Moments for L functions for GL r GL r Introduction

Transcription

1 Moments for L functions for GL r GL r. Diaconu P. Garrett D. Goldfeld 3 bstract: We establish a spectral identity for moments of Rankin-Selberg L functions on GL r GL r over arbitrary number fields, generalizing our previous results for r =.. Introduction. Background and normalizations 3. Moment expansion 4. Spectral expansion of Poincaré series 5. ppendix: half-degenerate Eisenstein series. Introduction Let k be an algebraic number field with adele ring. Fix an integer r and consider the general linear groups GL r (k, GL r ( of r r invertible matrices with entries in k,, respectively. Let Z + be the positive real scalar matrices in GL r. Let π be an irreducible cuspidal automorphic representation in L (Z + GL r (k\gl r (. Let π run over irreducible unitary cuspidal representations in L (Z + GL r (k\gl r (, where now Z + is the positive real scalar matrices in GL r. For brevity, denote a sum over such π by π. For complex s, let L(s, π π denote the Rankin-Selberg convolution L function. second integral moment over the spectral family GL r is described roughly as follows. For each irreducible cuspidal automorphic π of GL r, assign a constant c(π 0. Letting π be the archimedean component of π and π the archimedean factor of each π, let M(s, π, π be a function of complex s, whose possibilities will be described in more detail later. The corresponding second moment of π is c(π π Re s= L(s, π π M(s, π, π ds In fact, there are corresponding further correction terms corresponding to non-cuspidal parts of the spectral decomposition of L (Z + GL r (k\gl r (, but the cuspidal part presumably dominates. The theory of second integral moments on GL GL has a long history, although the early papers treated mainly the case that the groundfield k is Q. For example, see [Hardy-Littlewood 98], [Ingham 96], [Heath-Brown 975], [Sarnak 985], [Good 983,986], [Motohashi 997], [Jutila 997], [Petridis- Sarnak 00], [Bruggeman-Motohashi 00,003], [CFKRS 005], [Diaconu-Goldfeld-Hoffstein 005], [Diaconu-Goldfeld 006a,006b], [Diaconu-Garrett 009]. Second integral moments of level-one holomorphic elliptic modular forms were first treated in [Good 983,986], the latter using an idea that is a precursor of part of the present approach. The study of second integral moments for GL GL with arbitrary level, groundfield, and infinity-type is completely worked out in [Diaconu-Garrett 00]. The main aim of this paper is to establish an identity relating the second integral moment, described above, to the integral of a certain Poincaré series P against the absolute value squared f of a distinguished cuspform f π. cknowledging that the spectral decomposition of L (Z GL r (k\gl r ( also has a noncuspidal part generated by Eisenstein series and their residues, the identity we obtain takes the form The authors were partially supported by NSF grant DMS Department of Mathematical Sciences, Durham University, Durham DH 3LE UK University of Minnesota, 7 Vincent Hall, Church St. SE, Minneapolis, MN, 55455, US 3 4 Mathematics Building, Columbia University, New York, NY, 007 US

2 P(g, ϕ f(g dg = ρ(π π Re s= Z GL r(k\gl r( L(s, π π M(s, π, π, ϕ ds + (non-cuspidal part Here M(s, π, π, ϕ is a weighting function depending on the complex parameter s, on the archimedean components π and π, and on archimedean data ϕ defining the Poincaré series. The global constants ρ(π are analogues of the leading Fourier coefficients of GL cuspforms. The spectral expansion of the Poincaré series P relates the second integral moment to automorphic spectral data. Remarkably, the cuspidal data appearing in the spectral expansion of P comes only from GL. These new identities have some similarities to the Kuznetsov trace formula [Bruggeman 978], [Wallach 99], [Ye 000], [Goldfeld 006], in that they are derived via the spectral resolution of a Poincaré series, but they are clearly of a different nature. We have in mind application not only to cuspforms, but also to truncated Eisenstein series or wave packets of Eisenstein series, thus applying harmonic analysis on GL r to L functions attached to GL, touching upon higher integral moments of the zeta function ζ k (s of the ground field k. In connection to the present work, we mention the recent mean-value result of [Young 009], T ε T ε T <t j T L( + it, u j ϕ dt T 3+ε (for ε > 0 where ϕ is on GL 3, and where u j on GL has spectral data t j, as usual. From this the t-aspect convexity bound can be recovered. lso, [Li 009] obtains a t-aspect subconvexity bound for standard L functions for GL 3 (Q for Gelbart-Jacquet lifts. For context, we review the [Diaconu-Goldfeld 005] treatment of spherical waveforms f for GL (Q. In that case, the sum of moments is a single term Z GL (Q\GL ( P(g, z, w f(g dg = L(z + s, f L(s, f Γ(s, z, w, f ds πi Re (s= where Γ(s, z, w, f is a ratios of products of gammas, with arguments depending upon the archimedean data of f. Here the Poincaré series P(g = P(g, z, w is specified completely by complex parameters z, w, and has a spectral expansion P(g, z, w = π w Γ( w π w Γ( w E +z (g + + 4πi Re (s= ζ(z + s ζ(z + s ξ( s F on GL ρ F L( + z, F G( it F, z, w F (g G( s, z, w E s (g ds (for Re(z, Re(w where ξ(s = π s/ Γ(s/ζ(s, where G is essentially a product of gamma function values z+ s G(s, z, w = π (z+ w Γ( Γ( z+s Γ(z + w Γ( z s+w Γ( z+s +w and F is summed over (an orthogonal basis for spherical (at finite primes cuspforms on GL with Laplacian eigenvalues 4 + t F, and E s is the usual spherical Eisenstein series. The continuous part, the integral of Eisenstein series E s, cancels the pole at z = of the leading term, and when evaluated at z = 0 is

