INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS Adrian Diaconu Paul Garrett Abstract. WeobtainsecondintegralmomentsofautomorphicL–functionsonadelegroupsGL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adelegroupsGL1 andGL2. ThisrequiresreformulationofthenotionofPoincar´e series,replacingthe collection of classical Poincar´e series over GL2(Q) or GL2(Q(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms,distinguishingglobalandlocalissues,andallowinguniformapplicationtonumberfields. When specialized to the field of rational numbers Q, we recover the classical results on moments. 1. Introduction 2. Poincar´e series 3. Unwinding to an Euler product 4. Spectral decomposition of Poincar´e series 5. Asymptotics Appendix 1: Convergence of Poincar´e series Appendix 2: Mellin transform of Eisenstein Whittaker functions §1. Introduction For ninety years, mean values of families of automorphic L–functions have played a central role in analytic number theory. In the absence of the Riemann Hypothesis, or of the Grand Riemann Hypothesis referring to general L–functions, suitable mean value results often can substitute. Thus, asymptoticsorsharpboundsforintegralmomentsofautomorphicL–functionsareofintense interest. The study of integral moments was initiated in 1918 by Hardy and Littlewood [Ha-Li], who obtained the asymptotic formula for the second moment of the Riemann zeta-function Z T (1.1) |ζ(1 +it)|2 dt ∼ T logT 2 0 Eight years later, Ingham in [I] obtained the fourth moment Z T 1 (1.2) |ζ(1 +it)|4 dt ∼ ·T(logT)4 2 2π2 0 1991 Mathematics Subject Classification. 11R42, Secondary 11F66, 11F67, 11F70, 11M41, 11R47. Key words and phrases. Integral moments, Poincar´e series, Eisenstein series, L–functions, spectral decomposi- tion, meromorphic continuation Both authors were partially supported by NSF grant DMS-0652488.. Typeset by AMS-TEX 1 2 ADRIAN DIACONU PAUL GARRETT Since then, many authors have studied moments: for instance, see [At], [HB], [G1], [M1], [J1]. Most results are limited to integral moments of automorphic L–functions for GL (Q) and GL (Q). 1 2 No analogue of (1.1) or (1.2) was known over an arbitrary number field. The only previously known results for fields other than Q, are in [M4], [S1], [BM1], [BM2] and [DG2], all over quadratic extensions of Q. Here we obtain second integral moments of automorphic L–functions on adele groups GL 2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL and GL . This requires reformulation of the notion of Poincar´e series, 1 2 replacing the collection of classical Poincar´e series over GL (Q) or GL (Q(i)) with a single global 2 2 object which makes sense on an adele group over a number field. This is the first expression of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results on moments. More precisely, for f an automorphic form on GL and χ an idele class character of the number 2 field, let L(s,f ⊗χ) be the twisted L–function attached to f. We obtain asymptotics for averages ∞ Z X (cid:12) (cid:0) (cid:1)(cid:12)2 (1.3) (cid:12)L 12 +it, f ⊗χ (cid:12) Mχ(t)dt χ −∞ for suitable smooth weights M (t). We obtain the asymptotics from the appearance of this sum χ in moment expansions (3.12) ∞ Z Z (1.4) X L(cid:0)21 +it+v, f ⊗χ(cid:1)·L(cid:0)21 +it, f ⊗χ(cid:1)·Mχ(t)dt = P´ev ·|f|2 χ ZAGL2(k)\GL2(A) −∞ The Poincar´e series P´e has a spectral expansion, which, after an Eisenstein series is removed (see v (4.6)), has cuspidal components computed in (4.1) and continuous components computed in (4.10) (there are no residual spectrum components). Thus, using an auxiliary function ϕ introduced in Section 2, we have spectral expansions (see (4.8)) Z Z X (1.5) ZAGL2(k)\GL2(A)P´ev ·|f|2 = hEv+1,|f|2i N∞ϕ∞ + F ρ¯F GF∞(v,w)L(v+ 21,F)·hF,|f|2i + X χ(d) Z L(v+s¯,χ)·L(v+1−s¯,χ) |d|−(v−s¯+1/2)Z ϕ ·WE 4πiκ L(2s¯,χ2) ∞ s,χ,∞ χ <(s)=12 Z∞\G∞ where: F is summed over an orthonormal basis for spherical cuspforms on GL , E is an 2 v+1 Eisenstein series (see (4.6)), ρ = ρ (1) is (in effect) the first