 1 1 1 and M (A): this gives representations respectively denoted by π and π . As a factor of Q 1 2 (5.1.4.3), we get (5.1.4.4) E(g,ϕ ,λ ) E(g,ϕ ,λ ) dg 1 1 2 2 [G ] 1 280 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR where ϕ ∈ A (G ). Here the truncation is the usual Arthur’s truncation operator on i Q ,π d i i 1 the group G . It is clear from Langlands’ formula for the integral (5.1.4.4) (see [Art82]) that (5.1.4.4) vanishes unless there exists w ∈ W (Q , Q ) such that the contragredient 1 2 of π is isomorphic to wπ .But then π would be both (M ,(η ) )-distinguished and 2 1 2 Q d n+1 (M ,(η ) )-distinguished with η = η ◦ det and M = M ∩ G . This is not Q d d d Q Q 1 1 1 2 2 2 1 possible. 5.1.5. Relative characters. — Let (P,π) be a pair for which P be a standard parabolic subgroup of G and π be a cuspidal automorphic representation of its stan- dard Levi factor M . Building upon the truncation operator and the linear form J , P η we define the relative character I for any f ∈ S(G(A)) by P,π T T I (f ) = E(h, I (0, f )ϕ, 0) dh · J (ϕ) P η P,π r [H] ϕ∈B P,π where the K-basis B is defined in Section 2.8.3. Using Proposition 5.1.4.1,wehave P,π I (f ) = I (f ) P,π P,π where we define: I(I (0, f )ϕ, 0) · J (ϕ) if π is (M ,η )-distinguished; P η P G ϕ∈B P,π I (f ) = P,π 0 otherwise. Proposition 5.1.5.1. — Let χ ∈ X (G).Let (P,π) be a representative. The map f → I (f ) (and thus f → I (f )) is well-defined and gives a continuous linear form on S(G(A)).It P,π P,π depends only on χ and not on the choice of (P,π). Proof. — First we claim that ϕ → E(h,ϕ, 0) dh is a continuous map: this is [H] r an easy consequence of properties of Eisenstein series and the truncation operator (see Proposition 3.3.2.1). On the other hand ϕ → J (ϕ) is also continuous (see Section 4.1). Thus the first assertion results from an application of Proposition 2.8.4.1. The arguments of the proof of Lemma 4.3.7.2 give the independence on the choice of (P,π). 5.2. The ∗-regular contribution 5.2.1. Let χ ∈ X(G). Recall that we defined in Theorem 3.2.4.1 a distribution I on S(G(A)).Let (M,π) be a representative of χ where M is the standard Levi factor of the standard parabolic subgroup P of G. The following theorem is the main result of this section. Theorem 5.2.1.1. — Assume moreover χ ∈ X (G).Wehave − dim(a ) I = 2 I . χ P,π THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 281 In particular, we have I = 0 unless χ is relevant. The theorem is a direct consequence of the following proposition. ∗ + Proposition 5.2.1.2. — Assume moreover χ ∈ X (G) We have for T ∈ a n+1 T − dim(a ) (5.2.1.1) K (x, y)η (y)dxdy = 2 I (f ), χ G P,π [H] [G ] where the left-hand side is absolutely convergent (see Proposition 3.3.3.1). In particular, the left-hand side does not depend on T. Indeed, by Theorem 3.3.9.1,I is the constant term in the asymptotic expansion − dim(a ) in T of the left-hand side of (5.2.1.1)hence I = 2 I . χ P,π The rest of the section is devoted to the proof of Proposition 5.2.1.2. 5.2.2. Proof of Proposition 5.2.1.2.— We assume that χ ∈ X (G). The proof is a straightforward consequence of Theorem 4.3.3.1 and some permutations between in- tegrals, summations and the truncation. These permutations are provided by Lemmas 5.2.2.1 and 5.2.2.3 below. Lemma 5.2.2.1. — For all x∈[H],wehave ) * T T K )(x, y)η (y) dy = K (·, y)η (y) dy (x). χ G χ G r r [G ] [G ] Remark 5.2.2.2. — On the left-hand side we apply the truncation operator to the function K (·, y) (where y is fixed) and then we evaluate at x whereas on the right-hand side we apply the same operator to the function we get by integration of K (·, y)η (y) over y∈[G ] and then we evaluate at x. χ G Proof. — Since x is fixed, the operator is a finite sum of constant terms (see [Art78] lemma 5.1 for the finiteness). Then the lemma follows from Fubini’s theorem which holds because we have |K (nx, y)| dndy< ∞ [N ] [G ] for all parabolic subgroups Q of G containing B . Here we identify N with the sub- n+1 n Q group {1}× N of G = G × G . The convergence of the integral results from the Q n n+1 bound (3.3.2.3)above. Lemma 5.2.2.3. — We have ) * T − dim(a ) K (·, y)η (y) dy (h) dh = 2 I (f ). χ G P,π [H] [G ] 282 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Proof. — First, by Theorem 4.3.3.1,wehavefor any x∈[G]: − dim(a ) K (x, y)η (y) dy = 2 E(x, I (0, f )ϕ, 0) dh · J (ϕ) χ G P η [G ] ϕ∈B P,π where the notations are borrowed from Section 5.1.5. Then we want to apply the trunca- tion operator and evaluate at h∈[H]. We want to show that this operation commutes with the summation over the orthonormal basis. As in the proof of Lemma 5.2.2.1,it suffices to prove |E(ng, I (0, f )ϕ, 0)| dn·|J (ϕ)| < ∞ P η [N ] ϕ∈B P,π for any parabolic subgroups Q of G containing B , which is an easy consequence of n+1 n continuity properties of Eisenstein series. In this way, we get for h∈[H]: ) * K (·, y)η (y) dy (h) χ G [G ] − dim(a ) T = 2 ( E)(h, I (0, f )ϕ, 0) · J (ϕ). P η ϕ∈B P,π By integration over h∈[H],wehave: ) * K (·, y)η (y) dy (h) dh χ G [H] [G ] − dim(a ) T = 2 ( E)(h, I (0, f )ϕ, 0) dh · J (ϕ). P η [H] ϕ∈B P,π − dim(a ) The right-hand side is nothing else but 2 I (f ). Still we have to justify the change P,π of order of the integration and the summation. But it is easy to show that E(h, I (0, f )ϕ, 0)| dh·|J (ϕ)|< ∞. P η [H] ϕ∈B P,π 6. Spectral decomposition of the Flicker-Rallis period for ∗-regular cuspidal data The goal of this section is to give another proof of the spectral decomposition of the Flicker-Rallis period for the same cuspidal data as in Section 4.3.2. The main result of this section (obtained as a combination of Theorem 6.2.5.1 and Theorem 6.2.6.1)can THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 283 be used to get another version of Theorem 4.3.3.1 with a seemingly different relative character than J (this will actually be done in Section 8.2.4). Of course, these two P,π relative characters are the same. A direct proof of this fact will be given in Section 9. 6.1. Notation 6.1.1. In this section we adopt the set of notation introduced in Section 3.1: E/F is a quadratic extension of number fields, G = GL ,G = Res GL , (B , T ), n,F n E/F n,E n n n (B , T ) are the standard Borel pairs of G ,G and K ,K the standard maximal com- n n n n n n pact subgroups of G (A),G (A) respectively. Besides, we denote by N ,N the unipotent n n n n radicals of B ,B and we set ⎛ ⎞ ⎜ ⎟ w = ∈ G (F). ⎝ ⎠ . n We write e = (0,..., 0, 1) for the last element in the standard basis of F and we let ) * "" P = , P = P ∩ G be the mirabolic subgroups of G ,G respectively n n n n n n 0...01 (that is the stabilizers of e for the natural right actions). The unipotent radicals of P , P n n will be denoted by U and U respectively. For nonnegative integers m n,weembed G n m ) * in G (resp. G in G ) in the “upper left corner” by g → . Thus, in particular, m n n−m we have P = G U and P = G U . n n−1 n n n−1 n The entries of a matrix g ∈ G (A) are written as g ,1 i, j n, and the diagonal n i,j entries of an element t ∈ T (A) as t ,1 i n. n i × n 6.1.2. We fix a nontrivial additive character ψ : A/F → C .For φ ∈ S(A ),we define its Fourier transform φ ∈ S(A ) by φ(x ,..., x ) = φ(y ,..., y )ψ (x y +···+ x y )dy ... dy 1 n 1 n 1 1 n n 1 n the Haar measure on A being chosen such that φ(x) =φ(−x). We denote by c the nontrivial Galois involution of E over F. Then, c acts naturally on G (A) and thus on cuspidal automorphic representations of the latter. We denote this c × c × action by π → π .Wefix τ ∈ E such that τ =−τ and we define ψ : A /E → C by ψ(z) = ψ (Tr (τ z)), z ∈ A ,where A denotes the adèle ring of E and Tr : A → A E/F E E E/F E the trace map. We also define a generic character ψ :[N ]→ C by n n % & n−1 ψ (u) =ψ (−1) u , u∈[N ]. n i,i+1 n i=1 284 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR (The appearance of the sign (−1) is only a convention that will be justified a posteriori in Section 7.) Note that ψ is trivial on A and therefore ψ is trivial on N (A).Toany f ∈ T ([G ]), we associate its Whittaker function W defined by n f −1 W (g) = f (ug)ψ (u) du, g ∈ G (A). f n n [N ] 6.2. Statements of the main results 6.2.1. Let n 1 be a nonnegative integer. For f ∈ T ([G ]), φ ∈ S(A ) and s ∈ C we set FR s Z (s, f,φ) = W (h)φ(e h)|det h| dh f n N (A)\G (A) n n provided this expression converges absolutely. 6.2.2. Let χ ∈ X (G ) be a ∗-regular cuspidal datum (see Section 4.3.2 for the G (A) definition of ∗-regular) represented by a pair (M ,π) and set = Ind (π).Wecan P(A) write M = G ×···× G P n n 1 k where n ,..., n are positive integers such that n + ··· + n = n. Then, π decomposes 1 k 1 k accordingly as a tensor product π =π ... π 1 k where for each 1 i k, π is a cuspidal automorphic representation of G (A). i n 6.2.3. Let L(s,, As) be the Shahidi’s completed Asai L-function of [Sha90], [Gol94]. We have the decomposition L(s,, As) = L(s,π , As) × L(s,π ×π ). i i i=1 1i<jk c c (We emphasize that since L(s,π × π ) = L(s,π × π ), this decomposition does not i j j i depend on the order of the π ’s.) As χ is ∗-regular, the Rankin-Selberg L-functions L(s,π ×π ) are entire and non-vanishing at s = 1[JS81b], [JS81a], [Sha81]whereas by [Fli88], L(s,π , As) has at most a simple pole at s = 1. Therefore, L(s,, As) has a pole of order at most k at s = 1 and this happens if and only if L(s,π , As) has a pole at s = 1 for every 1 i k. We say that the cuspidal datum χ is distinguished if L(s,, As) has a pole of order k at s = 1. By [Fli88], it is equivalent to ask π to be M = M ∩ G -distinguished. P P n THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 285 6.2.4. For f ∈ C([G ]), we set W = W where f is defined as in Section n f , f 2.9.8. Then, W belongs to the Whittaker model W(,ψ ) of with respect to ψ . f , n n We define a continuous linear form β on W(,ψ ) as follows. For S a finite set n n of places of F and W ∈ W(,ψ ), we set β (W) = W(p )dp n,S S S N (F )\P (F ) S S n n the integral being convergent by (the same proof as) [BP21b, Proposition 2.6.1, Lemma 3.3.1] and the Jacquet-Shalika bound [JS81b]. By [Fli88, Proposition 3] and (2.3.2.3), for a given W ∈ W(,ψ ), the quantity S,∗ −1 S,∗ β (W) =( ) L (1,, As)β (W) n n,S is independent of S as long as it is sufficiently large (i.e. it contains all the Archimedean places as well as the non-Archimedean places where the situation is “ramified”) where S,∗ we recall that following our general convention of Section 2.1,L (1,, As) stands for the leading coefficient of the Laurent expansion of the partial L-function L (s,, As) at S,∗ s = 1 and we refer the reader to Section 2.3.3 for the definition of . This defines the linear form β . 6.2.5. For every f ∈ C([G ]), we set f (g) = f (ag)da, g ∈[G ]. Theorem 6.2.5.1. 1. Let N 0. There exists c > 0 such that for every f ∈ T ([G ]) and φ ∈ S(A ),the N N n FR expression defining Z (s, f,φ) is absolutely convergent for s ∈ H and the function s ∈ >c FR H → Z (s, f,φ) is holomorphic and bounded in vertical strips. Moreover, for every s ∈ >c N ψ FR H , (f,φ) → Z (s, f,φ) is a (separately) continuous bilinear form on T ([G ]) × >c N n N ψ S(A ). ∗ n 2. Let χ ∈ X (G ).For every f ∈ C ([G ]) and φ ∈ S(A ), the function s → (s − n χ n FR 0 1)Z (s, f,φ) admits an analytic continuation to H with a limit at s = 1.Moreover, >1 we have FR,∗ 0 FR 0 Z (1, f,φ) := lim(s − 1)Z (s, f,φ) ψ ψ s→1 1−k 2 φ(0)β (W ) if χ is distinguished, n f , 0 otherwise. 286 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 6.2.6. Theorem 6.2.6.1. — Let χ ∈ X (G ). The linear form P : f ∈ C([G ]) → f (h)dh G n [G ] is well-defined (i.e. the integral converges) and continuous. Moreover, for every f ∈ S ([G ]) and φ ∈ χ n S(A ) we have FR,∗ 0 (6.2.6.1) φ(0)P (f ) = Z (1, f,φ). 6.2.7. A direct consequence of Theorem 6.2.5.1 and Theorem 6.2.6.1 is the following corollary. Corollary 6.2.7.1. — Let χ ∈ X (G ) be represented by a pair (M ,π) and set = n P G (A) Ind (π). Then, for every f ∈ S ([G ]) we have χ n P(A) − dim(A ) ⎨ 2 β (W ) if χ is distinguished, n f , P (f ) = 0 otherwise. 