Documenta Math. 317 Stabilization of Periods of Eisenstein Series and Bessel Distributions on GL(3) Relative to U(3)

We study the regularized periods of Eisenstein series on GL(3) relative to U(3). A stabilization procedure is used to express the periods in terms of L-functions. This is combined with the rela- tive trace formula Jacquet and Ye to obtain new identities on Bessel distributions.


Introduction
Let G be a reductive group over a number field F and H the fixed point set of an involution θ of G.The period of a cusp form φ on G is defined as the integral Π H (φ) = H(F )\(H(A)∩G(A)1 ) φ(h) dh, known to converge by [AGR].Recall that a cuspidal representation π = ⊗π v of G is said to be distinguished by H if there exists an element φ in the space V π of π such that Π H (φ) = 0.It is known in some cases that the period factors as a product of local H v -invariant functionals, even when there is no local uniqueness for such functionals (cf.[J2]), and that it is related to special values of L-functions.Periods of more general automorphic forms, such as an Eisenstein series, are also of interest.Although the above integral need not converge, it should be possible to regularize it, as has been carried out in [JLR] and [LR] when (G, H) is a Galois pair, i.e., when H is the fixed point set of a Galois involution on G.The regularized period of cuspidal Eisenstein series was computed for the pair (GL(n) E , GL(n) F ) in [JLR].It is either identically zero or can be expressed as a ratio of Asai L-functions (up to finitely many local factors at the ramified places).In general, however, the result will be more complicated.
In this paper we study the pair (GL(3) E , U (3)) in detail in order to illustrate some phenomena which are likely to appear in the general case.Our goal is two-fold.First, we introduce a stabilization procedure to express the period of an Eisenstein series induced from a Borel subgroup as a sum of terms, each of which is factorizable with local factors given almost everywhere by ratios of L-factors.This is reminiscent of the procedure carried out in [LL] for the usual trace formula.Second, we define a stable version of the relative Bessel distributions occurring in the relative trace formula developed by Jacquet-Ye.We then use the comparison of trace formulae carried out in [JY] to prove some identities between our stable relative Bessel distributions on GL(3) E and Bessel distributions on GL(3) F .
The main motivation for this work comes form the relative trace formula (RTF), introduced by Jacquet to study distinguished representations.One expects in general that the distinguished representations are precisely those in the image of a functorial lifting from a group G determined by the pair (G, H).For example, G is GL(n) F for the pair (GL(n) E , U (n)).To that end one compares the RTF for G with the Kuznetzov trace formula (KTF) for G .This was first carried out in [Y] for the group GL(2).The cuspidal contribution to the RTF appears directly as sum of relative Bessel distributions (defined below) attached to distinguished representations of G.It should match term by term with the corresponding sum in the KTF of Bessel distributions attached to cuspidal representations of G .Examples ( [J1], [JY], [GJR]) suggest that the contribution of the continuous spectrum of G can also be written as integrals of relative Bessel distributions built out of regularized periods of Eisenstein series.However, these terms cannot be matched up with the continuous part of the KTF directly.Rather, as we show in our special case, the matching can be carried out using the stable relative Bessel distributions.
We now describe our results in greater detail.Assume from now on that (G, H) is a Galois pair, that is G = Res E/F H where E/F is a quadratic extension and θ is the involution induced by the Galois conjugation of E/F .We also assume that H is quasi-split.By abuse of notation, we treat G as a group over E, identifying it with H E .The regularized period of an automorphic form φ, also denoted Π H (φ), can be defined using a certain truncation operator Λ T m φ depending on a parameter T in the positive Weyl chamber.For T sufficiently regular, the integral Λ T m φ(h) dh is a polynomial exponential function of T , i.e., it has the form p j (T )e λj ,T for certain polynomials p j and exponents λ j .Under some restrictions on the exponents of φ, the polynomial p 0 (T ) is constant and Π H (φ) is defined to be its value.When φ = E(ϕ, λ) is a cuspidal Eisenstein series, Π H (φ) can be expressed in terms of certain linear functionals J(η, ϕ, λ) called intertwining periods ( [JLR], [LR]).To describe this, consider for simplicity the case of an Eisenstein series induced from the Borel subgroup B = T N .We assume that B, T and N are θ-stable.Given a character χ of T (E)\T (A E ), trivial on Z(A E ), and λ in the complex vector space a * 0,C spanned by the roots of G, the Eisenstein series H(γg)   converges for Re λ sufficiently positive.Here, as usual, ϕ : According to a result of T. Springer [S], each double coset in B(E)\G(E)/H(F ) has a representative η such that ηθ(η) −1 lies in the normalizer N G (T ) of T .Denoting the class of ηθ(η) −1 in the Weyl group W by ηθ(η) −1 , we obtain a natural map The intertwining period attached to η is the integral where dh is a semi-invariant measure on the quotient H η (A F )\H(A F ).The result of [LR] alluded to above is that for suitable χ and λ the integral defining J(η, λ, ϕ) converges and that where w is the longest element in W and c = vol the regularized period of a cuspidal Eisenstein series is either zero or is proportional to a single intertwining period.This intertwining period factors as a product of local integrals which are equal almost everywhere to a certain ratio of Asai L-functions ( [JLR]).For general groups, however, the sum in (1) is infinite and Π H (E(ϕ, λ)) cannot be expressed directly in terms of L-functions.This occurs already for G = SL(2) E and H = SL(2) F .This is related to the fact that base change is not necessarily one-to-one for induced representations.See [J2] for a discussion of the relation between non-uniqueness of local H-invariant functionals and the non-injectivity of base change for the pair (GL(3) E , U (3)).For the rest of this paper, let G = GL(3) E , G = GL(3) F , and let H = U (3) be the quasi-split unitary group in three variables relative to a quadratic extension E/F and the Hermitian form Let T and T be the diagonal subgroups of G and G , respectively, and Nm : T → T the norm mapping.We shall fix a unitary character χ of T (A E ) which is a base change lifting with respect to Nm of a unitary character of T (F )Z (A F )\T (A F ).We write B(χ) = {ν} for the set of four characters of The stable intertwining periods J st (ν, ϕ, λ) are defined in §8.Each functional J st (ν, ϕ, λ) is factorizable and invariant under H(A f ) where A f is the ring of finite adeles.Our first main result is that with a suitable normalization of measures, we have the following Theorem 1.
In particular, this expresses the left-hand side as a sum of factorizable distributions.
In Proposition 3, §7, we show that the local factors at the unramified places are given by ratios of L-functions.This is done by a lengthy calculation which can fortunately be handled using Mathematica.Hence we obtain a description of Π H (E(ϕ, λ)) in terms of L-functions.
We now describe our results on Bessel distributions.In general, if (π, V ) is a unitary admissible representation of G(A) and L 1 , L 2 ∈ V * are linear functionals, we may define a distribution on the space of compactly-supported, K-finite functions by the formula where {φ} is an orthonormal basis of V consisting of K-finite vectors.The sum is then finite and O is independent of the choice of basis.The distributions occurring in the KTF and RTF are all of this type.They are referred to as Bessel distributions and relative Bessel distributions in the two cases, respectively.In this paper, the representations of G and G that we consider are all assumed to have trivial central character.Correspondingly we will consider factorizable In particular, we define π(f ) = G(E)Z(E)\G(AE) f (g)π(g)dg and similarly for π (f ).The relative Bessel distribution attached to a cuspidal representation (π, V π ) of G is defined by and the Bessel distribution attached to a cuspidal representation (π , V ) of G is defined by Here {φ} and {φ } are orthonormal bases of V and V , respectively, and W(φ) and W (φ ) are the Fourier coefficients defined in §2.They depend on the choice of an additive character ψ which will remain fixed.Jacquet and Ye have studied the comparison between the relative trace formula on G and the Kuznetzov trace formula on G under the assumption that E splits at all real places of F ([J1], [JY]).They define a local notion of matching functions f v ↔ f v for all v and prove the identity for any cuspidal representation π of G (A F ) with base change lifting π on G(A E ).Our goal is to formulate and prove an analogous result for Eisensteinian automorphic representations.Assume that χ is unitary and let be an induced representation of G(A E ).In this case, we define a relative Bessel distribution in terms of the regularized period as follows: where {ϕ} runs through an orthonormal basis of I(χ, λ) and W(ϕ, λ) = W(E(ϕ, λ)).Throughout the paper we will use the notation W(•, •) for the complex conjugate of W(•, •).For ν ∈ B(χ), the Bessel distribution is defined by where W (ϕ , λ) is defined similarly.However, the equality (3) no longer holds.
In fact, it is not well-defined since there is more than one automorphic representation π whose base change lifting is I(χ, λ).However, for ν ∈ B(χ) we may define With this definition, Theorem 1 allows us to write Bst (f, ν, λ).
Our next main result is the following Theorem 2. Assume that the global quadratic extension E/F is split at the real archimedean places.Fix a unitary character χ and ν ∈ B(χ).Then for all matching functions f ↔ f .
There is a local analogue of this Theorem.The distributions Bst (f, ν, λ) are factorizable.Their local counterparts are defined in terms of Whittaker functionals and local intertwining periods.Let E/F be a quadratic extension of p-adic fields.For any character µ of F * , set Denote by ω the character of F * attached to E/F by class field theory.For any character ν with notation as in §2.
Theorem 3. Let E/F be a quadratic extension of p-adic fields.There exists a constant d E/F depending only on the extension E/F with the property: for all unitary characters ν of T (F ), whenever f ↔ f .Moreover, if E/F is unramified and p = 2 then d E/F = 1.
Documenta Mathematica 5 (2000) 317-350 We determine the constant d E/F for E/F unramified and p = 2 by taking f to be the identity in the Hecke algebra and directly comparing both sides of the equality.As remarked, this involves an elaborate Mathematica calculation.
Determining d E/F in general would require more elaborate calculations which we have not carried out.We remark, however, that in a global situation we do have d Ev/Fv = 1.