3 P(g, 0, w = (holomorphic at z =0 + ρ F L(, F G( it F, 0, w F (g F on GL + ζ(s ζ( s G( s, 0, w E s (g ds 4πi Re (s= ξ( s In this spectral expansion, the coefficient in front of a cuspform F includes G evaluated at z = 0 and s = ± it F, namely G( it F, 0, w = π w Γ( it F Γ( +it F Γ( w it F Γ( w +it F Γ( w The gamma function has poles at 0,,,..., so this coefficient has poles at w = ± it F, 3 ± it F,.... Over Q, among spherical cuspforms (or for any fixed level these values have no accumulation point. The continuous part of the spectral side at z = 0 is 4πi Re (s= ξ(s ξ( s ξ( s Γ( w s w +s Γ( Γ( w E s ds with gamma factors grouped with corresponding zeta functions, to form the completed L functions ξ. Thus, the evident pole of the leading term at w = can be exploited, using the continuation to Re(w > /. contour-shifting argument shows that the continuous part of this spectral decomposition has a meromorphic continuation to C with poles at ρ/ for zeros ρ of ζ, in addition to the poles from the gamma functions. lready for GL, over general ground fields k, infinitely many Hecke characters enter both the spectral decomposition of the Poincaré series and the moment expression. This naturally complicates isolation of literal moments, and complicates analysis of poles via the spectral expansion. Suppressing constants, the moment expansion is a sum of twists by Hecke characters χ, of the form P(s, z, w, ϕ f(g = L(z + s, f χ L( s, f χ M(s, z, χ, ϕ ds Z GL (k\gl ( χ Re (s= where M(s, z, χ, ϕ depends upon complex parameters s, z and archimedean components χ, f, and upon auxiliary archimedean data ϕ defining the Poincaré series. gain suppressing constants, the spectral expansion is P(g, z, ϕ = ( part E +z (g + ( part ρ F L( + z, F F (g F on GL + χ Re (s= L(z + s, χ L(z + s, χ L( s, χ G(s, χ E s,χ (g ds where the factor denoted -part depends only upon the archimedean data, as does G(s, χ. In the simplest case beyond GL, take f a spherical cuspform for GL 3 (Q generating an irreducible cuspidal automorphic representation π = π f. We can construct a weight function Γ(s, z, w, π, π with explicit asymptotic behavior, depending upon complex parameters s, z, and w, and upon the archimedean components π for π and for π irreducible cuspidal automorphic on GL, such that the moment expansion has the form P(g, z, w f(g dg = Z GL 3(Q\GL 3( ρ(π πi π on GL Re (s= L(s, π π Γ(s, 0, w, π, π ds + 4πi πi k Z Re (s = Re (s = L(s, π π (k E s ξ( it Γ(s, 0, w, π, E (k s ds, ds 3