6.3. Proof of Theorem 6.2.5.1.2 Part 1. of Theorem 6.2.5.1 will be established in Section 6.5.Here, we give the proof of part 2. of this theorem. Let f ∈ C ([G ]), φ ∈ S(A ) and (M ,π) be a pair χ n P representing the cuspidal datum χ as in Section 6.2. We make the identification ∗ k ia (iR) such that for every x =(x ,..., x ) ∈(iR) we have 1 k x x 1 k (6.3.0.1) π :=π |det| ... π |det| . x 1 k E E ∗ ∗ ∗ Let ia be the subspace of x ∈ ia such that n x +···+ n x = 0. We equip ia with 1 1 k k M,0 M M,0 the unique Haar measure such that the quotient measure on n x +···+ n x 1 1 k k ∗ ∗ ia /ia iR, x → M M,0 G (A) −1 ∗ is (2π) times the Lebesgue measure. For every x ∈ ia , we set = Ind (π ) and x x M P(A) f = f following the definition of Section 2.9.8 (so that in particular = and f = f x 0 0 with notation from the previous section). THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 287 ∞ ∗ We have an isomorphism A R , a → |det a| , sending the Haar measure on G + ∞ dt A to where dt is the Lebesgue measure. Therefore, by Theorem 2.9.8.1 and Fourier |t| inversion, we have n x +···+n x 1 1 k k (6.3.0.2) f = a · f dxda = |det a| f dxda = f dx x x x ∞ ∗ ∞ ∗ ∗ A ia A ia ia G M G M M,0 n n where the right-hand side is an absolutely convergent integral in T ([G ]) for some N > N n 0. Therefore, by the first part of Theorem 6.2.5.1, there exists c > 0such that for every s ∈ H we have >c FR 0 FR (6.3.0.3)Z (s, f,φ) = Z (s, f ,φ)dx. ψ ψ ia M,0 Let S be a finite set of places of F including the Archimedean ones and outside of which π is unramified and let S ⊂ S be the subset of finite places. Let I⊆{1,..., k} be the 0,f 0 subset of 1 i k such that L(s,π , As) has a pole at s = 1. We choose, for each 1 i k and v ∈ S , polynomials Q (T), Q (T) ∈ C[T] with roots in H and H −1 0,f i i,v ]0,1[ ]q ,1[ −s respectively such that s → Q (s)L (s,π , As) and s → Q (q )L (s,π , As) have no pole i ∞ i i,v v i in H .Finally,weset ]0,1[ −s−2x P(s, x) = (s + 2x )(s − 1 + 2x ) Q (s + 2x ) Q (q ) and i i i i i,v i∈I 1ik 1ik v∈S 0,f t −1 f (g) = f ( g ) x x for every x ∈ A , s ∈ C and g ∈ G (A). We will now check that the functions 0 n 1 1 FR (6.3.0.4) (s, x) ∈ C × A → P(s + , x)Z (s + , f ,φ) 0 x 2 2 and 1 1 FR (6.3.0.5) (s, x) ∈ C × A → P( − s, x)Z ( − s, f ,φ) 0 −1 x 2 2 satisfy the conditions of Corollary A.0.11.1. From thefirstpartofTheorem 6.2.5.1,Theorem 2.9.8.1 and Lemma A.0.9.1,we deduce that these functions satisfy the first condition of Corollary A.0.11.1.Tocheck that they also satisfy the second condition of Corollary A.0.11.1, we need to analyze more FR carefully the function s → Z (s, f ,φ) for a fixed x ∈ A . x 0 For S a sufficiently large finite set of places of F, that we assume to contain Archimedean places as well as the places where π , ψ or ψ are ramified (thus S ⊂ S), we have decompositions S S φ =φ φ and W = W W S f S,x x x 288 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR n S S n for every x ∈ A ,where φ ∈ S(F ), φ is the characteristic function of (O ) ,W ∈ 0 S S,x S F W( ,ψ ) (that is the Whittaker model of the representation with respect to the x,S n,S x,S S S S K S character ψ = ψ )and W ∈ W( ,ψ ) is such that W (1) = 1. By [Fli88, n,S n|N(F ) x x n x Proposition 3] and (2.3.3.4), we then have FR Z (s, W ,φ ) S,x S S,∗ FR −1 (6.3.0.6)Z (s, f ,φ) =( ) L(s, , As) x x ψ G L (s, , As) S x for s ∈ H where we have set >c FR s Z (s, W ,φ ) = W (h )φ (e h )|det h | dh . S,x S S,x S S n S S S N (F )\G (F ) S S n n FR Moreover, by [BP21b, Theorem 3.5.1] the function Z (s, W ,φ ) extends meromor- S,x S phically to the complex plane and satisfies the functional equation FR FR Z (1 − s, W ,φ ) Z (s, W ,φ ) −1 S,x S S,x S ψ ψ (6.3.0.7) =(s, , As) L (1 − s,( ) , As) L (s, , As) S x S x t −1 where W (g) = W (w g ),φ is the (normalized) Fourier transform ofφ with respect S,x S,x n S S to the bicharacter (u,v) → ψ (u v + ··· + u v ) and (s, , As) denotes the global 1 1 n n x epsilon factor of the Asai L-function L(s, , As). By (6.3.0.6), (6.3.0.7) as well as the meromorphic continuation and functional FR equation of L(s, , As) [Sha90, Theorem 3.5(4)], we conclude that Z (s, f ,φ) has a x x meromorphic continuation to C satisfying the functional equation FR FR (6.3.0.8)Z (1 − s, f ,φ) = Z (s, f ,φ). −1 x x ψ ψ On the other hand, we have the decomposition L(s, , As) = L(s + 2x,π , As) × L(s + x + x,π ×π ), x i i i j i i=1 1i<jk ∗ c and, asχ ∈ X (G ), the Rankin-Selberg L-functions L(s,π ×π ) are entire and bounded n i in vertical strips [Cog08, Theorem 4.1]. By the Jacquet-Shalika bound [JS81b]and the fact that the gamma function is of exponential decay in vertical strips, Q (s)L (s,π , As) i ∞ i −s and Q (q )L (s,π , As) are holomorphic and bounded in vertical strips of H for each i,v v i >0 1 i k and v ∈ S .By[FL17, Lemma 5.2], s → (s − 1)L (s,π , As),for i ∈ I, and 0,f i s → L (s,π , As),for i ∈/ I, are also holomorphic and of finite order in vertical strips of H . Therefore, by the definition of P and the functional equation, P(s, x)L(s, , As) is >0 x entire and of finite order in vertical strips. By (6.3.0.6), (6.3.0.8)and [BP21b,Theorem 3.5.2], it follows that the functions (6.3.0.4), (6.3.0.5) are entire and of finite order in THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 289 vertical strips in the first variable i.e. they also satisfy the second condition of Corollary A.0.11.1. Thus, the conclusion of this corollary is valid and in particular the map % & FR s → x → (s − 1 + 2x )Z (s, f ,φ) i x i∈I induces a holomorphic function H → S(A ) for some > 0. By (6.3.0.3)and >1− 0 9 FR 0 [BP21c, Lemma 3.1.1, Proposition 3.1.2], it follows that s → Z (s, f,φ) extends ana- lytically to H and that >1 1−k k FR 2 lim(s − 1) Z (s, f ,φ) if I={1,..., k}, FR 0 s→1 (6.3.0.9) lim(s − 1)Z (s, f,φ) = s→1 0 otherwise Recall that I={1,..., k} if and only if L(s,, As) has a pole of order k = rk(A ) at s = 1. Moreover, by [BP21b, Lemma 3.3.1] and the Jacquet-Shalika bound [JS81b], FR the integral defining Z (s, W ,φ ) is absolutely convergent in H for some ε> 0. S,0 S >1−ε Combining this with [BP21c, Lemma 2.16.3] and (6.3.0.6), in the case I ={1,..., k} identity (6.3.0.9) can be rewritten as FR 0 1−k S,∗ −1 S,∗ FR lim(s − 1)Z (s, f,φ) = 2 ( ) L (1,, As)Z (1, W ,φ ) S,0 S ψ G ψ s→1 S,∗ 1−k −1 S,∗ = 2 ( ) L (1,, As)β (W )φ (0) n,S S,0 S 1−k = 2 φ(0)β (W ) n f , and this ends the proof of Theorem 6.2.5.1.2. 6.4. Proof of Theorem 6.2.6.1 By (2.4.5.23), since any Siegel domain of [G ] is contained in a Siegel domain of G G 2 [G ] and ρ = 2ρ ,wehave (h) (h) for h∈[G ].Hence,by (2.4.5.24),the n B B n n linear form P is well-defined and continuous on C([G ]). This shows the first part of G n Theorem 6.2.6.1. ∞ ∞ ∞ Let f ∈ S ([G ]).Recallthat A = A but the Haar measure on A is twice the χ n G G G n n Haar measure on A (see Remark 3.1.3.1). Therefore, we have P (f ) = f (h)dh. [G ] 9 ∗ k More precisely, with the notation of Proposition 3.12 of loc. cit. we have an isomorphism ia (iR) , x → x M,0 0 ∗ n 1−k k given by x = n x and which sends the measure on ia to (2π) times the measure on (iR) used in loc. cit. i i i M,0 0 n ...n 1 k 290 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Let φ ∈ S(A ). We form the Epstein-Eisenstein series E(h,φ, s) = φ(e γ ah)|det(ah)| da, h∈[G ], s ∈ C. γ∈P (F)\G (F) n n n This expression converges absolutely for (s)> 1and the map s → E(φ, s) extends to a meromorphic function valued in T ([G ]) with simple poles at s = 0, 1 of respective residues φ(0) and φ(0) (cf. [JS81b, Lemma 4.2]). Consequently, the function FR 0 0 s → Z (s, f,φ) := f (h)E(h,φ, s)dh [G ] is well-defined for s ∈ C\{0, 1}, meromorphic on C with a simple pole at s = 1whose residue is FR 0 (6.4.0.1)Res Z (s, f,φ) = 2φ(0)P (f ). s=1 G n n Unfolding the definition, we arrive at the identity FR 0 0 s (6.4.0.2) Z (s, f,φ) = f (h)φ(e h)|det h| dh P (F)\G (A) n n valid for (s)> 1. More generally, for every 1 r n,let N be the unipotent radical of the standard r,n n−r parabolic subgroup of G with Levi component G ×(G ) ,N be its intersection with n r 1 r,n G and set 0 0 −1 f (g) = f (ug)ψ (u) du, g ∈ G(A), N ,ψ n r,n [N ] r,n FR 0 0 s Z (s, f,φ) = f (h)φ(e h)|det h| dh, s ∈ C, N ,ψ n r r,n P (F)N (A)\G (A) r r,n n provided the last expression above is convergent. The proof of the next lemma will be given in Section 6.5. Lemma 6.4.0.1. — For every 1 r n, there exists c > 0 such that the expression defining FR 0 Z (s, f,φ) converges absolutely for (s)> c . 0 FR 0 When r = 1, we have N = N and f = W so that Z (s, f,φ) = 1,n n N ,ψ f 1,n 1 FR 0 Z (s, f,φ). Therefore, by (6.4.0.1)and (6.4.0.2), the second part of Theorem 6.2.6.1 is a consequence of the following proposition. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 291 FR 0 Proposition 6.4.0.2. — For every 1 r n, the function s → (s− 1)Z (s, f,φ) extends to a holomorphic function on {s ∈ C| (s)> 1} admitting a limit at s = 1. Moreover, we have FR 0 FR 0 lim(s − 1)Z (s, f,φ) = lim(s − 1)Z (s, f,φ). n r + + s→1 s→1 Proof. — By descending induction on r, it suffices to establish the following: (6.4.0.3)Let 1 r n − 1. There exists a function F holomorphic on H for some r >1− > 0such that FR 0 FR 0 Z (s, f,φ) = Z (s, f,φ) + F (s) r+1 r for all s ∈ C satisfying (s)> max(c , c ). r r+1 Indeed, as P = G U ,wehave r+1 r r FR 0 0 s (6.4.0.4) Z (s, f,φ) = f (uh)duφ(e h)|det h| dh. N ,ψ n r+1 r+1,n G (F)N (A)\G (A) [U ] r r,n n r+1 By Fourier inversion on the locally compact abelian group U (F)U (A)\U (A),we r+1 r+1 r+1 have 0 0 0 (6.4.0.5) f (uh)du = ( f ) (γ h) +( f ) (h) N ,ψ N ,ψ U ,ψ N ,ψ U r+1,n r+1,n r+1 r+1,n r+1 [U ] r+1 γ∈P (F)\G (F) r r for all h ∈ G (A) wherewehaveset 0 0 −1 0 ( f ) (h) = f (uh)ψ (u) du = f (h), N ,ψ U ,ψ N ,ψ n N ,ψ r+1,n r+1 r+1,n r,n [U ] r+1 0 0 ( f ) (h) = f (uh)du. N ,ψ U N ,ψ r+1,n r+1 r+1,n [U ] r+1 By (6.4.0.4)and (6.4.0.5), we obtain FR 0 FR 0 Z (s, f,φ) = Z (s, f,φ) + F (s) r+1 r for all s ∈ C such that (s)> max(c , c ) and where we have set r r+1 0 s F (s) = ( f ) (h)φ(e h)|det h| dh. r N ,ψ U n r+1,n r+1 G (F)N (A)\G (A) r r,n n It only remains to check that F (s) extends to a holomorphic function on H for some r >1− > 0. 292 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Let P be the standard parabolic subgroup of G with Levi component M = G × r n r r G and set P = P ∩ G . We readily check that n−r r r n ) * 0 0 −1 ( f ) (h) = f ( h)ψ (u) du N ,ψ U P n r+1,n r+1 r [N ] n−r ) * −1 = f ( ah)daψ (u) du. P n [N ] A n−r Therefore, by the Iwasawa decomposition G (A) = P (A)K and since n r n ) * ) * 1/2 h h r r n−r −r δ =δ =|det h | |det h | P P r n−r h h n−r n−r for all h ∈ G (A), h ∈ G (A),wehave(for (s)> max(c , c ) and a suitable choice r n−r r r+1 r n−r of Haar measure on K ) ) ) ** (6.4.0.6) F (s) = f a da r P ,k,s uh n−r K ×[G ]×N (A)\G (A) [N ] A n−r n r n−r n−r G −1 ns/(n−r) ψ (u) du|det h | n n−r ×φ (e h )dh dh dk k,n−r n−r n−r n−r r −1/2+s/2(n−r) where f =δ (R(k)f ) | and φ stands for the composition of R(k)φ P ,k,s P M (A) k,n−r r P r r n−r n M with the inclusion A → A , x → (0, x).Let χ be the inverse image of χ in X(M ).By Corollary 2.9.7.2,wehave f ∈ C M([M ]) for every (k, s) ∈ K × H and the map P ,k,s χ r n >0 (k, s) ∈ K × H → f ∈ C M([M ]) n >0 P ,k,s χ r is continuous, holomorphic in the second variable. In particular, for (s)> 0 the integral ) ) ** a h f a dada P ,k,s uh ∞ ∞ n−r A [N ] A n−r is absolutely convergent and equals, by the obvious change of variable, to ) * a h f dada . P ,k,s uah n − r ∞ ∞ n−r A [N ] A n−r n−r It follows that (6.4.0.6) can be rewritten, for (s) 1, as ) * (6.4.0.7) F (s) = f da r P ,k,s uah n − r ∞ n−r K [G ]×N (A)\G (A) [N ] A n−r n r n−r n−r G −1 ns/(n−r) ×ψ (u) du|det h | φ (e h )dh dh dk n n−r k,n−r n−r n−r n−r r THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 293 n ns FR = (P ⊗Z ( ))(f ⊗φ )dk G P ,k,s k,n−r n−r r n − r n − r FR where Z (s) stands for the bilinear form n−r n−r FR 0 (f ,φ ) ∈ C([G ]) × S(A ) → Z r(s, f ,φ ). (−1) n−r On the other hand, by (2.9.6.10)wehave C ([M ]) = C ([G ])⊗C ([G ]) r χ r χ n−r 1 2 (χ ,χ )∈X(M )=X(G )×X(G )→χ 1 2 r r n−r and, asχ ∈ X (G ),for every(χ ,χ ) ∈ X(G )×X(G ) mapping toχ ∈ X(G ),wealso n 1 2 r n−r n have χ ∈ X (G ). Therefore, by the first part of Theorem 6.2.6.1,Theorem 6.2.5.1 2 n−r FR and (A.0.5.5), s → P ⊗Z (s) extends to an analytic family of (separately) continuous r n−r n−r bilinear forms on C ([M ]) × S(A ) for s ∈ H . Thus, by the first part of Theorem χ r >1 6.2.6.1, (A.0.5.4) and the equality (6.4.0.7), F (s) has an analytic continuation to { (s)> 1 − r/n}. This ends the proof of the proposition and hence of Theorem 6.2.6.1. 6.5. Convergence of Zeta integrals 6.5.1. Proof of Lemma 6.4.0.1.