Acknowledgement
The first author would like to thank the Institute for Advanced Study in Princeton for its kind hospitality he enjoyed during the academic year 1998-99.

Notation
Throughout, E/F will denote a quadratic extension of global or local fields of characteristic zero.In the local case we also consider E = F ⊕ F .In the global case we make the following assumption on the extension E/F : The character of F attached to E by class field theory will be denoted ω.In the local case, if E = F ⊕ F , then ω is trivial, and Nm : As in the introduction, H = U (3) denotes the quasi-split unitary group with respect to E/F and the Hermitian form Φ, and we set G = GL(3) E and G = GL(3, F ).We shall fix some notation and conventions for the group G.
Similar notation and conventions will be used for G with a prime added.
We write B for the Borel subgroup of G of upper triangular matrices and B = T N for its Levi decomposition where T is the diagonal subgroup.Let W be the Weyl group of G.The standard maximal compact subgroup of G(A) will be denoted K.In the local case we write K.We have the Iwasawa decompositions G(A) = T (A)N (A)K = N (A)T (A)K.We fix the following Haar measures.Let dn and dt be the Tamagawa measures on N (A) and T (A), respectively.Then vol(N (F )\N (A)) = 1.We fix dk on K by the requiring vol(K) = 1.Let dg the Haar measure dt dn dk.We define Haar measures for G and H similarly.
Let α 1 , α 2 be the standard simple roots and set Let a * 0 be the real vector space spanned by the roots, and let a 0 be the dual space.For λ ∈ a * 0 set If M is a Levi subgroup of G, π is an admissible representation of M (locally or globally) and λ ∈ a * M,C we write I(π, λ) for the representation of G unitary induced from the representation m → π(m)e λ,H(m) .In the global case, we let E(g, ϕ, λ) be the Eisenstein series on G(A) induced by ϕ.If χ is a unitary character of T (A), we identify the induced space I(χ) = I(χ, λ), with the pre-Hilbert space of smooth functions ϕ : We use the notation δ Q to denote the modulus function of a group Q.The scalar product is given by The representation I(χ, λ) is defined by It is unitary if λ ∈ ia * 0 .Similar notation will be used in the local case.Let w ∈ W and let wχ(t) = χ(wtw −1 ).The (unnormalized) intertwining operator It is absolutely convergent in a suitable cone and admits a meromorphic continuation in λ.Let B(χ) = {ν} be the set of four Hecke characters of the diagonal subgroup T (A F ) of G (A F ) such that ν is trivial on the center Z (A F ) and χ = ν • Nm.In the non-archimedean case, let H G be the Hecke algebra of compactlysupported, bi -K-invariant functions on G. Let f (χ, λ) be the Satake transform, i.e., f (χ, λ) is the trace of f acting on I(χ, λ).Define H G and f (ν, λ) similarly.We define the base change homomorphism bc : We fix a non-trivial additive character ψ of F \A F .The ψ-Fourier coefficient of an automorphic form on G is defined by

Documenta Mathematica 5 (2000) 317-350
The ψ-Fourier coefficient W (φ) of an automorphic form on G is defined in a similar way with respect to the character In this paper, we use a different Hermitian form from the one used in [JY].This forces us to modify the definition of matching functions, namely we have to take a left translate of f by τ where t τ τ = Φ.