4 where π runs over (an orthogonal basis for all level-one cuspforms on GL, with no restriction on the right K -types, E s (k is the usual level-one Eisenstein series of K -type k, and the notation E (k s, means that the dependence is only upon the archimedean component. Here and throughout, for Re(s = /, use s in place of s, to maintain holomorphy in complex-conjugated parameters. More generally, for an irreducible cuspidal automorphic representation π on GL r with r 3, whether over Q or over a numberfield, the moment expansion includes an infinite sum of terms L(s, π π for π ranging over irreducible cuspidal automorphic representations on GL r, as well as integrals of products of L functions L(s, π π... L(s, π π l for π,..., π l ranging over l-tuples of cuspforms on GL r... GL rl for all partitions (r,..., r l of r. Generally, the spectral expansion of the Poincaré series for GL r is an induced-up version of that for GL. Suppressing constants, using groundfield Q to skirt Hecke characters, the spectral expansion has the form P = ( part E r, z+ + ( part ρ F L( rz+r +, F Er, z+,f F on GL + Re (s= ( part rz+r ζ( + rz+r s ζ( + + s ζ( s E r,, z+, s z+ z+, s where F is summed over an orthonormal basis for spherical cuspforms on GL, and where the Eisenstein series are naively normalized spherical, with Es r, a degenerate Eisenstein series attached to the parabolic corresponding to the partition r,, and Es r,,,s,s 3,χ a degenerate Eisenstein series attached to the parabolic corresponding to the partition r,,. gain over Q, the most-continuous part of the moment expansion for GL r is of the form Re (s= t Λ L(s, π π E min M t (s ds dt = +it Λ Π l r L(s + it l, π Π j<l<n ζ( it j + it l ds M t (s ds dt where Λ = {t R r : t t r = 0} and where M t is a weight function depending upon π. More generally, let r = m b. For π irreducible cuspidal automorphic on GL m, let π = π... π on GL m... GL m. Inside the moment expansion we have (recall Langlands-Shahidi Re (s= Λ Π l b L(s + it l, π π L(s, π π Eπ, M +it π,t(s ds dt = Π j<l b L( it j + it l, π π M ds dt Replacing the cuspidal representation π on GL r (Q by a (truncated minimal-parabolic Eisenstein series E α with α C n, the most-continuous part of the moment expansion contains a term Taking α = 0 C r gives Π µ n, l r ζ(α µ + s + it l Π j<l<r ζ( it j + it l Λ For example, for GL 3, where Λ = {(t, t} R, Λ Π l r ζ(s + it l r Π j<l<r ζ( it j + it l ds dt M ds dt ζ(s + it 3 ζ(s it 3 R ζ( it M ds dt 4

5 and for GL 4 (s Λ ζ(s + it 4 ζ(s + it 4 ζ(s + it 3 4 ζ( it +it ζ( it +it 3 ζ( it +it 3 M ds dt. Background and normalizations We recall some facts concerning Whittaker models and Rankin-Selberg integral representations of L functions, and spectral theory for automorphic forms, on GL r. To compare zeta local integrals formed from vectors in cuspidal representations to local L functions attached to the representations, we specify distinguished vectors in irreducible representations of p-adic and archimedean groups. Locally at both p- adic and archimedean places, Whittaker models with spherical vectors have a natural choice of distinguished vector, namely, the spherical vector taking value at the identity element of the group. Even in general, for the specific purposes here, at finite places the facts are clear. t archimedean places the facts are more complicated, and, further, the situation dictates choices of data, and we are not free to make ideal choices. See [Cogdell 00], [Cogdell 003], [Cogdell 004] for detailed surveys, and references to the literature, mostly papers of Jacquet, Piatetski-Shapiro, and Shalika. The spectral theory is due to [Langlands 976], [Moeglin-Waldspurger 995], and proof of conjectures of [Jacquet 983] in [Moeglin- Waldspurger 989]. Fix an integer r and consider the general linear group G = GL r over a fixed algebraic number field k. For a positive integer l, in the following we use the notation l l to denote an l-by-l matrix, and let l denote the l l identity matrix. Then G = GL r has the following standard subgroups: ( (r (r P = P r, = { } 0 ( ( r (r (r 0 U = { } H = { } 0 0 N = {upper-triangular unipotent elements in H} = (unipotent radical of standard minimal parabolic in H Z = center of GL r Let = k be the adele ring of k. For a place v of k let k v be the corresponding completion, with ring of integers o v for finite v. For an algebraic group defined over k, let G v be the k v -valued points of G. For G = GL r over k, let K v be the standard maximal compact subgroup of G v : for v <, K v = GL r (o v for v R, K v = O r (R, and for v C, K v = U(r. standard choice of non-degenerate character on N k U k \N U is ψ(n u = ψ 0 (n + n n r,r ψ 0 (u r,r where ψ 0 is a fixed non-trivial character on /k. cuspform f has a Fourier-Whittaker expansion along NU f(g = W f (ξg where W f (g = ψ(nu f(nug dn du ξ N k \H N k U k \N k U The Whittaker function W f (g factors over primes, and a careful normalization of this factorization is set up below. Cuspforms F on H have corresponding Fourier-Whittaker expansions F (h = W F (ξh where W F (g = ψ(n F (nh dn ξ N k \H k N k \N where H GL r sits inside H as H sits inside G, N = N H, and ψ is restricted from NU to N. This Whittaker function also factors W F = v W F,v. 5