— We only treat the case 1 r n− 1. The case r = n can be dealt with in a similar manner, and is in fact easier. n−r Let Q be the standard parabolic subgroup of G with Levi component G × G r n r and set Q = Q ∩ G .Recallthat N is the unipotent radical of Q . Identifying A r r,n r r n G R , by the Iwasawa decomposition G (A) = Q (A)K , we need to show the convergence >0 n r n of ) * ah (6.5.1.1) (R(k)f ) |R(k)φ(t e )| N ,ψ n−r n r,n at K ×P (F)\G (A)×T (A)×R n r r >0 n−r ) * −1 s s |det h| |det t| δ dadtdhdk for (s) 1. We now apply Lemma 2.6.1.1. For this we note that ψ | =ψ ◦ where n [N ] r,n n−1 : N → G sends u ∈ N to Tr (τ u ) and τ ∈ E is the unique trace-zero r,n a r,n E/F i,i+1 i=r element such that ψ(z) =ψ (Tr (τ z)). We readily check that E/F n−r−1 ∗ −1 −1 Ad (m) ∼t e h r t t , V r A i A Q 1 i+1 i=1 294 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR ) * for m = ∈ G (A) × T (A). r n−r Therefore, by Lemma 2.6.1.1.1, we can find c > 0such that for everyN , N > 0we have 1 2 ) * ah (6.5.1.2) (R(k)f ) N ,ψ r,n at ) * n−r−1 −cN −N −N −N 2 −1 1 −1 1 ah t e h t t δ r i Q A A r G 1 i+1 i=1 for (k, h, t, a) ∈ K × G (A) × T (A) × R . On the other hand, for every N > 0, we >0 1 n r n−r have −N |R(k)φ(te )|t ,(k, t) ∈ K × A and it is easy to check that for some N > 0we have n−r n−r−1 N N N 2 −1 2 −1 2 e h r t t t e h t t , r A i A n−r r i A 1 A i+1 A i=1 i=1 (h, t) ∈ G (A) × T (A). r n−r ) * n−r n−r n+1−2(r+i) As δ =|det h| |t | for every (h, t) ∈ G (A) × T (A), combining Q i r n−r i=1 this with (6.5.1.2), we deduce the existence of c > 0such that for everyN , N > 0, 1 2 (6.5.1.1) is essentially bounded by the product of −N −N 2 1 s−(2cN +1)(n−r) (6.5.1.3) ah e h |det h| dadh P (F)\G (A)×R >0 r r and −N 1 s−(2cN +1)(n+1−2(r+i)) (6.5.1.4) t |t| dt for 1 i n − r. Let C , C > 0. By Lemma 2.6.2.1,for N sufficiently large the integral (6.5.1.4) 1 2 1 converges absolutely in the range 1+(2cN + 1)(n+ 1− 2(r+ i))< (s)< C +(2cN + 1)(n+ 1− 2(r+ i)) 2 1 2 and for N ,N sufficiently large the integral (6.5.1.3) converges absolutely in the range 1 2 1 +(2cN + 1)(n − r)< (s)< C +(2cN + 1)(n − r). 2 2 2 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 295 Since n + 1 − 2(r + i)< n − r for every 1 i n − r, by taking C = 2and C 2 1 2+(2cN + 1)(r + 2i − 1) for every 1 i n− r, it follows that if N 1and N 1 2 2 1 N the integrals (6.5.1.3)and (6.5.1.4) are convergent in the range 1 +(2cN + 1)(n − r)< (s)< 2 +(2cN + 1)(n − r). 2 2 The union of these open intervals for N sufficiently large as above is of the form ]c ,+∞[ 2 r FR 0 which shows that Z (s, f,φ) converges absolutely in the range (s)> c for a suitable c > 0. 6.5.2. Proof of Theorem 6.2.5.1.1. — Applying Lemma 2.6.1.1.2, the same manipulations as in the proof of Lemma 6.4.0.1 reduce us to showing the existence of c > 0such that for every C > c there exists N > 0 satisfying that the integral −N N −1 s (6.5.2.5) t t δ (t) |det t| dt i B A T T (A) i=1 converges in the range s ∈ H uniformly on compact subsets. But this follows again ]c ,C[ from Lemma 2.6.2.1 as there exists M> 0such that N −1 −1 M t δ (t) max(|t |,|t | ) , t ∈[T ]. B i i T n 1in 7. Canonical extension of the Rankin-Selberg period for H-regular cuspidal data This section is a continuation of Section 6 and we shall use the notation introduced there. The main goal is to show the existence of a canonical extension of corank one Rankin-Selberg periods to the space of uniform moderate growth functions for certain cuspidal data (see Theorem 7.1.3.1). Combining this with the results of Section 6, this will enable us to give an alternative proof of the spectral expansion of the Jacquet-Rallis trace formula for certain cuspidal data in Section 8. 7.1. Statements of the main results 7.1.1. Let n 1 be a positive integer. We set G = G × G and H = G that we n n+1 n consider as an algebraic subgroup of G via the diagonal inclusion H → G. We also set w = (w ,w ) ∈ G(F),K = K × K ,N = N × N and N = N .Put ψ = ψ n n+1 n n+1 n n+1 H n N n ψ (a generic character of [N]). We note that ψ is trivial on [N ] (see the convention n+1 N H 296 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR in the definition of ψ in Section 6.1.2). To any function f ∈ T ([G]), we associate its Whittaker function −1 W (g) = f (ug)ψ (u) du, g ∈ G(A). f N [N] For f ∈ T ([G]), we set RS s Z (s, f ) = W (h)|det h| dh ψ E N (A)\H(A) for every s ∈ C for which the above expression converges absolutely. We postpone the proof of the following lemma to Section 7.3. Lemma 7.1.1.1. — Let N 0. There exists c > 0 such that: RS • For every f ∈ T ([G]) and s ∈ H , the expression defining Z (s, f ) converges absolutely; N >c RS • For every s ∈ H , the functional f ∈ T ([G]) → Z (s, f ) is continuous; >c N RS • For every f ∈ T ([G]), the function s ∈ H → Z (s, f ) is holomorphic and bounded in N >c N ψ vertical strips. 7.1.2. H-regular cuspidal datum. — Let χ ∈ X(G) be a cuspidal datum represented by a pair (M ,π) where P = P × P is a standard parabolic subgroup of G and π = P n n+1 π π a cuspidal automorphic representation of M (A) (with central character trivial n n+1 P on A ). We have decompositions M = G ×···× G , M = G ×···× G P n n P m m n 1 k n+1 1 r and π , π decompose accordingly as tensor products n n+1 π =π ... π ,π =π ... π . n n,1 n,k n+1 n+1,1 n+1,r We say that χ is H-regular if it satisfies the following condition: (7.1.2.1)For every 1 i k and 1 j r,wehave π =π . n,i n+1,j 7.1.3. Theorem 7.1.3.1. — Let χ ∈ X(G) be a H-regular cuspidal datum. Then, RS 1. For every f ∈ T ([G]), the function s → Z (s, f ), a priori defined on some right half- plane, extends to an entire function on C; 2. The restriction of the linear form P : f ∈ S([G]) → f (h)dh [H] THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 297 to S ([G]) extends by continuity to T ([G]) and moreover for every f ∈ T ([G]), we have χ χ χ RS (7.1.3.2)P (f ) = Z (0, f ). 7.2. Proof of Theorem 7.1.3.1 Let f ∈ S ([G]).For 1 r n,let N and N be the unipotent radicals of the χ r,n r,n+1 n−r standard parabolic subgroups of G and G with Levi components G × (G ) and n n+1 r 1 n+1−r G ×(G ) respectively and set r 1 G H G N = N × N , N = N ∩ H = N , r,n r,n+1 r,n r r r −1 f G (g) = f (ug)ψ (u) du, for g ∈ G(A). N ,ψ N [N ] For 1 r n and s ∈ C, we also set, whenever this expression converges, RS s Z (s, f ) = f G (h)|det h| dh. N ,ψ r E P (F)N (A)\H(A) RS RS Note that we have Z (s, f ) = Z (s, f ). The proof of the next lemma will be given in 1 ψ Section 7.3. Lemma 7.2.0.1. — For every 1 r n, there exists c > 0 such that the expression defining RS Z (s, f ) converges absolutely for s ∈ H . >c r r For r = n + 1and every s ∈ C, we also set RS s Z (s, f ) = f (h)|det h| dh. n+1 E [H] Note that the above expression is absolutely convergent and defines an entire function RS of s ∈ C which is bounded in vertical strips, satisfying P (f ) = Z (0, f ). The following n+1 proposition is the crux to the proof of Theorem 7.1.3.1. Proposition 7.2.0.2. — For every 1 r n, we have RS RS (7.2.0.1)Z (s, f ) = Z (s, f ) r+1 r for (s) sufficiently large. Proof. — Let 1 r n− 1. As P = G U and N = U N ,for s ∈ H we r+1 r r r,n r r+1,n >c r+1 have RS s (7.2.0.2) Z (s, f ) = f (uh)du|det h| dh N ,ψ r+1 E r+1 G (F)N (A)\H(A) [U ] r+1 298 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR wherewehaveset U = U viewed as a subgroup of H = G (as always via r+1 n r+1 the embedding in “the upper-left corner”). Similarly, we set U = U × U r+1 r+1 r+1 viewed as a subgroup of G. By Fourier inversion on the compact abelian group G H G U (F)U (A)\U (A),wehave r+1 r+1 r+1 (7.2.0.3) f G (uh)du = (f G ) G (γ h) +(f G ) G (h) N ,ψ N ,ψ U ,ψ N ,ψ U r+1 r+1 r+1 r+1 r+1 [U ] r+1 γ∈P (F)\G (F) r r for every h ∈ H(A), where we have set −1 (f G ) G (h) = f G (uh)ψ (u) du = f G (h), N N ,ψ N ,ψ U ,ψ N ,ψ r+1 r+1 r+1 [U ] r+1 (f G ) G (h) = f G (uh)du. N ,ψ U N ,ψ r+1 r+1 r+1 [U ] r+1 By (7.2.0.2)and (7.2.0.3), we obtain RS RS (7.2.0.4)Z (s, f ) = Z (s, f ) + F (s) r+1 r for every s ∈ C such that (s)> max(c , c ) and where we have set r r+1 G G F (s) = (f ) (h)|det h| dh. N ,ψ U r+1 r+1 G (F)N (A)\H(A) By a similar argument, (7.2.0.4) still holds when n = r if we set f G (h) = f (uh)du and F (s) = f G (h)|det h| dh U U E n+1 n+1 [U ] [H] n+1 where U = 1 × U . n+1 n+1 From (7.2.0.4), we are reduced to showing that F (s) = 0 identically for every 1 r n and (s) sufficiently large. To uniformize notation, we set f G = f .Let N ,ψ n+1 P be the standard parabolic subgroup of G with Levi component L = G × G × ( ) r r r n−r (G × G ) and set P = P ∩ H. Then, we readily check that r n+1−r r ))) * ) ** * I I r r (f G ) G (h) = f , h N ,ψ U r r+1 r+1 u u [N ]×[N ] n−r n+1−r −1 −1 ×ψ (u) ψ (u ) du du n n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 299 for every h ∈ H(A) and 1 r n. Therefore, by the Iwasawa decomposition H(A) = P (A)K ,wehave )) * F (s) = f k, r P uh n−r K [G ]×N (A)\G (A) [N ]×[N ] n r n−r n−r n−r n+1−r ) * * ) * −1 h h r r −1 −1 k ψ (u) ψ (u ) du duδ H n n+1 P u h h n−r n−r s s ×|det h | |det h | dh dh dk. r n−r n−r r E E By a painless calculation, left to the reader, we have ) * −1 s s δ |det h | |det h | P r n−r E E n−r )) * ) ** 1 − +α (s) h h r r s+2rα (s) =δ , |det h | P n−r r E h h n−r n−r 2s+1 L where α (s) = .Let χ be the inverse image of χ in X(L ). By Corollary 2.9.7.2, r r 4n−4r+2 − +α (s) for (α (s)) > 0and every k ∈ K the function f := δ R(k)f | belongs to r n P ,k,s P [L ] r P r r C ([L ]). On the other hand, as L = G × G × G × G ,by(2.9.6.10)wehave χ r r r n−r r n+1−r the decomposition C ([L ]) = C ([G × G ]) χ r χ r r (χ ,χ )∈X(G )×X(G ×G )→χ 1 2 n−r n+1−r ⊗C ([G × G ]) χ n−r n+1−r and the above equality can be rewritten as RS (7.2.0.5)F (s) = P ⊗Z (s + 2rα (s)) (f )dk r G r P ,k,s n−r r RS where P denotes the period integral over the diagonal subgroup of G × G and Z (s) r r n−r stands for the continuous linear form RS f ∈ C([G × G ]) → Z r(s, f ). n−r n+1−r (−1) Sinceχ is H-regular, by (7.1.2.1) for every preimage(χ ,χ ) ∈ X(G )×X(G × G ) 1 2 n−r n+1−r 2 ∨ of χ with χ =(χ ,χ ) ∈ X(G ) we have χ =(χ ) . Hence, by definition of C ([G × 1 r χ r 1 1 1 1 1 G ]),P vanishes identically on C ([G × G ]). This implies that F (s) = 0 whenever r G χ r r r (s) 1 and this ends the proof of the proposition. 300 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR We can now finish the proof of Theorem 7.1.3.1. From the proposition, we deduce that RS s (7.2.0.6)Z (s, f ) = f (h)|det h| dh [H] for (s) 1and every f ∈ S ([G]). In particular, it follows from this equality that s → RS Z (s, f ) extends to an entire function on C that is bounded in vertical strips and satisfies the functional equation RS RS Z (−s, f ) = Z (s, f ) ψ ψ t −1 where f (g) = f ( g ). By Corollary A.0.11.2, we can now deduce the first part of the theorem from the above functional equation, (2.5.10.8), (2.9.5.9), Lemma 7.1.1.1. This corollary also entails that the linear map f ∈ T ([G]) → Z (0, f ) is continuous. As, by χ ψ (7.2.0.6), this functional coincides with P on S ([G]), this proves the second part of the H χ theorem. 7.3. Convergence of Zeta integrals Proof of Lemma 7.2.0.1.— The argument is very similar to the proof of Lemma 6.4.0.1 so we only sketch it. Let 1 r n and Q be the standard parabolic subgroup of n−r n+1−r G G with Levi component (G × (G ) ) × (G × (G ) ) so that N is the unipotent r 1 r 1 G H G H radical of Q .Set Q = Q ∩ H. By the Iwasawa decomposition H(A) = Q (A)K ,we r r r r need to show the convergence of ) * (7.3.0.1) (R(k)f ) G N ,ψ K ×P (F)\G (A)×T (A) n r r n−r ) * −1 s s ×|det h| |det t| δ H dtdhdk E E for (s) 1. We apply Lemma 2.6.1.1.1 to ψ =ψ and : N → G , % & n−1 n n n+1 (u, u ) → Tr (−1) τ u +(−1) τ u . E/F i,i+1 i,i+1 i=r i=r We readily check that there exists N > 0such that ) * n−r ∗ N (7.3.0.2) e h t Ad , for (h, t) ∈ G (A) × T (A). r A i A r n−r E V E G i=1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 301 Therefore, from (7.3.0.2) and Lemma 2.6.1.1, there exists c > 0such that for every N , N > 0, (7.3.0.1) is essentially bounded by 1 2 ) * n−r −cN −N −N −N 2 1 1 h e h r t δ G r i Q G A A r E r P (F)\G (A)×T (A) r r n−r i=1 ) * −1 s s ×δ H |det h| |det t| dtdh. E E Now, the convergence of the above expression for (s) 1, N 1and N 1can 2 s 1 s,N be shown as in the end of the proof of Lemma 6.4.0.1 using Lemma 2.6.2.1. Proof of Lemma 7.1.1.1.— Applying Lemma 2.6.1.1.2 in a similar way, we are re- duced to showing the existence of c > 0such that for everyC > c there exists N > 0 N N satisfying that the integral −N N −1 s t t δ (t) |det t| dt i B A T E E n T (A) i=1 converges in the range s ∈ H uniformly on any compact subsets. This is exactly what ]c ,C[ was established in the proof of Theorem 6.2.5.1.1 (up to replacing the base field F by E). 