Part I
The regularized Period Then H = U (3) is the fixed-point set of θ.Note that θ preserves B, T , N and K.We shall consider the space and its quotient modulo scalars Remark 1.The space S 0 is a translate by Φ −1 of the space of non-degenerate Hermitian forms.Indeed, s ∈ S 0 if and only if Φ −1 t s = sΦ −1 , i.e. sΦ −1 is Hermitian.
The group G acts on S via s → gsθ(g) −1 .This is compatible with the action on Hermitian forms.The stabilizer of s is the subgroup Thus H s is the unitary similitude group of the Hermitian form sΦ −1 .There are only finitely many equivalence classes of Hermitian forms modulo scalars and hence G has finitely many orbits in S. We obtain a bijection Set w = Φ and regard w as an element of W .We shall be interested in

This yields the following
Lemma 1.There is a bijection (depending on the choice of w) In the local case, ι −1 (w) consists of the open orbits.Set and where E 1 is the group of norm one elements in E * .The subgroup B η is thus independent of η.

Fourier inversion and stabilization
For E/F a quadratic extension of local fields or number fields, or for E = F ⊕F , we set Documenta Mathematica 5 (2000) 317-350 By Lemma 1, A(F ) parameterizes the B-orbits in ι −1 (w).Note that In the global case, we define A(A F ) as the direct sum of the corresponding local groups where v ranges over all places of F .View A(F ) as a subgroup of For an absolutely summable function g on A(A F ), we may define the Fourier transform for any character κ of A(A F ). Let X be the set of four characters of A(A F ) trivial on A(F ).Then the following Fourier inversion formula holds Suppose in addition that g is of the form where g v is a function on A(F v ) for all v and the infinite product converges absolutely.Define the local Fourier transform for any character κ of A(F v ).We shall write κ v for the restriction of a character κ ∈ A(A F ) to A(F v ).Then we have the following Proof.Let {S n } ∞ n=1 be an ascending sequence of finite sets of places of F whose union is the set of all places of F .For any finite set of places S let A S (A F ) be the subgroup of elements and since v ∈Sn g v (1) converges to 1 as n → ∞.
By (7), we have for all α ∈ T (E) and the expression depends only on the double coset BηH and the measure on H η (F )\H(F ).The stable local period is defined to be the distribution e λ,H(ηh) ϕ(ηh) dh.
Let ν be a unitary character of T .Its base change to Proposition 1.We have Proof.By definition as required.

Meromorphic Continuation of Local Periods in the p-adic case
Let E/F be quadratic extension of p-adic fields and let q = q F be the cardinality of the residue field of F .
It suffices to show that each integral J(η, ϕ, λ) is a rational function in q λ .We shall follow the discussion in [GPSR], pp.126-130, where a similar assertion is established for certain zeta integrals.
The key ingredient is a theorem of J. Bernstein, which we now recall.Let V be a vector space of countable dimension over C, Y an irreducible variety over C with ring of regular functions C [Y ], and I an arbitrary index set.By a system of linear equations in V * indexed by i ∈ I we mean a set of equations for ∈ V * of the form (v i ) = a i where v i ∈ V and a i ∈ C. Consider an algebraic family Ξ of systems parameterized by Y .In other words, for each i ∈ I we have functions , then L(v(y)) may be viewed as an element of K. We will say that L ∈ Hom C (V, K) is a meromorphic solution of the family Ξ if L(v i (y)) = a i (y) for all i ∈ I. Then Bernstein's Theorem is the following statement.
Theorem 4. In the above notation, suppose that the system Ξ y has a unique solution y ∈ V * for all y in some non-empty open set Ω ⊂ Y (in the complex topology).Then the family Ξ has a unique meromorphic solution To use the Theorem, we need the following input.Recall that there are 4 open B-orbits in G/H.The next lemma shows that generically each one supports at most one H-invariant functional.
Proof.To prove (a), first note that J(η, ϕ, λ) defines a non-zero H-invariant functional whenever the integral defining it converges.To prove uniqueness, let {η i } be a set of representatives for the open orbits in B\G/H and let V be the H-invariant subspace of ϕ ∈ I(χ, λ) whose support is contained in the union ηi Bη i H.The argument in [JLR], pp.212-213 shows that an H-invariant functional vanishing on V is identically zero if Re λ is sufficiently positive.On the other hand, V decomposes as a direct sum over the η i 's of ind H Hη i (χ ηi λ ) where It remains to show that the space of H-invariant functionals on ind H Hη i (χ ηi λ ) is at most one-dimensional.However, the dual of ind H Hη (χ η λ ) is Ind H Hη ((χ η λ ) −1 ) and the dimension of H-invariant vectors in the latter is at most one by Frobenius reciprocity.
To prove (b), let V be a small open subgroup of G.The map B × H → G defined by (b, h) → bηh is proper since the stabilizer B η is compact.We infer that the set U = ηH ∩ BηV is a small neighborhood of B η η and hence the weight function H(x) takes the constant value H(η) on U .Then we may take for ϕ 0 any non-negative, non-zero function supported in BηV .
To finish the proof of Proposition 2, let V = I(χ) and Y = C * .Then C[Y ] may be identified with the ring of polynomials in q ±λ .Fix ϕ 0 , c and n as in Lemma 3 and consider the following conditions on a linear functional ∈ V * : 1. is H-invariant, i.e. (I(h, χ, λ)ϕ − ϕ) = 0 for all ϕ and h ∈ H.
These conditions form an algebraic family of systems of linear equations as above.The functional J(η, ϕ, λ) converges for Re λ > λ 0 and it is the unique solution for the system by Lemma 3. The proposition now follows from Bernstein's Theorem.