6 t finite places v, given an irreducible admissible representation π v of G v admitting a Whittaker model, [Jacquet-PS-Shalika 98] shows that there is an essentially unique effective vector Wπ eff v, generalizing the characterization of newform in [Casselman 973], as follows. For π v spherical, Wπ eff v is the usual unique spherical Whittaker vector taking value at the identity element of the group, as in [Shintani 976], [Casselman-Shalika 980]. For non-spherical local representations, define effective vector as follows. Let ( Uv opp r 0 (l = { : x = 0 mod p l } x Let K H v GL r (o v be the standard maximal compact of H v. Define a congruence subgroup of K v by K v (l = K H v (U v K v Uv opp (l For a non-spherical Whittaker model π v there is a unique positive integer l v, the conductor of π v, such that π v has no non-zero vectors fixed by K v (l for l < l v, and has a one-dimensional space of vectors fixed by K v (l v. The remaining ambiguous constant is completely specified by requiring that local Rankin-Selberg integrals Z v (s, W eff π v W o π v = ( det Y s W eff Y π v N v\h v Wπ o (Y dy v produce the correct local factors L v (s, π v π v of GL r GL r Rankin-Selberg L functions for every spherical representation π v of the local GL r, with normalized spherical Whittaker vector Wπ o in v π v. That is, Z v (s, W eff π v W o π v = L v(s, π v π v with no additional factor on the right-hand side. See Section 4 of [Jacquet-PS-Shalika 983], and comments below. Cuspidal automorphic representations π v π v of G admit local Whittaker models at all finite places, so locally at all finite places have a unique effective vector. Facts concerning archimedean local Rankin-Selberg integrals for GL m GL n for general m, n are more complicated than the non-archimedean cases. See [Stade 00], [Stade 00], [Cogdell-PS 003], as well as the surveys [Cogdell 00], [Cogdell 003], [Cogdell 004]. The spherical case for GL r GL r admits fairly explicit treatment, but this is insufficient for our purposes. Fortunately, for us there is no compulsion to attempt to specify the archimedean local data for Rankin-Selberg integrals. Indeed, the local archimedean Rankin-Selberg integrals will be deformed into shapes essentially unrelated to the corresponding L-factor, in any case. Thus, in the moment expansion in the theorem below we can use any systematic specification of distinguished vectors e πv in irreducible representations π v of G v, and e π v in π v of H v, for v archimedean. For v and π v a Whittaker model representation of G v with a spherical vector, let the distinguished vector e πv be the spherical vector normalized to take value at the identity element of the group. Similarly, for π v a Whittaker model representation of H v with a spherical vector, let the distinguished vector e π v be the normalized spherical vector. nticipating that cuspforms generating spherical representations at archimedean places make up the bulk of the space of automorphic forms, we do not give an explicit choice of distinguished vector in other archimedean representations. Rather, we formulate the normalizations below, and the moment expansion, in a fashion applicable to any choice of distinguished vectors in archimedean representations. Let π be an automorphic representation of G, factoring over primes as π v π v admitting a global Whittaker model. Each local representation π v has a Whittaker model, since π has a global Whittaker model. t each finite place v, let Wπ eff v be the normalized effective vector, and e πv the distinguished vector at v. Let f π be a moderate-growth automorphic form on G corresponding to a monomial tensor in π, consisting of the effective vector at all finite primes, and the distinguished vector e πv at v. Then the global Whittaker function of f is a globally-determined constant multiple of the product of the local functions: = ρ f W f v e πv v< where ρ f is a general analogue of the leading Fourier coefficient ρ f ( in the GL (Q theory. 6 W eff π v

7 Let π be an automorphic representation of H spherical at all finite primes, admitting a global Whittaker model. Let π factor as j : v π v π. Certainly each π v admits a Whittaker model. t each finite v, let Wπ o be the normalized spherical vector in v π v, and at archimedean v let e π v be the distinguished vector. For a moderate-growth automorphic form F π corresponding to a monomial vector in the factorization of π, at all finite places corresponding to the spherical Whittaker function Wπ o, and to v the distinguished vector e π v at archimedean places, again specify a constant ρ F by = ρ F W F v e π v v< When π occurs discretely in the space of L automorphic forms on H, each of the local factors of π is unitarizable, and uniquely so up to a constant, by irreducibility. For an arbitrary vector ε = ε in π, let F ε be the automorphic form corresponding to v< W π o v ε by the isomorphism j. Define ρ F ε by W F ε W o π v = ρ F ε Wπ o v ε By Schur s Lemma, the comparison of ρ F and ρ F ε depends only upon the comparison of archimedean data, namely, ρ F ε ε π = ρ F v e π v π with Hilbert space norms on the representation π at archimedean places. The ambiguity of these norms by a constant disappears in taking ratios. Indeed, the global constants ρ F and ρ F ε can be compared by a similar device (and induction for F and F ε in any irreducible π occurring in the L automorphic spectral expansion for H. We do not do carry this out explicitly, since this would require setting up normalizations for the full spectral decomposition, while our main interest is in the cuspidal (hence, discrete part. With f cuspidal and F moderate growth, corresponding to distinguished vectors, as above, the Rankin- Selberg zeta integral is the finite-prime Rankin-Selberg L function, with global constants ρ f and ρ F, and with archimedean local Rankin-Selberg zeta integrals depending upon the distinguished vectors at archimedean places: ( det Y s Y F (Y f dy = ρ H k \H f ρ F L(s, π π Z v (s, e πv e π v The finite-prime part of the Rankin-Selberg L function appears regardless of the archimedean local data. The global constants ρ f and ρ F do depend partly upon the local archimedean choices, but are global objects. We need a spectral decomposition of part of L (H k \H, as follows. Let Kfin H be the standard maximal compact GL r (ô of H fin, where as usual ô is v< o v, with o v the local integers at the finite place v of k. First, there is the Hilbert direct-integral decomposition by characters ω on the central archimedean split component Z + of H: let v< v i : y (y d,..., y d,,,... (for y > 0, with d = [k : Q] be the diagonal imbedding of the positive real numbers in the archimedean factors of the ideles of k. The central archimedean split component is Z + = {j(y = i(y /(r... The point of our parametrization is that (with idele norms i(y /(r det j(y = i(y = y (with y > 0 7 H : y > 0}