8. Contributions of ∗-regular cuspidal data to the Jacquet-Rallis trace formula: second proof In this section, we adopt the set of notation introduced in Section 5.Inparticular, n 1 is a positive integer, G = G × G ,G = G × G ,H = G with its diagonal n n+1 n n n+1 embedding in G, K = K × K and K = K × K are the standard maximal compact n n+1 n n+1 subgroups of G(A) and G (A) respectively and η :[G]→{±1} is the automorphic character defined in Section 3.1.6. We will also use notation from Sections 6 and 7: N = N × N and N = N are the standard maximal unipotent subgroups of G and n n+1 H n H, ψ = ψ ψ is a generic character of [N] (where ψ and ψ aredefinedasin N n n+1 n n+1 Section 6.1.2). We also set P = P ×P (resp. P = P ×P )where P and P (resp. n n+1 n n+1 n n+1 P and P ) stand for the mirabolic subgroups of G and G (resp. of G and G ), n n+1 n n+1 n n+1 T = T × T for the standard maximal torus of G and N = N × N for the standard n n+1 n n+1 maximal unipotent subgroup of G .Finally,asinSection 7.1.1,for every f ∈ T ([G]) we set −1 W (g) = f (ug)ψ (u) du, g ∈ G(A). f N [N] 302 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 8.1. Main result 8.1.1. Let χ ∈ X (G) be a ∗-regular cuspidal datum (see Section 5.1.3)repre- G(A) sented by a pair (M ,π). We set = Ind (π). We have decompositions P = P × P , P n n+1 P(A) π = π π and = where: P ,P are standard parabolic subgroups of n n+1 n n+1 n n+1 G ,G respectively with standard Levi components of the form n n+1 M = G ×···× G , M = G ×···× G , P n n P m m n 1 k n+1 1 r π and π are cuspidal automorphic representations of M (A),M (A) decomposing n n+1 P P n+1 into tensor products π =π ... π ,π =π ... π n n,1 n,k n+1 n+1,1 n+1,r G (A) G (A) n+1 respectively and we have set = Ind (π ), = Ind (π ). We write χ ∈ n n n+1 n+1 n P (A) P (A) n n+1 ∗ ∗ X (G ) and χ ∈ X (G ) for the cuspidal data determined by the pairs (M ,π ) and n n+1 n+1 P n (M ,π ) respectively. P n+1 n+1 The representation is generic and we denote by W( ψ ) its Whittaker model , N with respect to the character ψ . Also, for every φ ∈ we define −1 W (g) := W (g) = E(ug,φ)ψ (u) du, g ∈ G(A). φ E(φ) N [N] Note that W ∈ W(,ψ ). φ N 8.1.2. We now define two continuous linear forms λ and β as well as a contin- uous invariant scalar product .,. on W(,ψ ).Let W ∈ W(,ψ ). Whitt N N • By [JPSS83][Jac09], the Zeta integral (already encountered in Section 7) RS s Z (s, W) = W(h)|det h| dh, N (A)\H(A) converges for (s) 0 and extends to a meromorphic function on C with no pole at s = 0. We set RS λ(W) = Z (0, W). • For S a sufficiently large finite set of places of F, we put S,∗ −1 S,∗ β (W) =( ) L (1,, As ) W(p )η (p )dp η G S G S S N (F )\P (F ) S S n+1 n (−1) (−1) wherewehaveset L(s,, As ) = L(s, , As )L(s, , As ). G n n+1 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 303 • Similarly, for S a sufficiently large finite set of places of F, we put S,∗ −1 S,∗ 2 W, W =( ) L (1,, Ad) |W(p )| dp Whitt S S N(F )\P(F ) S S ∨ ∨ where we have set L(s,, Ad) = L(s, × )L(s, × ). n n+1 n n+1 That the above expressions converge and are independent of S as soon as it is chosen suf- ficiently large (depending on the level of W) follow from [Fli88]and [JS81b]. Moreover, the inner form .,. is G(A)-invariant by [Ber84]and [Bar03]. Whitt The next result follows from works of Jacquet-Shalika [JS81b], Shahidi [Sha81] and Lapid-Offen [FLO12, Appendix A]. For completeness, we explain the deduction (see Section 2.7.2 for our normalization of the Petersson inner product). Theorem 8.1.2.1 (Jacquet-Shalika, Shahidi, Lapid-Offen).— We have φ,φ =W , W Pet φ φ Whitt for every φ ∈. Proof. — Let φ ∈ . By the Iwasawa decomposition, for a suitable Haar measure on K we have 2 −1 φ,φ = |φ(mk)| δ (m) dmdk. Pet P K [M ] P 0 Set N = N ∩ M and −1 φ (g) = φ(ug)ψ (u) du, g ∈ G(A). N ,ψ N [N ] k r Let P be the product of mirabolic groups P × P . It is a subgroup of M . n m P i=1 i j=1 j According to Jacquet-Shalika [JS81b, §4] (see also [FLO12, p. 265] or [Zha14a,Propo- sition 3.1] ), for S a sufficiently large finite set of places of F we have 2 −1 (8.1.2.1) |φ(mk)| δ (m) dm [M ] P 0 k r S,∗ −1 S ∨ S ∨ =( ) Res L (s,π ×π ) Res L (s,π ×π ) s=1 n,i s=1 n+1,j M n,i n+1,j i=1 j=1 2 −1 × |φ P (p k)| δ (p ) dp N ,ψ S P S S P P N (F )\P (F ) S S Note that our normalization of the Petterson inner product if different from loc. cit. 304 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR for every k ∈ K. On the other hand, by [FLO12, Proposition A.2] we have 2 −1 (8.1.2.2) |φ (p k)| δ (p ) dp dk N ,ψ S P S S P P K N (F )\P (F ) S S vol (K ) G(A ) S S vol S (K ∩ M (A )) M (A ) P 2 −1 × |φ P (p g )| δ (p ) dp dg N ,ψ S S P S S S P P P(F )\G(F ) N (F )\P (F ) S S S S S,∗ −1 S,∗ 2 =( ) |W (p ,φ P )| dp , S S N ,ψ S G M N(F )\P(F ) S S G(F ) where W : Ind (W(π ,ψ )) → W( ,ψ ) stands for the Jacquet functional,de- S S N,S S N,S P(F ) fined as the value at s = 0 of the holomorphic continuation of G G s −1 W (g ,φ ) = φ (w u g )δ (w u g ) ψ (u ) du , S,s S S S P S S N S S P P G G −1 P (w ) N (F )w \N(F ) S S P P (s) 1 G(F ) S G P G P G for g ∈ G(F ) and φ ∈ Ind (W(π ,ψ )) where w = w w with w (resp. w ) S S S N,S P(F ) P the permutation matrix representing the longest element in the Weyl group of T in M (resp. in G). Finally, by [Sha81,Sect. 4],wehave S ∨ S ∨ (8.1.2.3) W (φ ) = L (1,π ×π ) L (1,π ×π )W | S N ,ψ n,i n+1,i φ G(F ) n,j n+1,j S 1i<jk 1i<jr where W | stands for the restriction of the Whittaker function W to the subgroup φ G(F ) φ G(F ) ⊂ G(A). (Note that, as χ is regular, the Rankin-Selberg L-functions L(s,π ×π ) S n,i n,j and L(s,π ×π ) are all regular at s = 1.) As, for every s ∈ R, n+1,i n+1,j k r S ∨ S ∨ S ∨ L (s, × ) = L (s,π ×π ) × L (s,π ×π ) n,i n+1,j n,i n+1,j i=1 j=1 S ∨ × L (s,π ×π ) n,i n,j 1i<jk S ∨ × L (s,π ×π ) , n+1,i n+1,j 1i<jr we deduce from (8.1.2.1), (8.1.2.2)and (8.1.2.3) the identity of the Theorem. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 305 8.1.3. Relative characters. — Let B be a K-basis of as in Section 2.8.3.We P,π define the relative character I of as the following functional on S(G(A)): λ(R(f )W )β (W ) φ η φ I (f ) = , f ∈ S(G(A)), W , W φ φ Whitt φ∈B P,π where the series converges, and does not depend on the choice of B ,byProposition P,π 2.8.4.1. 8.1.4. For every f ∈ S(G(A)), we set K (g) = K (h, g)dh and f ,χ f ,χ [H] K (g) = K (g, g )η (g )dg , g ∈[G], f ,χ G f ,χ [G ] where the above expressions are absolutely convergent by Lemma 2.10.1.1.3. Recall that the notion of relevant ∗-regular cuspidal datum has been defined in Section 5.1.3 andthatwehavedefinedfor any χ ∈ X a distribution I (see Theorem 3.2.4.1). Theorem 8.1.4.1. — Let f ∈ S(G(A)) and χ ∈ X (G).Then, 1. If χ is not relevant, we have K (g) = 0 for every g ∈[G] and moreover f ,χ I (f ) = 0. 2. If χ is relevant, we have I (f ) = K (g )η (g )dg χ G f ,χ [G ] where the right-hand side converges absolutely and moreover − dim(A ) I (f ) = 2 I (f ). The rest of this section is devoted to the proof of Theorem 8.1.4.1. Until the end, we fix a function f ∈ S (G(A)). 8.2. Proof of Theorem 8.1.4.1 8.2.1. We fix a character η of [G] whose restriction to [G ] is equal to η (such G G a character exists as the idèle class group of F is a closed subgroup of the idèle class group ∨ ∗ ∗ of E) and we set χ '=η ⊗χ ∈ X (G). We can write χ ' as (χ ',χ ' ) where χ ' ∈ X (G ) G n n+1 k k 306 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR for k = n, n + 1. For every g ∈[G],wedenoteby K (g,.) the function η K (g,.).By f ,χ G f ,χ Lemma 2.10.1.1.2 and (2.9.6.10), we have (8.2.1.1) K (g,.) ∈ S ([G]) = S ([G ])⊗S ([G ]) f ,χ χ ' χ ' n χ ' n+1 n n+1 for all g ∈[G]. Moreover, with the notation of Theorem 6.2.6.1,wehave (8.2.1.2)K (g) = P ⊗P (K (g,.)). G G f ,χ f ,χ n n+1 8.2.2. The non-relevant case. — Assume that χ is not relevant. By definition of a relevant cuspidal data (see Section 5.1.3), at least one of χ ' , χ ' is not distinguished n n+1 (see Section 6.2.3 for the definition of distinguished). Hence, by Theorem 6.2.5.1 and Theorem 6.2.6.1,P vanishes identically on S ([G ]) for k = n or k = n + 1. Thus, by G χ ' k (8.2.1.1)and (8.2.1.2), the function K vanishes identically. By Theorem 3.3.9.1 applied f ,χ to the expression (3.3.5.5), this implies I (f ) = 0. This proves part 1. of Theorem 8.1.4.1. 8.2.3. Regularized Rankin-Selberg period and convergence. — From now on, we assume that χ is relevant. By Lemma 2.10.1.1.2, for every g ∈[G] the function K (., g) belongs f ,χ to S ([G]). Since χ is relevant, it is H-regular in the sense of Section 7.1.2 (this follows from the dichotomy of Section 4.1.2). Therefore, by Theorem 7.1.3.1,P extends to a continuous linear form on T ([G]) that we shall denote by P . By definition of this extension and of the linear form λ (see Section 8.1.2), for every φ ∈ we have (8.2.3.3)P (E(φ)) =λ(W ). By Lemma 2.10.1.1.3 there exists N 0 such that the function g ∈[G]→ K (., g ) ∈ T ([G]) f ,χ N is absolutely integrable. As 1 ∗ K (g) = P (K (., g)) = P (K (., g)), H f ,χ f ,χ f ,χ H combined with Theorem 3.3.9.1 applied to the expression (3.3.5.6), this shows at once that the expression (8.2.3.4) K (g )η (g )dg f ,χ [G ] converges absolutely, is equal to I (f ) and that ) * ∗ ∗ 2 (8.2.3.5)I (f ) = P K (., g )η (g )dg = P (K ). χ f ,χ [G ] H H f ,χ [G ] Let us point out here that the absolute convergence of (8.2.3.4), together with Theorem 3.3.7.1.1, implies that the exponential-polynomial T → I (f ) of Theorem 3.2.4.1.2 is χ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 307 actually constant (but we caution the reader that this is not necessarily true for cuspidal data that aren’t ∗-regular and relevant). 2 ∨ ' ' 8.2.4. Spectral expression of K .— Set = ⊗η . We may write as a tensor f ,χ ' ' product and we let n n+1 ' ' ' β =β ⊗β : W(,ψ ) = W( ,ψ )⊗W( ,ψ ) → C n n+1 N n n n+1 n+1 be the (completed) tensor product of the linear forms β , β defined in Section 6.2.4. n n+1 Fix g ∈[G] and set f = K (g,.). Since χ is relevant, χ ' and χ ' are both distinguished. g f ,χ n n+1 Note that the linear map f ∈ S([G]) → W := W ∈ W(,ψ ) f, f N is the (completed) tensor product of the continuous linear maps f ∈ S([G ]) → W ∈ k f, W( ,ψ ) for k = n, n + 1 (as can be checked directly on pure tensors). Therefore, by k k (8.2.1.1), (8.2.1.2), Theorem 6.2.5.1 and Theorem 6.2.6.1 we have 2 − dim(A ) (8.2.4.6)K (g) = 2 β(W ). f , f ,χ g Let B be a K-basis as in Section 2.8.3. Then, we have f = f ,η E(φ) η E(φ) P,π g, g G G G φ∈B P,π where the sum converges absolutely in T ([G]) for some N 0. Hence, W = f ,η E(φ) η W g G G G φ f , φ∈B P,π in W(,ψ ). On the other hand, we easily check that β(η W ) =β (W ) and N G φ η φ f ,η E(φ) =K (g,.), E(φ) = E(R(f )φ)(g) g G G f ,χ G for every φ ∈ B . Therefore, by (8.2.4.6), we obtain P,π 2 − dim(A ) (8.2.4.7)K (g) = 2 E(R(f )φ)(g)β (W ). η φ f ,χ φ∈B P,π Note that by Proposition 2.8.4.1, the series above is actually absolutely convergent in T ([G]) for some N 0 (and not just pointwise). 8.2.5. End of the proof. — By (8.2.3.5), (8.2.3.3)and (8.2.4.7), we obtain − dim(A ) I (f ) = 2 λ(R(f )W )β (W ). χ φ η φ φ∈B P,π 308 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Using Theorem 8.1.2.1 and since B is an orthonormal basis of , this can be rewritten P,π as λ(R(f )W )β (W ) φ η φ − dim(A ) − dim(A ) P P I (f ) = 2 = 2 I (f ) W , W φ φ Whitt φ∈B P,π and this ends the proof of Theorem 8.1.4.1 in the relevant case. 9. Flicker-Rallis functional computation The goal of this section is to prove Theorem 1.3.2.3 of the introduction that states that two natural functionals are equal. This is established in Theorem 9.2.5.1. The bulk of the work is in proving its local avatar. The case of split algebra E/F amounts to comparing scalar products which was done in Appendix A of [FLO12], which is an inspiration for this section. 9.1. Local comparison 9.1.1. Let E/F be an etale quadratic algebra over a local field F. Let Tr : E/F E → F bethe tracemap.AsinSection 6.1.2,let ψ : F → C be a non-trivial additive × × character, τ ∈ E an element of trace 0 and we set ψ : E → C to beψ(x) =ψ (Tr(τ x)). We use ψ and ψ to define autodual Haar measures on F and E respectively. The duality F × E/F → C given by (x, y) →ψ(xy) defines a unique Haar measure on E/F dual to the one on F. This measure on E/F coincides with the quotient measure. 