Unramified Computation
Suppose now that E/F is an unramified extension of p-adic fields with p = 2 and that χ is unramified.Let ϕ 0 be the K-invariant section of I(χ, λ) such that ϕ 0 (e) = 1.Recall that s i = λ, α ∨ i for i = 1, 2, 3. Recall our convention that the Haar measure dh on H(F ) is defined via the Iwasawa decomposition and assigns measure one to K H .In the following Proposition, we assume that the measure on H η (F )\H(F ) is the quotient of the dh by the measure on H η (F ) such that vol(H η (F )) = 1.
Without loss of generality we may assume that χ = 1.Let η 1 , η 2 , η 3 , η 4 be the matrices respectively, where ∈ F * − N E * , e.g., has odd valuation.They form a set of representatives for the double cosets in B\G/H over w.The matrices By definition, where e λ+ρ,H(ηj h) dh.
Since H ηj (F ) has measure one, we may use Iwasawa decomposition to write I j as ∆ ν,λ (η j ) −1 times where is a uniformizer of E and q E is the cardinality of the residue field of E.
The I j can be evaluated explicitly.For j = 2, 3, 4, the integrands depend only on v(x), v(β) and n, but for j = 1 it depends also on v( xx 4 ± 1).In any case, the integrations and the summation over β, x and n can be computed as geometric series.This is tedious to carry out by hand, especially in the first case, but we did it using Mathematica with the following results.The term I 1 is independent of ν and is equal to while respectively.Summing up the contributions (also done with Mathematica) we get 5 (2000) 317-350 as required.
Remark 2. We have also computed J st (ν, ϕ 0 , λ) when E/F is ramified but ν is unramified and −1 ∈ Nm E * .In this case the matrices η i still provide representatives for B\G/H.We find that J st (ν, ϕ 0 , λ) is equal to However, this is not sufficient to evaluate d E/F .We would also need to determine the function f on G that matches ϕ 0 .Finally if −1 ∈ Nm E * then the representatives are more difficult to write down and the computation is more elaborate.We have not attempted to do this.

Stabilization of Periods
We return to the global situation.Let The following identity is proved in [LR].It is valid whenever Re λ is positive enough.From now on, we assume that the Haar measure on H η (A) is the Tamagawa measure.This measure has the property that vol(H η (F v )) = 1 for almost all v and furthermore, c = 4 (11) by Ono's formula for the Tamagawa number of a torus [O].Fix a character ν 0 ∈ B(χ) to serve as a base-point.Recall from §4 that the set of double cosets over w is parameterized by the group A(F ) both locally and globally.For a ∈ A(F ), let η a be a representative for the double coset corresponding to a such that η a θ(η a ) −1 = tw with t ∈ T (F ).Fix λ and ϕ = ⊗ϕ v ∈ I(χ, λ) = ⊗I(χ v , λ), and let g v be the function on A(F v ) defined as follows: The product function g = v g v on A(A) is integrable over A(A F ) for Re λ positive enough.Indeed, Proposition 3 applied to |ν 0 | shows that almost all factors of the integral are local factors of a quotient of products of L-functions.
For the same reason, we may define the global stable intertwining period for Re λ positive enough as the absolutely convergent product For any character κ of A(A) trivial on A(F ) we have g(κ) = J st (κν 0 , ϕ, λ).
Remark 3. It seems unlikely that the individual terms J(η, ϕ, λ) have a meromorphic continuation to the entire complex plane.This is motivated by the following old result of Estermann ([E]).Let P (x) be a polynomial with integer coefficients with P (0) = 1.Then either P (x) is a product of cyclotomic polynomials or else the Euler product p P (p −s ) has the imaginary axis as its natural boundary.The function J(η, ϕ, λ) has Euler product where the factors in the inert places are almost always (10).