Fourier coefficient of the cuspform F, F F G is expressed in terms of gamma functions in (4.2) and (4.3), d is a differental idele ([W2], page 113, Definition 4) with component 1 at archimedean places, κ is a volume constant (see (4.7)), and WE is a normalization of Whittaker function attached to Eisenstein series (see Appendix 2). The sum in (1.3), (1.4), and (1.5) over idele class characters χ is infinite, in general. For general number fields, (1.3) is the correct structure of the second integral moment of GL automorphic 2 L–functions. This was first pointed out by Sarnak in [S1], where such an average was studied over the Gaussian field Q(i); see also [DG2]. Section 3 shows that this structure reflects Fourier inversion on the idele class group of the field. INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 3 Meanwhile, joint work [DGG2] with Goldfeld extends these ideas to produce integral moments for GL over number fields, exhibiting explicit kernels P´e again giving identities of the form r Z moment expansion = P´e·|f|2 = spectral expansion ZAGLr(k)\GLr(A) for cuspforms f on GL . The moment expansion on the left-hand side is of the form r Z X |L(s,f ⊗F)|2 M (s) ds + ... F F <(s)=1 2 summed over F in an orthonormal basis for cuspforms on GL , with corresponding continuous- r−1 spectrum terms. The specific choice gives a kernel with a surprisingly simple spectral expansion, with only three parts: a leading term, a sum induced from cuspforms on GL , and a continuous 2 partagaininducedfromGL . Inparticular, nocuspformsonGL with2 < ‘ ≤ r contribute. Since 2 ‘ the discussion for GL with r > 2 depends essentially on the GL case, the latter merits careful r 2 attention. Thus we give complete details for GL here. For GL over Q and square-free level, the 2 2 sum of moments can be arranged to have a single summand, recovering a classical integral moment ∞ Z |L(1 +it,f)|2M(t)dt 2 −∞ Asanon-trivialexample,considerthecaseofacuspformf onGL overQ. Weproduceaweight 3 function Γ(s,w,f ,F ) depending upon complex parameters s and w, and upon the archimedean ∞ ∞ data for both f and cuspforms F on GL , such that Γ(s,w,f ,F ) has explicit asymptotic 2 ∞ ∞ behavior similar to that discussed in Section 5 below, and such that the moment expansion above becomes Z Z X P´e·|f|2 = |L(s,f ⊗F)|2·Γ(s,w,f ,F )ds ∞ ∞ ZAGL3(Q)\GL3(A) F onGL2 <(s)=1 2 Z Z + X |L(s ,f ⊗E(k) )|2·Γ(s ,w,f ,E(k) ) ds ds 1 1−s2 1 ∞ 1−s2,∞ 2 1 k∈Z <(s1)=12 <(s2)=12 where L(s ,f ⊗E(k) ) = L(s1−s2+ 21,f)·L(s1+s2− 12,f) 1 1−s2 ζ(2−2s ) 2 Here F runs over an orthonormal basis for all level-one cuspforms on GL , without restriction on 2 (k) the right K –type. Similarly, the Eisenstein series E run over all level-one Eisenstein series for ∞ s GL (Q) with no restriction on K –type, indexed by k. 2 ∞ The discussion below makes several points clear. First, our sum of moments of twists of L– functions has a natural integral representation on the adele group, of a form insensitive to the underlying number field. Second, the kernel for this adele-group integral arises from a collection of local data, wound up into an automorphic form, and the computation proceeds by unwinding. This presentation requires a reformulation of the notion of Poincar´e series, replacing a weighted 4 ADRIAN DIACONU PAUL GARRETT sumofclassicalPoincar´eseriesforGL (Q)orGL (Q(i))withasingle, coherent, globalobjectthat 2 2 makes sense over a number field. Third, the ramifications of the choices of local data are subtle. In the present treatment we take local data at finite primes so as to avoid complications away from archimedean places. Fourth, some subtlety resides in choices of archimedean data. Good’s original idea in [G2] for GL (Q) can be viewed as a choice of local data for real places, which 2 can be improved as in the classical formulation of [DG1]. Similarly, we interpret the treatment of GL (Q[i]) in [DG2] as a choice of local data for complex primes. 