9.1.2. Let k = E or F. Let S be a closed subgroup of GL (k) equipped with a right-invariant Haar measure denoted by ds.Wedenoteby δ the modular character −1 such that δ (s) ds is a left-invariant Haar measure on S. Let R ⊂ S be a closed subgroup equipped with a right-invariant Haar measure dr.Wedenoteby the right S-invariant R\S −1 linear form on the space of left (R,δ δ )-equivariant functions on S such that −1 f (rt)δ (r)δ (r) drdt = f (s)ds S R R\S R S for all continuous and compactly supported functions f on S. We normalize the measures on GL (k) and its subgroups as follows: • On GL (k) we set dx ij dx = | det x| where x =(x ). ij THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 309 • On standard Levi subgroups of GL (k) we set the product measure using the measure defined above. • On (semi) standard unipotent subgroups N(k) ⊂ GL (k) we set the additive measure dn where n run through coordinates of N. ij ij • Let P is a standard parabolic subgroup of GL (k) with the standard Levi decom- position NM. We have the right-invariant measure dp := dndm on P(k) and the −1 × left invariant measure δ dp where δ : P(k) → R is the Jacobian homomor- P k >0 phism for the adjoint action of P(k) on N(k). With this normalization, we have for all f ∈ C (GL (k)) −1 f (g) dg = f (pn)δ (p) dpdn, GL (k) P(k) N(k) where N is the unipotent radical of the opposite parabolic to P. The linear form P(k)\GL (k) is given in this case by the integration over either N(k) or the standard maximal compact subgroup for a suitable Haar measure. Assume P is moreover maximal of type (n− 1, 1). Let P ⊂ P be the mirabolic subgroup (see Section 9.1.5 below). We have P(k) = P(k) × GL (k) and this gives the normalization of the measure on P(k). Moreover the modular character δ coincides with | det| on P(k). P(k) k 9.1.3. We will use the notation introduced in Section 4 with some changes. All groups considered in this section are subgroups of G = Res GL . We write simply n E/F n Gfor G ,P for the fixed minimal parabolic subgroup of G and N for its unipotent n 0 0 radical. In order to be as compatible with Appendix A of [FLO12] as possible, instead of G = GL (defined over F) we write G = GL and for any subgroup H of G we write H n F n F for H ∩ G . We will often identify a group with its F points in this section. 9.1.4. We define the character ψ : N → C as follows. Write n ∈ N as 0 0 ⎛ ⎞ 1 n n ... n 12 13 1n ⎜ ⎟ 01 n ... n 23 2n ⎜ ⎟ ⎜ ⎟ . . . . . . ⎜ ⎟ . . . n = 0 n , n ∈ E 2n ij ⎜ ⎟ ⎜ ⎟ . . . . ⎝ ⎠ . . 0 1 n n−1n 0 ... 00 1 and set ψ(n) =ψ((−1) (n + n +···+ n )). This is the same character as the one 12 23 n−1n from 6.1.2. By restriction,ψ defines a character of N ∩ M for all standard Levi subgroups M. 9.1.5. We denote by P = P the mirabolic subgroup of G defined as the stabi- lizerofthe rowvector 0 ... 01 . 310 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR ∞ ∞ For ϕ an element of C (N \P,ψ)={f ∈ C (P) | f (nx) =ψ(n)f (x), n ∈ N , x ∈ 0 0 P} we define β(ϕ) =β (ϕ) = ϕ(p) dp N \P 0,F F when it is absolutely convergent. Note that ψ is trivial on N . In the same way, we define 0,F β for all standard Levi subgroups M of G. 9.1.6. Let (G) be the set of irreducible generic complex representations of gen G = GL (E).Let W(π) = W (π) be the space of the Whittaker model of π ∈ (G) n gen with respect to the character ψ.Let δ = δ : W(π) → C be the evaluation at g ∈ G. The group G acts on W(π) by right multiplication. Fix P = MN ∈ F (P ).Let w be the element in the Weyl group of G such 0 M −1 w that w Mw is a standard Levi subgroup and the longest for this property. Let P = w w G w −1 N M ∈ F (P ) be the parabolic subgroup whose Levi component is M =w Mw . 0 M For σ ∈ (M) let Ind (W(σ)) be the normalized (smooth) induction to G, from gen W(σ), seen as a representation of P via the natural map P → M. As in Section 8.1.2 we define for ϕ ∈ Ind (W(σ)) the Jacquet’s integral. σ −1 −1 W(g,ϕ) = δ (ϕ(w u g))ψ (u ) du . e M We have then that W (ϕ) := W(e,ϕ) is a Whittaker functional on Ind (W(σ)). 9.1.7. Let σ ∈ (M) and ϕ ∈ Ind (W(σ)). We assume that σ is unitary. Then gen β (W(ϕ)) and β (ϕ(g)) are given by absolutely convergent integrals (cf. [Fli88, Lemma G M 4] and [BP21b, Proposition 2.6.1, proof of Lemma 3.3.1]). We shall say that σ is distin- guished if σ admits a non-zero continuous linear form invariant under G .Fromnow on we assume that σ is distinguished. Theorem 9.1.7.1. — Under the assumptions above, the linear map β is M -invariant and M F we have (9.1.7.1) β (W(ϕ)) =β (ϕ) where (9.1.7.2) β (ϕ) = β (ϕ(g)) dg. P \G F F Remark 9.1.7.2. — The integral (9.1.7.2) makes sense since β is invariant and sinceδ restricted to P equalsδ . This latter observation pertains also to similar integrals P F in the proof below. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 311 Proof. — If E is split then the theorem follows from [Bar03, corollary 10.4] and [FLO12, proposition A.2]. We assume from now on that E/F is a field extension. Let’s prove the first assertion. Working on factors of M we are reduced to the case M = G. If F is p-adic, the result follows from [GJR01, proposition 2]. More precisely, according to this proposition, any non zero G -invariant continuous linear form on (the Whittaker model of) a distinguished unitary generic representation of G should be (up to a non zero constant) β . We claim that the result also holds for F = R. Indeed, following the proof of [GJR01, proposition 2] and using multiplicity one property [AG09, theorem 8.2.5], we see that it suffices to show that any Z(g)-finite distribution on the symmetric space G/G left-invariant by P is also left-invariant by G . But this is a variant of [Bar03, F F F theorem 10.4] where the adjoint action of G on itself is replaced by the action of G on F F G/G . It can be proved as in [Bar03, section 9]: indeed the bulk of the argument there is a descent to the centralizer of semi-simple elements where the use of the exponential reduces everything to an analogous theorem for the adjoint action of G on g .Infact F F such a descent can be performed for G/G . The main observation is that the tangent space of G/G at the origin can be identified as a G -representation to the adjoint action F F of G on g . F F Once we have the first assertion, we observe that β is a non-zero G -invariant linear form on Ind (W(σ)) (since β is non-zero as a consequence of [GK72], [Jac10, Proposition 5] and [Kem15]).Still by thesameargumentasbeforewededucethatthe equality (9.1.7.1) holds up to a non zero constant. We have to show that this constant is 1. Thus it suffices to prove (9.1.7.1) for one specific ϕ such that β (ϕ) = 0. Thereisaneasyreduction to thecasewhere P ismaximal.Let Q = LV ⊃ Pbe maximal and suppose the assertion holds for M = M .Then −1 β (ϕ) = β (ϕ(g)) dg = δ (q)β (ϕ(qg)) dqdg. M M P \G Q \G P \Q F F F F F F The inner integral on the RHS by induction hypothesis equals β (g.W (ϕ)). L L G L If we let ϕ (g) = g.W (ϕ) ∈ W(Ind (W(σ))) then ϕ ∈ Ind (W(Ind (W(σ)))) and L∩P Q L∩P so by assumption and transitivity of Jacquet’s integral we obtain β (g.W (ϕ)) dg =β (W(ϕ)). L G Q \G F F From now on we assume that P = MN is maximal of type (n , n ).In[FLO12], the 1 2 authors use U instead of N. We will consequently use N in place of U here. Write M = M × M with M Res GL ,M being in the upper and M in the lower diagonal. 1 2 i E/F n 1 2 Let ) * 0I −1 n w =w = . I 0 2 312 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR Let P = M N be of type (n , n ) so that M = M × M with M Res GL ,M 2 1 E/F n 2 1 i i 2 being in the upper and M in the lower diagonal. Let P be the mirabolic subgroup of 1 i M .Let N be the maximal upper triangular unipotent of M , similarly without .Note i i i −1 P = wP w etc. We identify N with the group of n × n matrices. We write that i 2 1 σ = σ ⊗ σ where σ and σ are respectively viewed as representations of M and M . 1 2 1 2 1 2 We view σ as a representation of M as well. Let C ={I +ξ | ξ column vector of size n in the i-th column}⊂ N , i n 2 i = n + 1,..., n. Let R ={I +ξ | ξ row vector of size n in the i-th row}⊂ N , i n 2 i = n + 1,..., n. We can identify C and R with E which induces a pairing between C and R that we i j i i−1 will denote ·,· . We note some obvious facts • The groups R (resp. C ) commute with each other and are normalized by M . i i • The commutator set [C , R ] is contained in N for j < i. i j We define the following groups 1. X = C ··· C .Itisnormalizedby N . i i+1 n 2. Y = R ··· R .Itisnormalizedby N . i n +1 i−1 2 1 3. V = N X Y . This is a unipotent group. i i i 4. V = N X Y ⊃ V . This is a unipotent group. i−1 i i i 1 M V , i > n , i 2 S = P N N , i = n 2 1 6. S = M V for i > n . i 2 i Note that • S = C S for i > n as well as S = R S for i > n + 1. i i 2 i−1 i−1 2 i i • Let δ and δ be modular characters of S and S respectively. It follows that i i i i −1 (δ ) =| det| δ and (δ ) =| det| δ in the above range. |S E i |S i−1 i i i i−1 E • Let δ and δ be modular characters of S and S respectively. i,F i,F i,F i,F n+n −2i+1 • We have δ =| det| for i ≥ n . i 2 E THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 313 Let us define A = Ind (W(σ ) ⊗ψ ), i = n + 1,..., n. i 2 i 2 A = Ind ψ, i = n . i 2 Here, ψ is the character of V - the unipotent radical of S - whose restriction to X Y is i i i i i trivial and that coincides with ψ on N . Explicitly, for i > n we have ) * 1/2 δ (m) A ={ϕ : P → W(σ ) | ϕ(mvg) = ψ (v)σ (m)ϕ(g), i 2 i 2 | det m| g ∈ P, m ∈ M ,v ∈ V }. 2 2 We also denote A the L -induction version of the above as in [FLO12]. Note that • A = Ind (Ind ((W(σ ) ⊗ψ ))) for i > n . i 2 i 2 S S • A = Ind (Ind ((W(σ ) ⊗ψ ))) for i > n + 1. i−1 2 i−1 2 S S i−1 n +1 P 2 • A = Ind (Ind ψ). 2 S N n +1 For any i > n the restriction map to C identifies Ind ((W(σ ) ⊗ ψ )) with 2 i 2 i C (C , W(σ )) because S /S = C . Let us denote ϕ → ϕ| the restriction map and i 2 i i C i i ι the map in the reverse order. Similarly, restriction to R identifies Ind ((W(σ )⊗ C i−1 2 i S i−1 ψ )) with C (R , W(σ )). Let us denote ϕ →ϕ the restriction map and ι the i−1 i−1 2 R R i−1 i−1 map in the reverse order. Given that C and R are in duality we have a Fourier transform i i−1 2 2 F : L (C , W(σ )) → L (R , W(σ )) i 2 i−1 2 where W(σ ) is the L completion of W(σ ). 2 2 Lemma 9.1.7.3. — For i = n,..., n + 2, the above Fourier transform induces a map 2 P i 2 B : A = Ind (Ind ((W(σ ) ⊗ψ ))) → A i 2 i i S S i−1 P i = Ind (Ind ((W(σ ) ⊗ψ ))) 2 i i−1 S S i i induced from the equivalence Ind ((W(σ ) ⊗ ψ )) → Ind ((W(σ ) ⊗ ψ )) given by ϕ → 2 i 2 i−1 S S i i−1 ι (F (ϕ| )). It is an equivalence of unitary representations. R C i−1 i i Similarly, we have the map S S n +1 n +1 2 2 F : Ind ((W(σ ) ⊗ψ )) → Ind ψ n +1 2 n +1 2 S 2 N n +1 0 2 314 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR given by 1/2 F ϕ(vm) =ψ(v)| det m| ϕ( ˆ χ )(m), m ∈ M ,v ∈ V = N N n +1 m 2 E 2 n +1 1 where × −1 χ : C → C ,χ (c) =ψ(mcm ), ϕ( ˆ χ) = ϕ(c)χ(c) dc. m n +1 m n +1 Lemma 9.1.7.4. — The above Fourier transform induces the equivalence of unitary represen- tations n +1 2 P 2 B : A = Ind (Ind ((W(σ ) ⊗ψ ))) n +1 2 n +1 2 n +1 S S 2 2 n +1 n +1 n +1 2 P 2 → A = Ind (Ind ψ). n N S 0 n +1 From now on we take ϕ ∈ Ind (W(σ ⊗σ )) supported on the big cell PwP .We 1 2 shall show (9.1.7.1) for such functions which suffices to conclude. For m ∈ M ,let δ : W(σ ⊗σ ) → W(σ ) be the evaluation map in the first vari- 1 1 2 2 able. Define for p ∈ P ϕ (p) =δ ϕ(wp) ∈ A . n n We have then 2 2 ϕ(g) dg =ϕ . 2 n L (W(σ ⊗σ )) n 1 2 P\G ) * 1/2 −1 Let i ∈{n,..., n + 1}. The restriction of to S is equal to δ δ . 2 i,F i,F | det| −1 Thus for φ ∈ A ,the map p ∈ P → β (φ(p)) is (S ,δ δ )-equivariant. We may i F M i,F i,F 2 F introduce (at least formally) β (φ) = β (φ(p)) dp. i M S \P i,F F Lemma 9.1.7.5. — We have β (ϕ) =β (ϕ ). n n Proof. — Indeed by various changes of variables we get the equality of absolutely convergent integrals: β (ϕ) = β (ϕ(g)) dg = β (ϕ(gw)) dg M M P \G P \G F F F F THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 315 = β (ϕ(uw)) du = β (ϕ(wu )) du M M N N = β (δ (ϕ(wu ))) dm du M 1 2 m N N \P 1,F 1,F −1/2 = β (δ (ϕ(m wu )))δ (m ) dm du M 1 1 1 2 e P N N \P 1,F 1,F 1/2 = β (δ (ϕ(wm u )))δ (m ) dm du e 1 P 1 1 N N \P F 1,F 1,F = β (δ (ϕ(wp))) dp. S \P n,F F Define recursively ϕ = B (ϕ ) for i = n,..., n + 1. As shown at the end of the i−1 i i 2 Appendix A.3 of [FLO12], we have (9.1.7.3) ϕ = W(ϕ). Lemma 9.1.7.6. — For i = n,..., n + 2 we have β (ϕ ) =β (ϕ ). i i i−1 i−1 Proof. — Observe that the map (9.1.7.4) g ∈ P → β (ϕ (rg)) dr F M i−1 i−1,F −1 is (S,δ δ )-equivariant. We have i i,F P β (ϕ ) = β (ϕ (rg)) drdg. i−1 i−1 M i−1 S \P R F i−1,F i,F Let g ∈ P . By construction we have: ϕ (rg) = ϕ (cg)ψ(c, r )dc. i−1 i i Since the duality ·,· between C and R restricts to a duality between C /C and i i i−1 i i,F R we getthat(9.1.7.4) is equal to i−1,F β (ϕ (cg)) dc. M i i,F 316 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR But we have β (ϕ ) = β (ϕ (cg)) dcdg. i i M i S \P C F i,F i,F Note that the inner function in c is compactly supported and the convergence of the integrals we have considered follows. Lemma 9.1.7.7. — We have β (ϕ ) =β (W(ϕ)). n +1 n +1 G 2 2 Proof. — Recall that W(ϕ) =ϕ = B (ϕ ), see (9.1.7.3). Thus the right-hand n n +1 n +1 2 2 2 side is B (ϕ )(p) dp = B (ϕ )(mp) dmdp. n +1 n +1 n +1 n +1 2 2 2 2 N \P S \P N \M 0,F F F n +1,F 2,F 2,F Let us fix p ∈ P and let us work on the inner integral. By definition of B , it is equal F n +1 to −1 ϕ (cp)(m)ψ(mcm )dc| det m| dm n +1 F N \M C n +1 2,F 2,F 2 We denote by P the standard mirabolic subgroup of M .Let P ⊂ Q ⊂ M be the n 2 n 2 2 2 maximal parabolic subgroup of type (n − 1, 1).Let N the unipotent radical of the 2 Q opposite parabolic subgroup. The integral above is also equal to −1 2 ϕ (cp)(qλn¯)ψ(qλnc¯ (qλn¯) )dqdc|λ| d nd ¯ λ n +1 2 F GL N C N \P 1,F Q,F n +1 2 2,F n ,F whereweidentify λ with the diagonal matrix diag(1,..., 1,λ) ∈ M . A first observation is that we have −1 −1 ψ(qλnc¯ (qλn¯) ) =ψ(λnc¯ (λn¯) ). Hence the inner integral over q is: ϕ (cp)(qλn¯)dq =β (ϕ (cp)(·λn¯)) n +1 M n +1 2 2 N \P 2,F n ,F =β (ϕ (cp)). M n +1 For the latter equality, we use the fact that σ is distinguished and thus β is left M - 2 M 2,F invariant. By a change of variables we get −1 β (ϕ (cp))ψ(n¯λc(n¯λ) )dqdc|λ| d nd ¯ λ. M n +1 GL N C 1,F Q,F n +1 2 THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 317 Now |λ| dλ is the additive measure. We can identify c ∈ C with an element X ∈ E n +1 F 2 n−1 × and n¯λ with an element Y ∈ F × F ⊂ E insuchawaythatwehave −1 ψ(n¯λc(n¯λ) ) =ψ(X, Y ) where X, Y is the obvious pairing. By Fourier inversion, we deduce that the previous integral is β (ϕ (cp))dc. M n +1 n +1,F Hence β (W(ϕ)) = β (ϕ (cp))dcdp G M n +1 S \P C F n +1,F n +1,F =β (ϕ ). n +1 n +1 2 2 Note that the last integrals are absolutely convergent and our computations are justified. Theorem 9.1.7.1 then follows from Lemmas 9.1.7.5, 9.1.7.7, 9.1.7.6 for our spe- cific functions ϕ and thus holds in general. 