8 The corresponding spectral decomposition is L (H k \H R L (Z + H k \H, ω it dt where L (Z + H k \H, ω it is the isotypic component of functions Φ with Φ in L (Z + H k \H transforming by Φ(j(y h = y it Φ(h (for y > 0 and h H under Z +. The projections and spectral synthesis along Z + can be written as F (h = ( F (j(y h y R 0 it dy Each isotypic component L (Z + H k \H, ω it has a direct integral decomposition as a representation of H, of the form L (Z + H k \H, ω it Ξ y dt π det it dπ where Ξ is the set of irreducibles π occuring in L (Z + H k \H, ω 0. That is, the irreducibles for general archimedean split-component character ω it differ merely by a determinant twist from the trivial splitcomponent character case. The measure is not described explicitly here, apart from the observation that the discrete part of the decomposition, including the cuspidal part, has counting measure. For our applications, we are concerned with the subspaces L (Z + H k \H /Kfin H, ω of right KH fin -invariant functions. Since each π factors over primes as a restricted tensor product π v π v of irreducibles π v of the local points H v, the decomposition of L (Z + H k \H /K H fin, ω only involves the subset Ξo consisting of irreducibles π Ξ such that for every finite place v the local representation π v is spherical. Let π be the archimedean factor of π, and π fin the finite-place factor, so π π π fin. Let π o fin be the one-dimensional space of K H fin -fixed vectors in π fin. s a representation of the archimedean part H of H, L (Z + H k \H /K H fin, ω it Ξ o ( π π o fin det it dπ n automorphic spectral decomposition for F in L (Z + H k \H /K H fin, ω it can be written in the form F = Ξ o j F, Φ π j det it Φ π j det it dπ where Ξ o indexes spherical automorphic representations π with trivial archimedean split-component character entering the spectral expansion, for each of these j indexes an orthonormal basis in the archimedean component π, and Φ π j is the corresponding moderate-growth spherical automorphic form in the global π. The pairing is the natural one, namely, F, Φ π j det it = H k \H F (h Φ π j(h det h it dh 3. Moment expansion We define a Poincaré series P = P z,ϕ depending on archimedean data ϕ and a complex equivariance parameter z. With various simplifying choices of archimedean data ϕ depending only on a complex parameter w, the Poincaré series P = P z,w is a function of the two complex parameters z, w. By design, 8