2 Recipe for spectral identities involving second moments: Spectral identities involving second integralmomentsofautomorphicL-functionsandotherperiodscanbeproducedsystematically, as Z P follows. We suppress secondary details, even where non-trivial, writing as an ad hoc device to indicate integration against a suitable automorphic spectral measure not specified in detail. Thus, for a reductive group Θ over a number field k, with center Z, write the decomposition of functions Ψ in L2(ZAΘk\ΘA) as Z P Ψ = hΨ,Fi · F Θ F onΘ where the automorphic forms F generate irreducible representations of ΘA, or nearly so. In the discrete part of this spectral decomposition, the automorphic forms F are genuinely orthonormal. Themorecontinuouspartofthedecompositionbehavesinalesselementaryfashion,althoughmost ofitadmitsanexplicitdescriptionintermsofEisensteinseries. Wewillnotworryaboutunresolved issues concerning the residual spectrum. Further, we imagine that this L2 spectral decomposition is extended to suitable distributions and whatever non-L2 functions we need. Regularization of implied integrals is a genuine issue, but not our immediate concern. Let B be a subgroup of Θ containing Z, and let u be a left ZAΘk-invariant distribution on Θ supported on the image ZABk\BA inside ZAΘk\ΘA. Distributions u involving no derivatives transverse to ZABk\BA inside ZAΘk\ΘA admit spectral expansions Z P u = hu, Fi · F B F onB Now let Θ be of the form Θ = G×G, and let Z be the center of G (rather than of Θ). Let H be a k-subgroup of G containing Z. Consider two chains of subgroups inside G×G, pictorially, G×G % - H ×H G∆ - % H∆ where the superscript ∆ denotes diagonal copies, and where ascending arrows are inclusions. Consider the left ZAGk ×ZAGk-invariant distribution u on G×G Z Z u(f ⊗f ) = f ⊗f = f ·f 1 2 1 2 1 2 ZA∆Hk∆\HA∆ ZAHk\HA INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 5 The spectral expansion of u along H ×H has a special diagonal property, namely Z Z P P u = hu, F ⊗F i ·F ⊗F = F ⊗F 1 2 H×H 1 2 F1,F2 onH F onH The diagonal feature of this expansion is completely analogous to the fact that the Fourier series expansion of the distribution that integrates along the diagonal circle in a product of two circles has Fourier coefficients only along the diagonal. Then Z Z u(f ⊗f) = P hf ⊗f,F ⊗Fi = P |hf, Fi |2 H×H H F onH F onH The positivity of the summands is a virtue of this relation. On the other hand, the spectral expansion of u along G∆ is Z Z P P u = hu, Fi ·F = F ·F G∆ H F onG∆ F onG∆ where F = hu,Fi is the period of F along H. Thus, H Z Z u(f ⊗f) = P F ·hf ⊗f, Fi = P F · hF, |f|2i H G∆ H G F onG∆ F onG The period non-vanishing condition F 6= 0, for F to appear in the expansion, is non-trivial. H Equating these two expansions gives Z Z P |hf,Fi |2 = u(f ⊗f) = P F · hF, |f|2i H H G F onH F onG Diagrammatically, this is (moment side) (spectral side) Z Z P |hf,Fi |2 ←− f ⊗f −→ P F · hF, |f|2i H H G F onH F onG G×G ↑ % - ↑ Z Z P F ⊗F H ×H G∆ P F ·F H F onH F onG∆ - % H∆ u ∼ 1 The evaluation along H ×H is especially interesting when H is an Euler-Gelfand subgroup of G, in the sense that restrictions from G to H of (possibly a restricted class of) irreducibles on v v G are multiplicity-free. This makes hf,Fi tend to have an Euler product, possibly including a v H period. Thus, the left-hand (moment) side of the spectral identity is a sum (and integral) of second 6 ADRIAN DIACONU PAUL GARRETT integral moments of L-functions. By contrast, the right-hand (spectral) side inevitably involves integrals of three eigenfunctions, for automorphic forms F with non-vanishing periods F . H Theexponentialdecayofthearchimedeancontributionsinthisidentitymakethesumsconverge too well, so u must be deformed to extract more information. Specifically, call any deformation of theinitialdistributionutoa(classical, ratherthangeneralized)functiononZ∆G∆\G∆ aPoincar´e A k A series. A simple family of Poincar´e series is constructed by winding up local data, created by local N deformations, as follows. Let ϕ = ϕ where ϕ is a left