9.2. Global comparison 9.2.1. We go back to the global setting and notation introduced in Section 3.1. 9.2.2. We normalize all local and global measures as in Section 2.3, with respect × × × × to a fixed character ψ : F\A → C . We have the quadratic character η : F \A → C associated to E/F and the associated character η of G (A) as defined in Paragraph 3.1.6. 9.2.3. As in Section 6.1.2, we also fix a non-trivial additive character ψ : E\A → C ,trivial on A which is then used to define a non-degenerate character ψ E N of the maximal unipotent subgroup of G(A) as in the beginning of Section 8. 9.2.4. Let χ ∈ X (G) be a relevant ∗-regular cuspidal datum (cf. Section 5.1.3) G(A) and let (M,π) represent χ.Set = Ind (π). P(A) 318 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 9.2.5. The comparison. Theorem 9.2.5.1. — For all φ ∈ we have J (φ) =β (W ) η η φ where • J is defined in 5.1.2.1; • β is defined in 8.1.2. • W ∈ W(,ψ ) is defined in 8.1.1. φ N Proof. — The proof is essentially the same as of Theorem 8.1.2.1. The only differ- ence is that the natural analogue of (8.1.2.1) is provided by Proposition 3.2 of [Zha14a] and the analogue of (8.1.2.2) is established invoking Theorem 9.1.7.1. Corollary 9.2.5.2. — We have the equality of distributions on S(G(A)) I = I P,π where 1. I is defined in Section 5.1.5. P,π 2. I is defined in Section 8.1.3. Proof. — Looking at definitions of I and I , taking into consideration Theorem P,π 8.1.2.1 and Theorem 9.2.5.1 above, we see that we need to establish for all φ ∈ λ(W ) = I(φ, 0) RS where λ = Z (0,·) is defined in Section 8.1.2 and I(φ, 0) is given by Proposition 5.1.4.1. This equality is precisely Theorem 1.1 of [IY15]. 10. Proofs of the Gan-Gross-Prasad and Ichino-Ikeda conjectures 10.1. Identities among some global relative characters 10.1.1. Besides notation of Sections 2 and 3, we shall use notation of Section 1. We fix an integer n 1 and we will omit the subscript n: we will write H for H . 10.1.2. Relative characters for unitary groups. — Let h ∈ H be a Hermitian form. Let σ be an irreducible cuspidal automorphic subrepresentation of the group U .Wedefine the relative character J by J (f ) = P (π(f )ϕ)P (ϕ), ∀f ∈ S(U (A)) h h h ϕ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 319 where ϕ runs over a K -basis (see 2.8.3) for some maximal compact subgroup K ⊂ h h U (A). The periods P are those defined in 1.1.5. For any subset X ⊂ X(U ) of cuspidal h h 0 h data which do not come from proper Levi subgroups (that is they are represented by pairs (U ,τ) where τ is a cuspidal automorphic representation) we define more generally h h (10.1.2.1) J (f ) = J (f ) X σ χ∈X where the inner sum is over the set of the constituents σ of some decomposition of L ([U ]) (see Section 2.9.2.1) into irreducible subrepresentations. One can show that the double sum is absolutely convergent (see e.g. [BP21a, Proposition A.1.2]). 10.1.3. Let V ⊂ S ⊂ V be a finite set of places containing all the places that F,∞ 0 F are ramified in E. For every v ∈ V , we set E = E ⊗ F and whenv/∈ V we denote F v F v F,∞ by O ⊂ E its ring of integers. Let H ⊂ H be the (finite) subset of Hermitian spaces of E v rank n over E that admits a selfdual O -lattice for everyv/∈ S . E 0 ◦ 0 For each h ∈ H ,the group U is naturally defined over O and we fix a choice of such a model. Since we are going to consider invariant distribution, this choice is irrelevant. We define the open compact subgroups K = U (O ) and h v h v/∈S ◦ S S 0 0 K = G(O ) respectively of U (A ) and G(A ). v h v/∈S ◦ ◦ Letv/∈ S .Wedenoteby S (U (F )), resp. S (G(F )), the spherical Hecke alge- 0 h v v bra of complex functions on U (F ) (resp. G(F ))thatare U (O )-bi-invariant (resp. h v v h v G(O )-bi-invariant) and compactly supported. We have the base change homomorphism ◦ ◦ BC : S (G(F )) → S (U (F )). h,v v h v ◦ S ◦ S ◦ 0 0 We denote by S (U (A )), resp. S (G(A )), the restricted tensor product of S (U (F )), h h v resp. S (G(F )),forv/∈ S . We have also a global base change homomorphism given by v 0 BC =⊗ BC . v/∈S h,v h 0 ◦ ◦ ◦ We also denote by S (G(A)) ⊂ S(G(A)) and S (U (A)) ⊂ S(U (A)),for h ∈ H , h h ◦ ◦ the subspaces of functions that are respectively bi-K -invariant and bi-K -invariant. ◦ h 10.1.4. Transfer. — Let h ∈ H .Weshall saythat f ∈ S(G(F )) and f ∈ S S 0 0 S S(U (F )) are transfers if the functions f and f have matching regular orbital inte- h S S 0 0 S grals in the sense of Definition 4.4 of [BPLZZ21]. The Haar measures on the F -points of the involved groups are those defined in Section 2.3.3. The product structure is given by the convolution where the Haar measure is normalized so that the characteristic functions of U (O ) and G(O ) are units. h v v 320 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR 10.1.5. Let P be a standard parabolic subgroup of G and π be a cuspidal auto- morphic representation of M .Let χ ∈ X(G) be the class of the pair (M ,π). We assume P P henceforth that χ is a regular relevant cuspidal datum in the sense of Section 5.1.3. Set = Ind (π) for the corresponding parabolically induced representation. The assumption that χ is regular and relevant means exactly that is a Hermitian Arthur parameter (see Section 1.1.3). Moreover,weassume,aswemay,that S has been chosen such that admits K -fixed vectors. Attached to these data, we have three distributions denoted by I ,I and I .The χ P,π first is constructed as a contribution of the Jacquet-Rallis trace formula and it is defined in Theorem 3.2.4.1. The second and third are relative characters built respectively in Section 5.1.5 and Section 8.1.3. The bulk of the paper was devoted to the proof of the following identities (see Theorem 5.2.1.1,Theorem 8.1.4.1 and Corollary 9.2.5.2) − dim(a ) − dim(a ) P P (10.1.5.2)I = 2 I = 2 I . χ P,π 10.1.6. Let S be the union of S \ V and the set of all finite places of F that 0 F,∞ are inert in E. We define X ⊂ X(U ) as the set of equivalence classes of pairs (U ,σ) where σ is h h a cuspidal automorphic representation of U (A) that satisfies the following conditions: • σ is K -unramified; • for allv/∈ S ∪ V the (split) base change of σ is . F,∞ v v ◦ h ◦ ◦ Proposition 10.1.6.1. — Let f ∈ S (G(A)) and f ∈ S (U (A)) for every h ∈ H .As- sume that the following properties are satisfied for every h ∈ H : S ,∗ S ,∗ 0 0 S S ◦ S 0 0 0 1. f =( )f ⊗ f with f ∈ S(G(F )) and f ∈ S (G(A )). S S S H G 0 0 0 h 0 2 h h,S h h,S ◦ S 0 0 0 2. f =( ) f ⊗ f with f ∈ S(U (F )) and f ∈ S (U (A )). h S h S S 0 U 0 0 3. The functions f and f are transfers. 0 S h,S 0 S 0 0 4. f = BC (f ). 5. The function f is a product of a smooth compactly supported function on the restricted product G(F ) by the characteristic function of G(O ). v v v/∈S v∈S \S 0 0 Then we have: h h − dim(a ) − dim(a ) P P (10.1.6.3) J (f ) = 2 I (f ) = 2 I (f ). h P,π h∈H Remark 10.1.6.2. — If the assumptions hold for the set S , they also hold for any large enough finite set containing S : this follows from the Jacquet-Rallis fundamental lemma (see [Yun11]and [BP]) and the simple expression of the transfer at split places (see [Zha14b, proposition 2.5]). We leave it to the reader to keep track of the different choices of Haar measures in these references. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 321 Proof. — Theproof follows thesamelinesasthe proof of [BPLZZ21,Theorem 1.7]. For the convenience of the reader, we recall the main steps. In Theorem 3.2.4.1 we defined a distribution I on S(G(A)): this is the “Jacquet- Rallis trace formula” for G. We have an analogous distribution J on S(U (A)) for each h ∈ H:itisdefinedin[Zyd20, théorème 0.3] for compactly supported functions and extended to the Schwartz space in [CZ21, §1.1.3 and théorème 15.2.3.1]. Note that, by the Jacquet-Rallis fundamental lemma [Yun11], [BP], for every h ∈ H \ H there exists a place v ∈ S \ S such that the characteristic function 1 admits the zero function 0 G(O ) 0 v on U (F ) as a transfer. Therefore, by [CZ21, théorème 1.6.1.1], the hypotheses of the h v proposition imply: h h (10.1.6.4) I(f ) = J (f ). h∈H S S 0 0 We will denote by M (G(A)), resp. M (U (A)), the algebra of S -multipliers defined in [BPLZZ21, definition 3.5] relatively to the subgroup G(O ), resp. v/∈S S S 0 0 U (O ). Any multiplier μ ∈ M (G(A)), resp. μ ∈ M (U (A)), gives rise to a h v h v/∈S ◦ ◦ linear operator μ∗ of the algebra S (G(A)), resp. S (U (A)) and for every admissible irreducible representation π of G(A), resp. of U (A), there exists a constant μ(π) ∈ C ◦ ◦ such that π(μ ∗ f ) =μ(π)π(f ) for all f ∈ S (G(A)), resp. f ∈ S (U (A)). Let ξ be the infinitesimal character of .By[BPLZZ21, Theorem 4.12 (4)], for ◦ h every h ∈ H and (U ,σ) ∈ X , the base-change of the infinitesimal character of σ is ξ . However, the universal enveloping algebras of the complexified Lie algebras of U are all canonically identified for h ∈ H (since these are inner forms of each other) and base- change is injective at the level of infinitesimal characters. As, by [GRS11], there exists at ◦ h least one h ∈ H such that the set X is nonempty (we may even take for h any quasi-split Hermitian form unramified outside S ), there exists a common infinitesimal character ξ h ◦ of all (U ,σ) ∈ X ,for h ∈ H , whose base-change is ξ . By the strong multiplicity one theorem of Ramakrishnan (see [Ram18]) and The- orem 3.17 of [BPLZZ21], one can find a multiplier μ ∈ M (G(A)) such that i. μ() = 1; ii. For all χ ∈ X(G) represented by a pair (M ,π ) such that the central character 1 1 of π is trivial on A and χ =χ we have 1 G K = 0 μ∗f ,χ where the kernel K is defined as in Section 2.10.1. μ∗f ,χ By Theorem 3.6 and Theorem 4.12 (3) of [BPLZZ21], for every h ∈ H there exists h S a multiplier μ ∈ M (U (A)) such that h h iii. μ (σ) = 1for all (U ,σ) ∈ X ; 0 322 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR iv. For all χ ∈ X(U ) such that χ ∈/ X and for all parabolic subgroups P of U , h h we have K = 0 h h P,μ ∗f ,χ where the left-hand side is the kernel of the operator given by the right convo- h h 2 lution of μ ∗ f on L ([U ] ) (see (2.9.2.1)). h P Moreover, by [BPLZZ21, Proposition 4.8, Lemma 4.10], we may choose μ and h h h ◦ μ such that the functions μ ∗ f and μ ∗ f ,for h ∈ H , still satisfy the assumptions of the proposition. So, in particular, from (10.1.6.4) applied to the functions μ ∗ f and h h h (μ ∗ f ) ◦ instead of f and (f ) ◦,weget h∈H h∈H h h h (10.1.6.5) I(μ ∗ f ) = J (μ ∗ f ). h∈H Note that by conditions i. and iii. we have: I (μ ∗ f ) = I (f ), I (μ ∗ f ) = I (f ) and P,π P,π (10.1.6.6) h h h h h ◦ J (μ ∗ f ) = J (f ), for every h ∈ H . h h X X 0 0 Let χ ∈ X(G) be such that χ = χ . It’s easy to see that the integral (3.3.3.4)at- tached to χ vanishes for any f ∈ S(G(A) if χ does not satisfy the hypothesis in condition ii. If it does, the integral (3.3.3.4) attached to χ and μ ∗ f vanishes by condition ii. Thus we can conclude by Theorem 3.3.9.1 that I (μ∗ f ) = 0 in any case. By Theorem 3.2.4.1 assertion 4 and by the equality (10.1.5.2), we see that the left-hand side of (10.1.6.5)re- duces to − dim(a ) − dim(a ) P P I (μ ∗ f ) = 2 I (μ ∗ f ) = 2 I (μ ∗ f ). χ P,π On the other hand, by iv. and the very definition of J given in [Zyd20], the right-hand side of (10.1.6.5) reduces to h h h J (μ ∗ f ). h∈H Therefore, (10.1.6.5)and (10.1.6.6) give the identity of the proposition. 