9 for a cuspform f of conductor l on G = GL r over a number field k, for any choice of data for the Poincaré series sufficient for convergence, the integral Z G k \G f P is an integral moment of L functions attached to f, in the sense that it is a sum and integral over a spectral family, namely, a weighted average over spectral components with respect to L (GL r (k\gl r (. Subsequently, we will obtain a spectral expansion of the more-simply parametrized Poincaré series P = P z,w, giving the meromorphic continuation of this integral in the complex parameters z, w. For z C, let ϕ = ϕ v v where z C specifies an equivariance property of ϕ, as follows. For v finite, ( (det /d ϕ v (g = r z 0 (for g = mk with m = in Z v 0 d v H v and k K v 0 (otherwise For v archimedean require right K v -invariance and left equivariance z ϕ v (mg = det d r ϕ v (g (for g G v, for m = v ( 0 Z 0 d v H v Thus, for v, the further data determining ϕ v consists of its values on U v. simple useful choice of archimedean data parametrized by a single complex parameter w is ( x r x ϕ v = ( + x x r [kv:r]w/ (where x =., and w C x r The norm x x r is normalized to be invariant under K v. Thus, ϕ is left Z H k -invariant. We attach to any such ϕ a Poincaré series P(g = P ϕ (g = γ Z k H ϕ(γg 3. Remark: There is an essentially unique choice of (parametrized archimedean data ϕ = ϕ z,w, such that the associated Poincaré series at z = 0 has a functional equation (as in [Diaconu-Garrett-Goldfeld 008]. For instance, when G = GL 3 over Q this choice is ϕ ( I u ( I u = ϕ 0,w, with F the usual hypergeometric function F (α, β; γ; x = Γ(γ Γ(αΓ(β = w π Γ( w ( + u w F ( w, w ; w; + u m=0 Γ( w+ (for z = 0 Γ(α + mγ(β + m x m (for x < m! Γ(γ + m The functional equation of the Poincaré series P 0,w (g attached to this choice of ϕ = ϕ w when z = 0 is: the function ( πw π ζ(w ζ( w sin P 0,w (g + ( w π w Γ(w ζ(w E,,( g, w 3, w 3 9

10 is invariant as w w, where E,, (g, s, s = E,, s,s (g is the minimal parabolic Eisenstein series. fter our discussion of the spectral expansion of the Poincaré series, we give a general prescription for archimedean data producing Poincaré series admitting a functional equation: with suitable archimedean data, the functional equation is visible from the spectral expansion. With subscripts denoting the archimedean parts of various objects, for h, m H, define K(h, m = K ϕ (h, m = ϕ (u ψ (huh ψ (mum du U Let π π v be a cuspidal automorphic representation of G, with finite set S of finite primes such that π v is spherical for finite v S, and π v has conductor l v for v S. We say a cuspform f in π is a newform if it is spherical at finite v S and is right K v (l v -fixed for v S. s above, the global Whittaker function W f of f factors as W f = ρ f Wπ eff v e πv v v< Let e π = v e πv. Let π be an automorphic representation of H admitting a global Whittaker model, with unitarizable archimedean factor π, with orthonormal basis ε π j for π. Recalling that K(h, m = K z,ϕ (h, m depends on the parameter z and the data ϕ, the gamma factors appearing in the moment expansion below are = j N \H N \H Γ(e π, π, s = Γ z,ϕ (e π, π, s K e π (hkε π j(h det h z+s eπ (mkε π j(m det m s K(h, m dm dh dk The sum over the orthonormal basis for π is simply an expression for a projection operator, so is necessarily independent of the orthonormal basis indexed by j. Thus, the sum indeed depends only on the archimedean Whittaker model π. For each automorphic representation π of H occurring (continuously or discretely in the automorphic spectral expansion for H, and admitting a global Whittaker model, and spherical at all finite primes, let F π be an automorphic form in π corresponding to the spherical vector at all finite places and to the distinguished vector e π in the archimedean part. 3. Theorem: Let f be a cuspform, as just above. For Re(z and Re(w, we have the moment expansion f P = ρ f Z G k \G ρ Fπ Ξ o L( + it + z, π π L( it, π π Γ(e π, π, + it dt dπ R Proof: The typical first unwinding is P(g f(g dg = ϕ(g f(g dg Z G k \G Z H k \G Express f in its Fourier-Whittaker expansion, and unwind further: ϕ(g W f (ηg f(g dg = ϕ(g W f (g f(g dg Z H k \G η N k \H Z k N k \G Use an Iwasawa decomposition G = (HZU K everywhere locally to rewrite the whole integral as N k \H U K ϕ(huk W f (huk f(huk dh du dk 0