H -invariant function on the ν-adic ν ν ν ν points G of G. Form a Poincar´e series ν X P´e (g) = ϕ(γ ·g) ϕ γ∈Hk\Gk To have ϕ be a classical function on G at non-archimedean places ν, we have the option to make ν ν an extremely simple choice  1 (for g ∈ H ·K )  ν ν ϕ (g) = ν 0 (otherwise) where K is (maximal) compact in G . Archimedean places do not usually allow such trivial local ν ν deformations. In this recipe, taking H = GL and G = GL gives the L-functions L(s,f ⊗χ) considered in 1 2 this paper, with f on GL and grossencharacters χ. Extending the GL ×GL case, we obtain 2 2 1 convolution L-functions L(s,f ⊗F) on GL ×GL by taking H = GL and G = GL . On n n−1 n−1 n the spectral side are standard L-functions attached to F on GL , along with triple integrals n−1 of eigenfunctions. Due to vanishing of GL periods, the spectral side involves no cuspidal n−1 data on GL for r > 2. Rankin-Selberg convolutions for GL ×GL arise by taking H = GL∆ r n n n and G = GL × GL . In the smallest case n = 2, the subgroup G∆ is also Euler-Gelfand n n in G × G. Rankin-Selberg L-functions given by doubling arise as follows. Let Φ be a form (orthogonal, symplectic, or hermitian) on a suitable vectorspace, and U(Φ) its isometry group. Let H = U(Φ)×U(−Φ) and G = U(Φ⊕−Φ) and take Siegel-type Eisenstein series on G. The period non-vanishing condition on the spectral side is again a stringent condition in this example. Triple-product L-functions appear in at least two ways. First, one may take G = GSp and 3 H = (GL ×GL ×GL )\, where the \ means to take the subgroup where the three determinants 2 2 2 are equal. Take Siegel-type Eisenstein series on G. A smaller family appears by taking G = GSp 2 and H = (GL ×GL )\, with an Eisenstein series on G with cuspidal data. 2 2 Ofcourse,thegeneralideaofexploitingdecompositionsofautomorphicformsorrepresentations along subgroups is a main technical device in the theory of automorphic forms, and is several decades old. Indeed, restriction and decomposition along simple inclusions A ⊂ B for Euler- Gelfand subgroups A of groups B have been studied for decades. By contrast, consideration of larger configurations of subgroups and iterated spectral decomposition along them is much less clich´ed, due in part to technical complications, and due also to a paucity of suggestive examples. For us, the chief desired effect is a positivity property of a sum-and-integral of Euler products on one side of the identity, attached to an Euler-Gelfand subgroup H of G. Our recipe above accomplishes this for arbitrary inclusions H∆ ⊂ H ×H and G∆ ⊂ G×G, regardless of whether these diagonals are Euler-Gelfand subgroups. It may be that Reznikov’s notable recent work [R] is the only other current example of systematic use of iterated spectral decompositions along larger configurations of subgroups. Given the scope of possibilities in considering the spectral theory of largeconfigurations,itisnotsurprisingthat[R]offersasomewhatdifferentrecipe,undersomewhat INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 7 different hypotheses, achieving somewhat different ends than we have indicated here. Leaving a more detailed comparison and systematization to the interested reader, we take the viewpoint that, in any case, our present discussion and that of [R] offer persuasive evidence for the utility of iterated spectral decompositions in large configurations of subgroups. The structure of the paper is as follows. In Section 2, integral kernels we call Poincar´e series are described in terms of local data, reformulating classical examples in a form applicable to GL r over a number field. In Section 3, the integral of the Poincar´e series against |f|2 for a cuspform f on GL is unwound and expanded, yielding a sum of weighted moment integrals of L–functions 2 L(s,f ⊗χ) of twists of f by idele class characters χ. In Section 4, the spectral decomposition of the Poincar´e series is exhibited: the leading term is an Eisenstein series, and there are cuspidal