10.2. Proof of Theorem 1.1.5.1 10.2.1. Let = Ind (π) be a Hermitian Arthur parameter of G. Note that by properties 1 and 2 of Section 1.1.3, the cuspidal datum χ associated to the pair (M ,π) is regular and relevant in the sense of Section 4.3.2.For h ∈ H and σ a cuspidal auto- morphic representation of U (A), it is readily seen that the linear form P is nonzero on h h THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 323 σ if and only if J is not identically zero. On the other hand, the linear form J or β is η η always nonzero (this follows either from the fact that χ is relevant or is an easy conse- quence of [GK72], [Jac10, Proposition 5] and [Kem15]) whereas the linear form I, from Proposition 5.1.4.1,or λ,fromSection 8.1.2, is nonzero if and only if L( ,) = 0(as fol- lows either from the work of Ichino and Yamana, see [IY15, corollary 5.7], or of Jacquet, Piatetski-Shapiro and Shalika [JPSS83], [Jac04]). Therefore, we similarly deduce that the distribution I or I is non-zero if and only if L( ,) = 0. P,π As a consequence, Theorem 1.1.5.1 amounts to the equivalence between the two assertions: (A) The distribution I or I is non-zero. P,π (B) There exist h ∈ H, f ∈ S(U (A)) and a cuspidal subrepresentation σ of U h h such that BC(σ) = and J (f ) = 0. 10.2.2. Proof of (A) ⇒ (B).— We choose the S of Section 10.1.3 such that I is not identically zero on f ∈ S (G(A)). Then Assertion (B) above is a consequence of h ◦ Proposition 10.1.6.1: it suffices to take functions f and f for h ∈ H satisfying the hy- potheses of that theorem and such that I (f ) = 0. That it is possible is implied by a combination of a result of [Xue19] and the existence of p-adic transfer [Zha14b]. 10.2.3. Proof of (B) ⇒ (A).— We may choose the set S so that there exist h ∈ 0 0 ◦ ◦ H , f ∈ S (U (A)) and a cuspidal representation σ of U such that for v/∈ S (see h 0 h 0 0 0 h 0 ◦ h Section 10.1.6)BC(σ ) = and J (f ) = 0. For any other h ∈ H we set f = 0. Up 0,v v σ 0 to enlarging S , we may assume that the family (f ) ◦ satisfies conditions 2. and 5. of 0 h∈H h h h 0 0 Proposition 10.1.6.1. Moreover, we have (see [Zha14b, §2.5]) J (f ∗ f ) 0for every σ 0 0 h h h 0 h 0 0 σ ∈ X and J (f ∗ f )> 0. In particular, the left hand side of (10.1.6.3)for thefamily 0 σ 0 0 h h (f ∗ f ) ◦ is nonzero. Once again by [Xue19] and the existence of p-adic transfer h∈H 0 0 ◦ h ◦ [Zha14b], this implies that we can find test functions f ∈ S (G(A)) and f ∈ S (U (A)), for h ∈ H , satisfying all the conditions of Proposition 10.1.6.1 and such that the left hand side of (10.1.6.3) is still nonzero. The conclusion of this proposition immediately gives Assertion (A). 10.3. Proof of Theorem 1.1.6.1 10.3.1. Let h ∈ H and σ be a cuspidal automorphic representation of U (A) which is tempered everywhere. By [Mok15], [KMSW], σ admits a weak base-change to G. Moreover, by these references is also a strong base-change of σ : for every place v of F, the local base-change of σ (defined in [Mok15]and [KMSW]) coincides with . v v In particular, it follows that is also tempered everywhere. We choose a finite set of places S as in Section 10.1.3 such that h ∈ H and σ as well as the additive character ψ used to normalize local Haar measures in Section 2.3 are unramified outside of S . 0 324 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR For each place v of F, we define a distribution J on S(U (F )) by σ h v h h h J (f ) = Trace(σ (h )σ (f ))dh , f ∈ S(U (F )), σ v v v v h v v v v v U (F ) where h h σ (f ) = f (g )σ (g )dg v v v v v v v U (F ) h v and the Haar measures are the one defined in Section 2.3.3.Moreoverby[Har14], and since the representations σ are all tempered, the expression defining J is absolutely v σ convergent and for everyv/∈ S we have L( , ) −2 2 J (1 ) = . σ U (O ) v h v U ,v L(1,σ , Ad) 10.3.2. By [Zha14a, Lemma 1.7] and our choice of local Haar measures, The- orem 1.1.6.1 is equivalent to the following assertion: for all factorizable test function h h 0 2 h f ∈ S(U (A)) of the form f =( ) f × 1 ,wehave h U (O ) 0 v h v U v∈S v/∈S 0 0 S 1 L ( ,) h h −1 2 h (10.3.2.1) J (f )=|S | J (f ). σ v v L (1,σ, Ad) v∈S 10.3.3. For every place v of F, we define a local relative character I on G(F ) by λ ( (f )W )β (W ) v v v v η,v v I (f ) = , f ∈ S(G(F )), v v v W , W v v Whitt,v W ∈W( ,ψ ) v v N,v where the sum runs over a K -basis of the Whittaker model W( ,ψ ) (in the sense of v v N,v Section 2.8.3)and λ , β , .,. are local analogs of the forms introduced in Section v η,v Whitt,v 8.1.2 given by λ (W ) = W (h )dh , v v v v v N (F )\H(F ) H v v β (W ) = W (p )η (p )dp , η,v v v v G ,v v v N (F )\P (F ) v v and W , W = |W (p )| dp . v v Whitt,v v v v N(F )\P(F ) v v Note that the above expressions, and in particular λ (W ), are all absolutely convergent v v due to the fact that is tempered (see [JPSS83, Proposition 8.4]). The above defini- v THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 325 tion also implicitly depends on the choice of an additive character ψ of A /Etrivial on A (through which the generic character ψ is defined, see beginning of Section 8 and Section 6.1.2) and up to enlarging S , we may assume that ψ is unramified outside of S . Then, it follows from the definition of I that for every factorizable test function S ,∗ S ,∗ 0 0 f ∈ S(G(A)) of the form f = f × 1 ,wehave v G(O ) H G v∈S v/∈S v 0 0 S 1 L ( ,) (10.3.3.2) I (f ) = I (f ) − v L (1,, As ) v∈S n n+1 S − S (−1) S (−1) 0 0 0 where we have set L (s,, As ) = L (s, , As )L (s, , As ). n n+1 10.3.4. Let f be a test function as in Section 10.3.2. Then, as both sides of (10.3.2.1) are continuous functionals in f for v ∈ V , by the main result of [Xue19]we F,∞ may assume that for every v ∈ V the function f admits a transfer f ∈ S(G(F )).On F,∞ v v the other hand, by [Zha14b], for everyv ∈ S \ V , the function f admits a transfer f ∈ 0 F,∞ v S(G(F )). Moreover, by the results of those references we may also choose the transfers such that for every h ∈ H with h = h, the zero function on U (F ) is a transfer of h S S ,∗ S ,∗ 0 0 h f = f . We set f = f × 1 . Then, setting f = 0for every S v S G(O ) 0 v∈S H G 0 v/∈S v 0 0 ◦ h h ∈ H \{h}, the functions f and (f ) satisfy the assumptions of Proposition 10.1.6.1. h ∈H Therefore, we have h h − dim(a ) (10.3.4.3) J (f ) = 2 I (f ). σ ∈X 10.3.5. If there exists a place v ∈ S such that σ does not support any nonzero 0 v continuous U (F )-invariant functional, both sides of (10.3.2.1) are automatically zero. Assume now that for every v ∈ S , the local representation σ supports a nonzero 0 v continuous U (F )-invariant functional. By the local Gan-Gross-Prasad conjecture [BP20], and the classification of cuspidal automorphic representations of U in terms h h of local L-packets [Mok15], [KMSW], it follows that all the terms except possibly J (f ) in the left hand side of (10.3.4.3)are zero.Moreover, by [BP21c, Theorem 5.4.1] and since is the local base-change of σ , there are explicit constants κ ∈ C for v ∈ S v v v 0 satisfying κ = 1and such that v∈S (10.3.5.4) I (f ) =κ J (f ) v v σ v v v for every v ∈ S . Combining this with (10.3.3.2), we get h h − dim(a ) − dim(a ) P P J (f ) = 2 I (f ) = 2 I (f ) P,π L ( ,) − dim(a ) 2 = 2 I (f ) L (1,, As ) v∈S 0 326 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR L ( ,) − dim(a ) 2 h = 2 J (f ). v v L (1,, As ) v∈S S S − dim(a ) 0 0 P As L (s,, As ) = L (s,σ, Ad) and |S |= 2 , this exactly gives (10.3.2.1)and ends the proof of Theorem 1.1.6.1. Acknowledgements The project leading to this publication of R. B.-P. has received funding from Excellence Initiative of Aix-Marseille University-A*MIDEX, a French “Investissements d’Avenir” programme. We thank Harald Grobner for valuable comments on a prelimi- nary version of this paper. Finally we would like to gratefully thank the referee for care- fully reading our work and for his very helpful comments. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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Appendix A: Topological vector spaces A.0.1 In this paper, by a locally convex topological vector space (LCTVS) we mean a Hausdorff locally convex vector space over C. Most LCTVS encountered in this paper will be Fréchet or LF (that is a countable inductive limit of Fréchet spaces) or even strict LF (that is countable inductive limit lim V of Fréchet spaces with closed embeddings −→ THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 327 V → V as connecting maps) spaces. Let V and W be LCTVS. We denote by V the n n+1 topological dual of V and by Hom(V, W) the space of continuous linear mappings V → W both equipped with their weak topologies (i.e. topologies of pointwise convergence). Recall that a total subspace H ⊂ V is a subspace such that Ker(λ) = 0. A bounded λ∈H subset B ⊆ V is one that is absorbed by any neighborhood of 0. If B ⊆ Vis bounded and absolutely convex, we denote by V the subspace generated by B equipped with the { } norm v = inf λ 0 |v ∈λB . Then, the natural inclusion V → V is continuous. B B The space V is said to be quasi-complete if every closed bounded subset of it is complete. Fréchet spaces and strict LF spaces are quasi-complete. A.0.2 We recall the notion of integral valued in a LCTVS in the form we use it in the core of the paper. Let X be a σ -compact locally compact topological space, dx be a Radon measure on X and V be a LCTVS. Let f : X → V be a continuous function. We say that f is absolutely integrable if for every continuous semi-norm p on V the integral p(f (x))μ(x) converges. If f is absolutely integrable and V is quasi-complete, there exists an unique element f (x)μ(x) in V such that λ, f (x)μ(x) = λ, f (x) μ(x) X X for every λ ∈ V . This notion applies in particular to series v valued in a quasi- i∈I complete LCTVS V in which case we will rather use the terminology absolutely summable: afamily (v ) of vectors in V is absolutely summable if for every continuous semi-norm p i i∈I on V, the series p(v ) converges. i∈I We will also use the following weaker notion: a family (v ) of vectors in a LCTVS i i∈I Vis said to be summable if for every continuous semi-norm p on V and every> 0, there exists a finite subset J ⊆ Isuch that p( v)< i∈K for every finite subset K ⊆ I \ J. If (v ) is a summable family in V and V is quasi- i i∈I complete then the partial sums v converge to some limit v ∈ V along the filter i∈J associated to the inclusion order on finite subsets of I. In this case, we call v the sum of the family (v ) . i i∈I Note that absolutely summable families are automatically summable but the con- verse is not true e.g. if V is a Hilbert space with a Hilbert decomposition V = V and i∈I v ∈ V then the family of orthogonal projections (v ) of v to the subspaces V is always i i∈I i summable but not always absolutely summable. Recall that a subset S ⊆ V is said to be absolutely convex is it is convex and circled i.e. λS ⊆ S for every complex number λ with |λ| 1. 