11 t finite primes v S, the right integral over K v can be dropped, since all the functions in the integrand are right K v -invariant. t finite primes v S, using the newform assumption on f, the one-dimensionality of the K v (l v -fixed vectors in π v implies that the K v -type in which the effective vector lies is irreducible. Thus, by Schur orthogonality and inner product formulas, a diagonal integral of f(xk v f(yk v over k v K v is a positive constant multiple of f(xf(y, for all x, y G. Thus, the integrals over K v for v finite can be dropped entirely, and, up to a positive constant depending only upon the right K v -type of f at v S, the whole integral is N k \H U K ϕ(huk W f (huk f(huk dh du dk Since f is left H k -invariant, it decomposes along H k \H. The function h f(huk with u U and k K is right Kfin H -invariant. Thus, f decomposes as f(huk = Φ π j(h det h it Φ π R Ξ H j(m det m it f(muk dm dπ dt o k \H j Unwind the Fourier-Whittaker expansion of f f(huk = Φ π j(h det h it Φ π j(m det m it Ξ o H k \H = Then the whole integral is = R Ξ o j N k \H U j Φ π j(h det h it Ξ o η N k \H k W f (ηmuk dm dk dπ N k \H Φ π j(m det m it W f (muk dm dk dπ Z G k \G P(g f(g dg K ϕ(hukφ π j(h det h it W f (huk N k \H W f (mukφ π j(m det m it dm dh du dk dπ dt The part of the integrand that depends upon u U is ϕ(huk W f (huk W f (muk du = ϕ(h W f (hk W f (mk ϕ(u ψ(huh ψ(mum du U U The latter integrand and integral visibly factor over primes. We need the following: 3.3 Lemma: Let v be a finite prime. For h, m H v such that Wπ eff v (h 0 and Wπ eff v (m 0, ϕ v (h ψ v (huh ψ v (mum du = du U v U v K v Proof: t a finite place v, ϕ v (u 0 if and only if u U v K v, and for such u ψ v (huh W πv (h = W eff π v (huh h = W eff π v (hu = W eff π v (h by the right U v K v -invariance, since f is a newform, in our present sense. Thus, for Wπ eff v (h 0, ψ v (huh =, and similarly for ψ v (mum. Thus, the finite-prime part of the integral over U v is just the integral of over U v K v, as indicated. /// Returning to the proof of the theorem, the archimedean part of the integral does not behave as the previous lemma indicates the finite-prime components do, because of its non-trivial deformation by ϕ. Thus, with subscripts denoting the infinite-adele part of various objects, for h, m H, as above, let K(h, m = ϕ (u ψ (huh ψ (mum du U

12 The whole integral is P(g f(g dg Z G k \G = K(h, mϕ(hw f (hkφ π j(h det h it W f (mkφ π j(m det m it dm dh dπ dk dt R Ξ o j N k \H K N k \H Normalize the volume of N k \N to. For a left N k -invariant function Φ on H, using the left N - equivariance of W by ψ, and the left N -invariance of ϕ, ϕ(nh Φ(nh W f (nhk dn = ϕ(h W f (h ψ(n Φ(nh dn = ϕ(h W f (hk W Φ (h N k \N N k \N where W Φ (h = ψ(n Φ(nh dn N k \N (The integral is not against ψ(n, but ψ(n. That is, the integral over N k \H is equal to an integral against (up to an alteration of the character the Whittaker function W Φ of Φ, which factors over primes for suitable Φ. Thus, the whole integral is = R Ξ o j N \H N \H Z G k \G P(g f(g dg K K(h, mw f (hkw Φπ j (h det h it W f (mkw Φπ j (m det m it dm dh dπ dk dt For fixed π, j, t, the integral over m, h, k is a product of two Euler products, since the Whittaker functions factor over primes, normalized by global constants ρ f and ρ Φπ j. The functions {Φ π,j : j} correspond to an orthonormal basis {ε π j} in the local archimedean part π of π, so, as noted earlier, by Schur s lemma the global constant ρ Φπ j is independent of j. For each π, let F π be the finite-prime spherical automorphic form corresponding to distinguished vectors at archimedean places. The Φ π j s are normalized spherical at all finite places. Thus, for each π and j, ϕ(hw f (hkw (h det h it Φπ j W f (mkw (m det m it Φπ j dm dh dk N \H N \H K N \H N \H K = ρ f ρ Fπ L( + it + z, π π L( it, π π K e π (hukε π j(h det h it ε π j(me π (muk det m it dm dh dk This gives the assertion of the theorem. /// 3.4 Remark: utomorphic forms not admitting Whittaker models do not enter this expansion. 4. Spectral expansion of Poincaré series The Poincaré series admits a spectral expansion facilitating its meromorphic continuation. The only cuspidal data appearing in this expansion is from GL, right K v -invariant everywhere locally. In the Poincaré series P, let ϕ be the archimedean data, and z, w the two complex parameters. For a spherical GL cuspform F, let ( ( Φ s,f θ = det s det D (r s F (D (where θ K 0 D