and continuous-spectrum parts with coefficients which are values of L–functions. In Section 5, an asymptotic formula is derived for these integral moments. We note there that the length of the averagesinvolvedisamenabletosubsequentapplicationstoconvexitybreakinginthet–aspect. The first appendix discusses convergence of the Poincar´e series in detail, proving pointwise convergence and L2 convergence. The second appendix computes integral transforms necessary to understand some details in the spectral expansion. For the most immediate applications, such as subconvexity, refined choices of archimedean data must be combined with the generalization [HR] of [HL], invoking [Ba], as well as an extension of [S2](or[BR])tonumberfields. However, fornow, wecontentourselveswithlayingthegroundwork for applications and extensions. In subsequent papers we will address the extension of the identity to GL , and consider convexity breaking in the t–aspect. r §2. Poincar´e series Let k be a number field, G = GL over k, and define standard subgroups: r (cid:26)(cid:18) (cid:19)(cid:27) (r−1)-by-(r−1) ∗ P = Pr−1,1 = 0 1-by-1 (cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27) I ∗ (r−1)-by-(r−1) 0 U = r−1 H = Z = center of G 0 1 0 1 Let K denote the standard maximal compact in the k –valued points G of G. ν ν ν The Poincar´e series P´e(g) is of the form X (2.1) P´e(g) = ϕ(γg) (g ∈ GA) γ∈ZkHk\Gk for suitable functions ϕ on GA described as follows. For v ∈ C, let O (2.2) ϕ = ϕ ν ν where for ν finite  (cid:18)A 0(cid:19) (2.3) ϕ (g) =  (cid:12)(cid:12)(detA)/dr−1(cid:12)(cid:12)vν for g = mk with m = 0 d ∈ ZνHν and k ∈ Kν ν  0 otherwise 8 ADRIAN DIACONU PAUL GARRETT and for ν archimedean require right K –invariance and left equivariance ν (cid:12) (cid:12)v (cid:18) (cid:18) (cid:19) (cid:19) (cid:12)detA(cid:12) A 0 (2.4) ϕν(mg) = (cid:12)(cid:12) dr−1 (cid:12)(cid:12) ·ϕν(g) for g ∈ Gν and m = 0 d ∈ ZνHν ν Thus, for ν|∞, the further data determining ϕ consists of its values on U . The simplest useful ν ν choice is   x   (cid:18) (cid:19) 1 (2.5) ϕν Ir0−1 x1 = (cid:0)1+|x1|2+···+|xr−1|2(cid:1)−dν(r−1)wν/2 x =  ...  and wν ∈ C x r−1 with d = [k : R]. Here the norm |x |2 +···+|x |2 is invariant under K , that is, |·| is the ν ν 1 r−1 ν usual absolute value on R or C. Note that by the product formula ϕ is left ZAHk–invariant. Proposition 2.6. (Apocryphal) With the specific choice (2.5) of ϕ = ⊗ ϕ , the series (2.1) ∞ ν|∞ ν defining P´e(g) converges absolutely and locally uniformly for <(v) > 1 and <(w ) > 1 for all ν|∞. ν Proof: In fact, the argument applies to a much broader class of archimedean data. For a complete argument when r = 2, and w = w for all ν|∞, see Appendix 1. (cid:3) ν We can give a broader and more robust, though somewhat weaker, result, as follows. Again, for Q simplicity, take r = 2. Given ϕ , for x in k = k , let ∞ ∞ ν|∞ ν (cid:18) (cid:19) 1 x Φ (x) = ϕ ∞ ∞ 0 1 For 0 < ‘ ∈ Z, let Ω be the collection of ϕ such that the associated Φ is absolutely integrable, ‘ ∞ ∞ and such that the Fourier transform Φb∞ along k∞ satisfies the bound Φb∞(x) (cid:28) Q (1+|x|ν)−‘ ν|∞ For example, for ϕ to be in Ω it suffices that Φ is ‘+1 times continuously differentiable, with ∞ ‘ ∞ each derivative absolutely integrable. The simple explicit choice of ϕ above lies in Ω for every ∞ ‘ ‘ > 0, when <(w ) > 1 and <(v) > 1 for convergence. ν Theorem 2.7. (Apocryphal) Take r = 2, <(v) > 3, ‘ > <(v) + 5, and ϕ ∈ Ω . The series ∞ ‘ defining P´e(g) converges absolutely and locally uniformly in both g and v. Furthermore, up to an Eisenstein series, the Poincar´e series is square integrable on ZAGk\GA. Proof: See Appendix 1. (cid:3) Wedonotclaimthatthelowerboundsarethebestpossible, butonlythattheyariseasartifacts in the natural argument given in Appendix 1. The precise Eisenstein series to be subtracted from the Poincar´e series to make the latter square-integrable will be discussed in Section 4 (see formula 4.6). For our special choice (2.5) of archimedean data, both