Part II Relative Bessel Distributions
We now turn to the relative Bessel distributions, starting with the local case.Thus we assume that F is a local field.The Whittaker functional is defined by the integral H(wn) ϕ(wn)ψ N (n) dn for ϕ ∈ I(χ, λ).It converges absolutely for Re λ sufficiently large, and defines a rational function in q λ .Definition 1.Let ν be a unitary character of T which base changes to χ.
The stable relative Bessel distribution is defined by for λ ∈ ia * 0 where {ϕ} is an orthonormal basis for I(χ, λ).

Documenta Mathematica 5 (2000) 317-350
Similarly, the local Bessel distributions on G are defined in terms of the Whittaker functionals on G as follows: Remark 4. Let us check that Theorem 3 is compatible with a change of additive character.In doing this, we include the additive character in the notation.
Let ψ be the character where where It follows from the definition in [JY] that if f ↔ f with respect to ψ then f t0 ↔ f t0 with respect to ψ .On the other hand, It remains to note that χ(wt 0 w −1 )e wλ+ρ,H(t0 Our goal is to prove Theorems 2 and 3. Let us start with the split case E = F ⊕ F .By a special case of a result of Shahidi ([Sh]) we have the following local functional equations.For any ϕ ∈ I (ν, λ) where γ(ν, λ, ψ) is defined in (5).Recall that ω ≡ 1 in this case.
Proposition 4. We have where Documenta Mathematica 5 (2000) 317-350 Proof.The involution ϑ on G preserves B , T , N .It induces the principal involution on the space spanned by the roots.We also let (ϑϕ)(g) = ϕ(ϑ(g)).This is a self-adjoint involution on I (ν).By Proposition 1 and hence is equal to The following identity holds for any operator A, functional l and orthonormal basis {e i } on a Hilbert space: Hence ( 15) is equal to We have the following simple relations M (w, λ) * = M (w, −wλ) We get for any ϕ, 5 (2000) 317-350 In the global setup we define for f = ⊗f v .Note that this is compatible with (4).Similarly 9 The Relative Trace Formula The relative trace formula identity, established by Jacquet reads where ), Jacquet established the following spectral expansion for RT F (f ), at least for K-finite functions f .It is a sum over terms indexed by certain pairs Q = (M, π) consisting of a standard Levi subgroup M and a cuspidal representation π of M (A).
where {ϕ} is an orthonormal basis of the space V π of π.We will say that a Hecke character of GL(1) E is distinguished if it is trivial on E 1 (A), that is, it is the base change of a Hecke character of GL(1) F .If M is the Levi factor of a maximal parabolic subgroup, then π = σ ⊗ κ where σ is a cuspidal representation of of GL(2) E and κ is a Hecke character of GL(1) E .The pair Q contributes if σ is distinguished relative to some unitary group in two variables relative to E/F in GL(2) E and κ is distinguished.In this case, the contribution is Finally, if M is the diagonal subgroup, then π is a triple of characters which we denote χ = (χ 1 , χ 2 , χ 3 ).There are two kinds of contributions.The first one, Documenta Mathematica 5 (2000) 317-350 which we call fully continuous is in the case where each χ j is distinguished.In the notation of relative Bessel distributions, the contribution is On the other hand, the residual contribution comes from triplets χ such that χ 2 is distinguished, χ 3 (x) = χ 1 (x) but χ 1 , χ 3 are not distinguished.Up to a volume factor it is Moreover the sum over χ and the integrals are absolutely convergent.
The spectral decomposition of the KT (f ) is The fully continuous part is There is no contribution from the residual spectrum because the representations occurring in it are not generic, i.e., the ψ-Fourier coefficients of the residual automorphic forms all vanish.This follows from the description of the residual spectrum by Moeglin and Waldspurger ([MW]).We may change the orthonormal basis {ϕ} to {M (w, λ)ϕ} because M (w, λ) is unitary for λ ∈ ia * 0 .A similar remark applies to the other contributions.In particular, all the expressions above depend on Q only up to conjugacy.Recall that Bst (f, ν, λ).
We shall now prove Theorems 2 and 3 by isolating the term corresponding to ν, λ in the relative trace formula identity.We proceed in several steps.Proposition 6.For any χ and λ and matching functions f ↔ f , To prove this, we modify the usual linear independence of characters argument ( [L]).In the following lemma, let G be a reductive group over a global field F and S a set of places containing all the archimedean places and the "bad" places of G. Let X be a countable set of pairs (M, π) consisting of a Levi subgroup M and a cuspidal representation π of M (A) which is unramified outside S. For each (M, π) ∈ X let a subspace A π ⊂ ia * M and a continuous function g π (•) on A π be given.We make the following hypotheses: 1.If (M 1 , π 1 ), (M 2 , π 2 ) ∈ X with π 1 = π 2 and λ i ∈ A πi , i = 1, 2, then I(π 1 , λ 1 ) S and I(π 2 , λ 2 ) S have no sub-quotient in common.
Let H S denote the Hecke algebra of G S relative to hyperspecial maximal compact subgroups of G v for v / ∈ S. For f ∈ H S and σ S an unramified representation of G S , we set f S (σ S ) = tr(σ S (f S )).
Lemma 4 (Generalized linear independence of characters).Let µ π be the push-forward under T π,U of the measure g π (λ)dλ.The Stone-Weierstrass Theorem implies that the image of H U under the Satake transform gives a dense set of continuous functions on G un U relative to the sup norm.Therefore ( 21) vanishes for all continuous functions on Ĝun U .The Riesz Representation Theorem implies that π µ π = 0.