328 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR A.0.3 We will also freely use the notions of smooth or holomorphic functions val- ued in a LCTVS. For basic references on these subjects, we refer the reader to [Bou67, §2,§3],[Gro53,§2],[Gro73, Chap. 3, §8]. There are actually two ways to define smooth and holomorphic maps valued in V: either scalarly (that is after composition with any ele- ment of V ) or by directly requiring the functions to be infinitely (complex) differentiable. These two definitions coincide when the space V is quasi-complete and, fortunately for us, we will only consider smooth/holomorphic functions valued in such spaces so that we don’t have to distinguish. Let M be a connected complex analytic manifold. A function f : M → V is holo- morphic if and only if for every relatively compact open subset # ⊆ M, there exists a bounded absolutely convex subset B ⊆ Vsuch that f | factorizes through a holomor- phic map # → V see [Gro53, §2, Remarque 2]. We also record the following convenient criterion of holomorphicity [Bou67, §3.3.1]: (A.0.3.1) Assume that V is quasi-complete. A function ϕ : M → V is holomorphic if and only if it is continuous and for some total subspace H ⊆ V , the functions s ∈ M → ϕ(s),λ are holomorphic for every λ ∈ H. A.0.4 Assume that V is a LF space. As LF spaces are barreled [Trè67, Corol- lary 33.3] they satisfy the Banach-Steinhaus theorem [Trè67, Theorem 33.1] hence any bounded subset of Hom(V, W) is equicontinuous (as Hom(V, W) is equipped with the weak topology, a subset B ⊆ Hom(V, W) is bounded if and only if for every v ∈ V, {T(v) | T ∈ B} is a bounded subset of W). This implies in particular that for every bounded subset B ⊆ Hom(V, W), the restriction of the canonical bilinear map Hom(V, W)× V → WtoB× V is continuous. Also, by [Trè67, §34.3 Corollary 2], if W is quasi-complete then so is Hom(V, W). In particular, we get: (A.0.4.2) Assume that V is LF, W is quasi-complete and let K be a topological space. Let s ∈ M → T ∈ Hom(V, W) be holomorphic and (s, k) ∈ M× K → v ∈ Vbe s s,k a continuous map which is holomorphic in the first variable. Then, the map (s, k) ∈ M × K → T (v ) ∈ W is continuous and holomorphic in the first s s,k variable. Indeed, T has locally its image in a bounded set. Hence, by the above discussion, the map (s, s , k) ∈ M × M × K → T (v ) ∈ W is continuous. Moreover, this map is s s ,k separately holomorphic in the variables s, s . Thus, by Hartog’s theorem, this map is holomorphic in the variables (s, s ) which immediately implies the claim by “restriction to the diagonal”. (A.0.4.3) Assume that V is LF and W is quasi-complete. Let U ⊆ M be a nonempty open subset and s ∈ U → T ∈ Hom(V, W) be a holomorphic map. If, for every v ∈ Vthe map s → T (v) ∈ W extends analytically to M then T ∈ Hom(V, W) s s for every s ∈ Mand moreover s ∈ M → T ∈ Hom(V, W) is holomorphic. s THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 329 Indeed, the hypothesis implies that s → T induces a holomorphic map M → Hom(V, W) where Hom(V, W) stands for the space of all linear maps V → W(not nec- essarily continuous) equipped with the topology of pointwise convergence. Hence, for every relatively compact connected open subset # ⊆ Msuch that #∩ U = ∅ there exists a bounded subset B ⊆ Hom(V, W) such that s → T factorizes through a holomorphic map # → Hom(V, W) . By the Banach-Steinhaus theorem, Hom(V, W)∩ Hom(V, W) B B is closed in Hom(V, W) and it follows that s ∈ # → T factors through a holomorphic B s map # → Hom(V, W) ∩ Hom(V, W) . Indeed, by the Hahn-Banach theorem it suf- fices to show that for every continuous linear form λ : Hom(V, W) → C vanishing on Hom(V, W) ∩ Hom(V, W) we have λ(T ) = 0for every s ∈ #. Since T ∈ Hom(V, W) B s s for s ∈ U the equality λ(T ) = 0 holds at least for s ∈ # ∩ U. As s ∈ # → λ(T ) is holo- s s morphic # is connected and U ∩# non-empty, the claim follows. A.0.5 Let Bil (V, W) = Hom(V, Hom(W, C)) be the space of separately contin- uous bilinear mappings V × W → C equipped with the topology of pointwise conver- gence. Applying (A.0.4.2) and (A.0.4.3) twice, we get: (A.0.5.4) Assume that V and W are LF. Let s ∈ M → B ∈ Bil (V, W) be holomorphic s s and (s, k) ∈ M × K → v ∈ V, (s, k) ∈ M × K → w ∈ W be continuous s,k s,k maps which are holomorphic in the first variable. Then, the function (s, k) ∈ M × K → B (v ,w ) is continuous and holomorphic in the first variable. s s,k s,k (A.0.5.5) Assume that both V and W are LF. Let U ⊆ M be a nonempty open subset and s ∈ U → B ∈ Bil (V, W) be a holomorphic map. If for every (v,w) ∈ V × W s s the function s → B (v,w) extends analytically to M then B ∈ Bil (V, W) for s s s every s ∈ Mand moreover s ∈ M → B ∈ Bil (V, W) is holomorphic. s s A.0.6 We refer the reader to [Trè67, Definitions 47.2, 47.3] for the definition of nuclear mappings between LCTVS. The following lemma will be useful in proving that certain families are absolutely summable. Lemma A.0.6.1. — Let T : V → W be a nuclear mapping between LCTVS. Then, for every summable family (v ) in V,the family (T(v )) is absolutely summable in W. i i∈I i i∈I Proof. — First, by the very definition, T is a nuclear mapping if and only if it factor- izes through a nuclear mapping between Banach spaces. Thus, we may assume without loss in generality that V and W are Banach spaces. Recall [Trè67, Definition 47.2] that this means that T belongs to the image of the natural map V ⊗W → Hom (V, W) where V (resp. Hom (V, W)) is the topological dual of V (resp. the space of continuous linear maps V → W) equipped with the strong topology (aka norm topology) and ⊗ stands for the completed projective tensor product. Let (v ) be a summable family in V and set i i∈I M = sup v < ∞ i V i∈J 330 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR where the sup runs over all finite subsets J ⊆ I. It follows from the next elementary lemma that for (,w) ∈ V × W, the family ((v )w) is absolutely summable in W and more- i i∈I over (A.0.6.6) (v )w =w |(v )| 4M w i W W i V W i∈I i∈I where . (resp. . ) denotes the norm on W (resp. dual norm on V ). W V Lemma A.0.6.2. — Let (z ) be a summable family in C.Then, (z ) is absolutely i i∈I i i∈I summable and moreover (A.0.6.7) |z | 4sup| z | i i i∈I i∈J where the sup runs over all finite subsets of I. Let (I, W) be the vector space of absolutely summable families indexed by I in W. We equip (I, W) with the norm (w ) 1 = w . i i∈I i W i∈I It then becomes a Banach space. By (A.0.6.6), the bilinear map V × W → (I, W), (,w) → ((v )w) , is continuous hence induces a continuous linear map V ⊗W → i i∈I (I, W). Obviously, this map is the composite of the natural morphism V ⊗W → Hom (V, W) and of Hom (V, W) → W ,T → (T(v )) . The lemma follows. s s i i∈I A.0.7 We denote by V⊗W the completed projective tensor product [Trè67, Chap. 43]. It admits a canonical linear map V ⊗ W → V⊗W satisfying the following universal property: for every complete LCTVS U, precomposition yields an isomorphism Hom(V⊗W, U) Bil(V, W; U) where Bil(V, W; U) denotes the space of all continuous bilinear mappings V× W → U. In particular, if U ,U are two other LCTVS and T : V → U ,S : W → U are continuous 1 2 1 2 linear mapping, there is an unique continuous linear map T⊗S : V⊗W → U ⊗U which 1 2 on V ⊗ Wis given by v ⊗w → T(v) ⊗ S(w). Assume now that V and W are spaces of (complex valued) functions on two sets X, Y and that their topologies are finer than the topology of pointwise convergence. When V is moreover a complete nuclear LF space, the following result of Grothendieck [Gro55, Théorème 13, Chap. II, §3 n. 3] generally allows to describe V⊗W explicitly as a space of functions on X × Y. THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 331 (A.0.7.8)Let F(X× Y) be the space of all complex valued functions on X× Y equipped with the topology of pointwise convergence. Then the linear map V ⊗ W → F(X × Y), v ⊗ w → ((x, y) → v(x)w(y)), extends continuously to a linear embedding V⊗W → F(X × Y) with image the space of functions f : X × Y → C satisfying the two conditions: • For every x ∈ X, the function y ∈ Y → f (x, y) belongs to the comple- tion of W; • For every λ ∈ W , the function x ∈ X → f (x,.),λ belongs to V. A.0.8 Let C ∈ R∪{−∞} and f : H → V be a holomorphic function. We say >C −|z| that f is of order at most d in vertical strips if for every d > d the function z → e f (z) is bounded in vertical strips of H . We say that f is of finite order in vertical strips if it is of order at >C most d in vertical strips for some d > 0. Finally, we say that f is rapidly decreasing in vertical strips if for every d > 0 the function z → |z| F(z) is bounded in vertical strips. A.0.9 Let A be a real vector space. Denote by Diff(A) the space of complex poly- nomial differential operators on A (which can be identified with Sym(A )⊗ Sym(A )). C C When V is quasi-complete, we define the space of Schwartz functions on A valued in V, denoted by S(A, V), as the space of smooth functions f : A → Vsuch that for ev- ery D ∈ Diff(A), the function Df has bounded image. Note that if W is also quasi- complete and T : V → W is a continuous linear map then for every f ∈ S(A, V),we have T◦ f ∈ S(A, W). When V = C, we simply set S(A) = S(A, C) that we equip with its standard Fréchet topology. Lemma A.0.9.1. — Assume that V is quasi-complete and barreled (e.g. a strict LF space). Let C > 0,d > 0 and s ∈ H → Z ∈ V be a map such that such that for every v ∈ V,s ∈ H → >C s >C Z (v) is a holomorphic function of order at most d in vertical strips. Then, for every f ∈ S(A, V),the map (A.0.9.9) s ∈ H → λ ∈ A → Z (f ) ∈ S(A) >C s λ is holomorphic and of finite order in vertical strips. Proof. — Indeed, by the Banach-Steinhaus theorem, for every d > d , every vertical strip S ⊆ H and every bounded subset B ⊆ V the set >C −|s| e Z (v) | s ∈ S,v ∈ B ⊆ C is bounded and, by [Trè67, Corollary 33.1], for every s ∈ H ,Z converges uniformly 0 >C s on compact subsets to Z as s → s .Let f ∈ S(A, V). Moreover, for every D ∈ Diff(A), s 0 as the function λ ∈ A → Df is continuous and converges to 0 as λ→∞, the subset Df |λ ∈ A ∪{0} λ 332 RAPHAËL BEUZART-PLESSIS, PIERRE-HENRI CHAUDOUARD, AND MICHAŁ ZYDOR of V is compact. Therefore, for every s ∈ H ,Z (Df ) converges to Z (Df ) as s → s 0 >C s λ s λ 0 −|s| uniformly in λ ∈ A and e Z (Df ) | s ∈ S,λ ∈ A is bounded for every d > d and s λ every vertical strip S ⊆ H . This shows that the map (A.0.9.9) is continuous and of >C finite order in vertical strips. To conclude we apply the holomorphicity criterion (A.0.3.1) to the subspace H ⊆ S(A) generated by “evaluations at a point of A”. A.0.10 Lemma A.0.10.1. — Assume that V is quasi-complete. Let Z , Z : H → V be holo- + − >C morphic functions of finite order in vertical strips for some C > 0. Assume that there exists a total subspace H ⊂ V such that for every λ ∈ H, Z := λ ◦ Z and Z := λ ◦ Z extend to holo- +,λ + −,λ − morphic functions on C of finite order in vertical strips satisfying Z (s) = Z (−s) for every s ∈ C. +,λ −,λ Then, Z and Z extend to holomorphic functions C → V of finite order in vertical strips satisfying + − Z (s) = Z (−s) for every s ∈ C. + − Proof. — Let d > 0be such thatZ and Z are of order at most d in vertical + − strips of H . Then, by their functional equation and the classical Phragmen-Lindelöf >C principle, for every λ ∈ H, the holomorphic continuations of Z and Z are also of +,λ −,λ 4n+2 order at most d in vertical strips. Therefore, up to multiplying Z and Z by z → e + − for some n 0, we may assume that all these functions are rapidly decreasing in vertical strips. Let D> C. Then, for every s ∈ H and ∈{±}, we set ]−D,D[ ) * +∞ +∞ 1 Z (D + it) Z (D + it) $ (s) = dt − dt . 2π D + it − s D + it + s −∞ −∞ Note that, since Z and Z are rapidly decreasing in vertical strips and V is quasi- + − complete, the above integrals converge absolutely and define elements of V. By the usual holomorphicity criterion for parameter integrals, we readily check that the functions $ , $ are holomorphic. Moreover, by the uniform boundedness principle, $ and $ are − + − bounded in vertical strips. Finally, by Cauchy’s integration formula and the fact that the functions Z ,Z are rapidly decreasing in vertical strips, for every ∈{±} and λ ∈ H +,λ −,λ the functions λ ◦ $ and Z coincide on H . Therefore, as H is total, $ and Z ,λ ]−D,D[ coincide on H . This shows that Z and Z admit holomorphic extensions bounded ]C,D[ + − in vertical strips to H for every D > Chence to C. That the functional equation >−D Z (s) = Z (−s) holds for these extensions easily follows from the assumption. + − A.0.11 Let A be a real vector space. Specializing the previous lemma to V = S(A) and H the total subspace of V given by “evaluations at a point of A” yields the following corollary. Corollary A.0.11.1. — Let Z , Z : A × C → C be two functions such that: + − THE GLOBAL GAN-GROSS-PRASAD CONJECTURE FOR UNITARY GROUPS. . . 333 1. There exists C > 0 such that for every s ∈ H , the function Z (., s), Z (., s) belong to >C + − S(A) and the maps s ∈ H → Z (., s) ∈ S(A), ∈{±}, >C are holomorphic functions of finite order in vertical strips; 2. For every λ ∈ A,s ∈ C → Z (λ, s) and s ∈ C → Z (λ, s) are holomorphic functions + − of finite order in vertical strips satisfying the functional equation Z (λ, s) = Z (λ,−s) + − Then, for every s ∈ C the functions Z (., s), Z (., s) belong to S(A) the maps s ∈ C → Z (., s) ∈ + − S(A), ∈{±}, are holomorphic. Assume now that W is a LF space. As W is barreled, W is quasi-complete [Trè67, §34.3 Corollary 2]. Specializing Lemma A.0.10.1 to V = W and H a dense subset of V = W, we obtain the following. Corollary A.0.11.2. — Let W be a LF space, C > 0 and Z , Z : H × W → C be + − >C two functions. Assume that: 1. For every s ∈ H , Z (s,.) and Z (s,.) are continuous functionals on W; >C + − 2. There exists d > 0 such that for every w ∈ W and ∈{±},s ∈ H → Z (s,w) is a >C holomorphic function of order at most d in vertical strips; 3. For every f ∈ H and ∈{±},s → Z (s, f ) extends to a holomorphic function on C of finite order in vertical strips satisfying Z (s, f ) = Z (−s, f ). + − Then, Z and Z extend to holomorphic functions C → W of finite order in vertical strips satisfying + − Z (s,w) = Z (−s,w) for every s ∈ C and every w ∈ W. + − REFERENCES [AG09] A. AIZENBUD and D. GOUREVITCH, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem, Duke Math. 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