these convergence results apply with <(w ) > 1 for ν|∞ and <(v) large. ν A monomial vector ϕ as in (2.2) described by (2.3) and (2.4) will be called admissible when ϕ ∈ Ω , with <(v) > 3 and ‘ > <(v)+5. ∞ ‘ INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 9 §3. Unwinding to an Euler product Unlike classical contexts, where the Euler factorization of a Dirichlet series is visible only at the end, the present construction presents us with an Euler product almost immediately. From now on, take r = 2, so G = GL over a number field k, and 2 (cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27) ∗ ∗ 1 ∗ ∗ 0 ∗ 0 P = N = U = H = M = ZH = 0 ∗ 0 1 0 1 0 ∗ For a place ν of k, let K be the standard maximal compact subgroup. That is, at finite places ν K = GL (o ), at real places K = O(2), and at complex places K = U(2). ν 2 ν ν ν UsingthePoincar´eseriesdefinedby(2.1),weunwindacorrespondingglobalintegralandexpress it as an inverse Mellin transform of an Euler product. This produces a sum over Hecke characters of weighted integrals of corresponding L–functions over the critical line. Recall the definition X (3.1) P´e(g) = ϕ(γg) (g ∈ GA) γ∈Mk\Gk where the monomial vector O ϕ = ϕ ν ν is (cid:26) χ (m) for g = mk, m ∈ M and k ∈ K 0,ν ν ν (3.2) ϕ (g) = (for ν finite) ν 0 for g 6∈ M ·K ν ν and for ν infinite, we do not entirely specify ϕ , only requiring the left equivariance ν (3.3) ϕ (mnk) = χ (m)·ϕ (n) (for ν infinite, m ∈ M , n ∈ N and k ∈ K ) ν 0,ν ν ν ν ν Here, χ is the character of M given by 0,ν ν (cid:12)a(cid:12)v (cid:18) (cid:18)a 0(cid:19) (cid:19) (3.4) χ (m) = (cid:12) (cid:12) m = ∈ M , v ∈ C 0,ν (cid:12)d(cid:12) 0 d ν ν N Then χ0 = νχ0,ν is Mk–invariant, and ϕ has trivial central character and is left MA–equivariant by χ . Note that for ν infinite, the assumptions imply that 0 (cid:18) (cid:19) 1 x x −→ ϕ ν 0 1 is a function of |x| only. Let f1 and f2 be cuspforms on GA. Eventually we will take f1 = f2, but for now merely require the following compatibilities. Suppose that the representations of GA generated by f1 and f2 are irreducible, with the same central character. At all ν, require that f and f have the same right 1 2 K –type, that this K –type is irreducible, and that f and f correspond to the same vector in ν ν 1 2 the K–type, up to scalar multiples. Schur’s lemma assures that this makes sense, insofar as there 10 ADRIAN DIACONU PAUL GARRETT are no non-scalar automorphisms. Last, require that each f is a special vector locally everywhere i in the representation it generates, in the following sense. Let X (3.5) f (g) = W (ξg) i fi ξ∈Zk\Mk be the Fourier expansion of f , and let i O W = W fi fi,ν ν≤∞ be the factorization of the Whittaker function W into local data. By [JL], we may require that fi for all ν < ∞ the Hecke–type local integrals Z (cid:18) (cid:19) a 0 s−1 W |a| 2 da fi,ν 0 1 ν a∈kν× differ by at most an exponential function from the correct local L–factors for the representation generated by f . i Suppressing some details in the notation, the integral under consideration is Z (3.6) I(χ ) = P´e(g)f (g)f¯(g)dg 0 1 2 ZAGk\GA For χ (and archimedean data) in the range of absolute convergence, from the definition of the 0 Poincar´e series, the integral unwinds to Z ϕ(g)f (g)f¯(g)dg 1 2 ZAMk\GA Using the Fourier expansion X f (g) = W (ξg) 1 f1 ξ∈Zk\Mk this further unwinds to Z (3.7) ϕ(g)W (g)f¯(g)dg f1 2 ZA\GA Let J be the ideles, let C be the idele class group J/k× = GL1(A)/GL1(k), and let Cb be the dual of C. By Fujisaki’s Lemma (see Weil [W1], page 32, Lemma 3.1.1), the idele class group C is a product of a copy of (0,∞) and a compact group C0. By Pontryagin duality, Cb ≈ R×Cb0 with Cb0 discrete. For any compact open subgroup Ufin of the finite-prime part in C0, the dual of C /U isfinitelygeneratedwithrank[k : Q]−1. OnC thespectraldecomposition(Fourier-Mellin 0 fin inversion) for a suitable function F is Z Z (3.8) F(x) = F(y)χ(y)dy χ−1(x)dχ Cb C Z Z = X 1 F(y)χ0(y)|y|sdy χ0−1(x)|x|−sds 2πi C χ0∈Cb0 <(s)=σ