Suppose that
Let Z π,U be the image of T π,U .Then µ π1 (Z π,U ) = 0 unless Z π,U ⊃ Z π1,U .We now claim that for any subset Z ⊂ G un U we have We argue by induction on dim Z (i.e., dim A π where Z = Z π,U ).For Z zero dimensional this follows from the fact that the atomic part of µ is π:Aπ={0} µ π .
For the induction step, we can assume that there are no π with dim A π < dim Z.
The restriction of µ to Z is then given by the left-hand side of ( 23), and our claim is proved.
For each place v, let X v be the group of unramified characters of the maximal split torus T v in M 0 (F v ) where M 0 is a Levi factor of a fixed minimal parabolic subgroup P 0 of G(F v ) (contained in a globally defined minimal parabolic).We identify X v with the vector space a Attached to each unramified representation σ of M v is an orbit of characters in X v under the Weyl group of M v .Let λ σ be a representative of this orbit.Let W v be the Weyl group of G v and let W U be the Weyl group v∈U W v of G U .If σ is an unramified representation of G U , let λ σ be the element (λ σv ) in the product X U = X v .We observe that the natural map a * P → X U is injective since ln p 1 and ln p 2 are linearly independent over Q.We identify a * P with its image in X U .We now claim that if Z π,U = Z π0,U , then there exists an element w ∈ W U such that To prove this, observe that for each λ ∈ A π0 , there exist w λ ∈ W U and λ ∈ A π such that There are only finitely many possibilities for w λ , and hence there exists w ∈ W U such that w λ = w for a set of λ whose closure has a non-empty interior.
Since the condition w λ = w is a closed condition, it holds on an open set and hence everywhere on A π0 .Taking λ = 0 gives (1).Parts (2) and (3) follow.Set Z = Z π0,U for a fixed π 0 .We rewrite (23) as follows: where λ + λ π,U = w(x + λ π0,U ).The sum taken inside the integral is absolutely convergent for almost all λ by Fubini's theorem.It follows that the pushforward to G un U with respect to T π0,U of the measure  is zero.We conclude that for almost all λ 0 ∈ A π0 , we have g π (λ) = 0.
Proof of Proposition 6. Fix a character χ and let S be a finite set of places containing the archimedean places such that χ and E/F are unramified outside for some linear combination L of the L i 's.Fix a compact open K small enough so that L | V K = 0. Then (26) implies that for any f ∈ H(G, K).We can rewrite this as for some 0 = ϕ 0 ∈ V K and all f ∈ H(G, K).This implies that L| V K = 0. Thus, in order to prove the Lemma, it suffices to show that the functionals {J st v (ν v , ϕ v , λ)} ν∈B(χ) are linearly independent.However, it is clear that {J(η, ϕ v , λ)} η is a linearly independent set in the range of convergence, since they are given by integrals over disjoint open orbits.Confining ourselves to ϕ ∈ V K where K is small enough, the condition of linear independence can be expressed in terms of a non-vanishing of some determinant which is a meromorphic function in λ (in fact rational in q λ ).Thus, it holds for generic λ.Finally, in order to prove the same thing for the J st v , it suffices to check that the matrix of coefficients (∆ ν,λ (η) −1 ) ν,η is non-singular.However the determinant of this matrix is easily seen to be a non-zero constant multiple of a power of q λ times the determinant of the character table of A(F v ). of Proposition 6 as a decomposable distribution in f with three factors, where the first two factors are the components at u and u and the third factor is the product of all components away from u and u .Lemma 6 allows us to apply Lemma 5 to conclude that (27) holds with the permutation a-priori depending on λ ∈ ia * 0 .Each side of ( 27) is the restriction to ia * 0 of a meromorphic function on a * 0,C .Thus, the permutation does not depend on λ, because there are only finitely many of them.
12 Uniform distribution of Hecke characters We need to prove that τ χ (ν) = ν.We first prove a Lemma which is interesting in its own right.Let F be a number field and S a finite set of places including the archimedean ones.We write S = S ∞ ∪ S f .Embed R + in F ⊗ Q R by x → 1 ⊗ x.Proof.This is a simple application of the trace formula for L 2 (R + F * \I F ). Let where g(x) = R+ f (tx) d * t.By our choice of f , the sum in the right hand side extends over X k , and in the left hand side only γ = 1 contributes provided that k is sufficiently large.
By standard methods, this Lemma implies that as k → ∞, the set of restrictions S of the Hecke characters in X k is uniformly distributed in the dual of F * S .The Lemma carries over immediately to the torus T , which is a product of copies of the multiplicative group.We shall use this variant to prove the following Corollary.If Q is a finite set of finite places, we denote by U Q the space of unramified unitary characters of T (F Q ) with the usual topology.
Corollary 2. Given a place w ∈ S, a unitary character η = (η v ) v∈S f of T (F S f ) and an open set U ⊂ U S f there exists a Hecke character of T which is unramified outside S ∪ {w} such that −1 S f η ∈ U .
13 Proof of Theorems 2 and 3 We now finish the proofs of Theorem 2 and Theorem 3. We first prove Theorem 3 by choosing a favorable global situation.Suppose that we are given local data which consists of: • a quadratic extension E 0 /F 0 , and Documenta Mathematica 5 (2000) 317-350 • a unitary character µ of T (F 0 ).
In principle there is also an additive character of F 0 , but we are free to choose it at will by Remark 4. We can find a quadratic extension of number fields E/F and a place v 1 of F such that E v1 /F v1 E 0 /F 0 .By passing to EK/F K for an appropriate K we can assume in addition that 1.Every real and even place of F splits at E.
2. Let S 1 = {v i } l i=1 be the set (possible empty) of places of F which ramify over E. Then E vi /F vi E 0 /F 0 .We fix such isomorphisms.
Choose a non-trivial additive character ψ of F \A F .Let w 1 be an auxiliary place of F which is inert in E with residual characteristic p. Assume that p |q F 0 .Let S 2 = {w j } m j=1 be the places of F of residual characteristic p.We may also assume that ψ v is unramified for v ∈ S 2 .Set the subspace spanned by the co-roots of the split component of the center of M .The map H : G(A) → a 0 is characterized, as usual, by the condition e αi,H(ntk) = |α i (t)|.We write d(a, b, c) for the diagonal element d(a, b, c) {s} G/H s → S g → gsθ(g) −1 Documenta Mathematica 5 (2000) 317-350 where {s} is a set of orbit representatives.In fact, our assumption (6) implies that there is only one orbit.Consider the B-orbits in S. According to a result of Springer ([S]), every B-orbit in S intersects the normalizer N G (T ) of the diagonal subgroup T .Moreover, the map C → C ∩ N G (T ) is a bijection between B-orbits of S and T -orbits of S ∩ N G (T ).Thus we may define a map ι : B orbits of S → W sending C to the T -coset of C ∩ N G (T ).

Corollary 1 .
There exists a permutation τ χ of B(χ) such thatBst (f, ν, λ) = B (f , τ χ (ν), λ)(27)for all λ.Proof.First, as was noted by Jacquet in ([J2], §4) one may use the localization principle to infer that B (f , ν, λ) depends only on the orbital integrals used in the definition of matching functions, and hence only on f .Choose any two finite places u, u of F which are inert in E and view each term in the equality For any ideal I of O F let φ(I) = |(O F /I) * |.Lemma 7. Let {I k } be a family of ideals, disjoint from S whose norms N (I k ) tend to ∞.For each k let X k be the set of Hecke characters of F which are trivial on R + and whose conductor divides I k J for some ideal J whose prime factors lie in S. Then for any f S ∈ C ∞ c (F * S ) we have1/φ(I k ) ∈X k fS ( S ) → λ F R+ f S (t) d * twhere λ F = vol(F * R + \I F ).
is fixed and f k is the characteristic function of {x ∈ v ∈S O * v : x ≡ 1 (mod I k )}.The Poisson summation formula gives L p (η, s) = j L wj (η wj , s) for any Hecke character η.Lemma 8.There exists an open set U 2 of U S2 such that whenever ν is a Hecke character of T such that ν S2 ∈ U 2 we necessarily have τ χ (ν) = ν in the notations of Corollary 1.
Then the O i 's are linearly independent distributions if and only if the L i are linearly independent.Indeed, any relation among the O i 's would imply that