13 and define an Eisenstein series Diaconu-Garrett-Goldfeld: Moments for L functions for GL r GL r E r, s,f (g = γ P r, Φ s,f (γ g lso, for a Hecke character χ, with Φ s,s,s 3,χ( 0 m θ = det s m s χ(m m 3 s3 χ(m 3 (for θ K, GL r 0 0 m 3 define an Eisenstein series E r,, s,s,s 3,χ(g = Φ s,s,s 3,χ(γg γ P r,, 4. Theorem: With Eisenstein series as just above, the Poincaré series P has a spectral expansion P = ( N ϕ E r, z+ + F + χ χ(d 4πiκ ( W F, ρ F L( P GL (k ϕ rz+r +, π F Er, z+,f Re (s= ( ( P GL (k ϕ W E s,χ, rz+r L( + s, χ L( rz+r + s, χ rz+r ( Λ( s, χ d + s E r,, z+,s (r (z+, s (r (z+,χ where F runs over an orthonormal basis for everywhere-spherical cuspforms for GL, ρ F is the GL leading Fourier coefficient of F, χ runs over unramified grossencharacters, d is the differental ideal of k, κ is the residue of ζ k (s at s =, W F, and W Es,χ are the normalized archimedean Whittaker functions for GL, π F is the representation generated by F, L(s, χ is the usual grossencharacter L function, and Λ(s, χ is the grossencharacter L function with its gamma factor. 4. Remark: Notably, the spectral expansion of P contains nothing beyond the main term, the cuspidal GL part induced up to GL r, and the continuous GL part induced up to GL r. Proof: Rewrite the Poincaré series as summed in two stages, and apply Poisson summation, namely P(g = ϕ(γg = ϕ(βγg = ϕ γg (ψ Z k H β U k ψ (U k \U where Z k H k U Z k H k U ϕ g (ψ = ψ(u ϕ(ug du U (for g G The inner summand for ψ trivial gives the leading term in the spectral expansion of the Poincaré series. Specifically, it gives a vector from which a degenerate Eisenstein series for the (r, parabolic P r, = ZHU is formed by the outer sum. That is, g ϕ(ug du U is left equivariant by a character on P r,, and is left invariant by P r, k, namely, ϕ(upg du = ϕ(p p up g du = δ P r,(m ϕ(m u g du U U U = det d r z+ U ϕ(ug du (where p = (, m = 0 d ( 0, GL 0 d r ds 3

14 The normalization is explicated by setting g = : ϕ(u du = U ϕ U ϕ fin = U fin ϕ meas (U fin K fin = U U ϕ natural normalization is that this be, so the Eisenstein series includes the archimedean integral and finite-prime measure constant as factors: ( ϕ E r, z+ (g = ϕ(uγg du U U γ P r, The group H k is transitive on non-trivial characters of U k \U. For fixed non-trivial character ψ 0 on k\, let ψ ξ (u = ψ 0 (ξ u r,r (for ξ k The spectral expansion of P with its leading term removed is ϕ αγg (ψ ξ ξ k γ P r, α P r, k \H k where P r, is the corresponding parabolic subgroup of H GL r. Let U = { r } U = { r } Let γ P r,, Θ = { r U } GL Regrouping the sums, the expansion of the Poincaré series with its leading term removed is ψ ξ (u ϕ(u u γg du du ξ k U = γ P r, α P, \Θ k ξ k U ψ ξ (u U ϕ(u u αγg du du Letting ϕ(g = ϕ(u g du U the expansion becomes γ P r, α P, \Θ k ξ k U ψ ξ (u ϕ(u αγg du We claim the equivariance ϕ(pg = det z+ a z d (r z (r ϕ(g (for p = 4 a G, with GL r d

15 This is verified by changing variables in the defining integral: let x r and compute r x b c a = b c + xd a = b c a r d d d xd Thus, det z a z d (r z comes out of the definition of ϕ, and another det d r from the changeof-measure in the change of variables replacing x by x/d in the integral defining ϕ from ϕ. Note that Thus, letting we can write ( a z d (r z (r a = det Φ(g = P(g α P, k \Θ k γ P r, ( ξ k d U U ϕ(uγg du = (r (z+ a/d rz+(r ψ ξ (u ϕ(u αg du Φ(γg γ P r, The right-hand side of the latter equality is not an Eisenstein series for P r, in the strictest sense. Define a GL kernel ϕ ( for a Poincaré series as follows. s in the general case, we require right invariance by the maximal compact subgroups locally everywhere, and left equivariance ( ϕ ( a ( D = a/d β ϕ ( (D d The remaining ambiguity is the archimedean data ϕ (, completely specified by giving its values on the archimedean part of the standard unipotent radical, namely, ϕ ( ( x = ϕ r x ( ϕ as above Let U, be the unipotent radical of the standard parabolic P, in GL. Express ϕ ( in its Fourier expansion along U,, and remove the constant term: let ϕ (β, D = ϕ ( (β, D ϕ ( (β, ud du = U, ξ k The corresponding GL Poincaré series with leading term removed is Q(β, D = ϕ α P, k \GL(k (β, αd U, ψ ξ (u ϕ ( (β, ud du Thus, for g = ( D (with GL r ( and D GL ( the inner integral g U ψ(u ϕ(u g du 5

16 is expressible in terms of the kernel ϕ for Q, namely, ψ ξ (u ϕ(u g du ξ k U = det z+ det D (r (z+ ϕ ( rz+r, D Thus, α P, k \Θ k ξ k U ψ ξ (u ϕ(u αg du = det z+ det D (r (z+ Q ( rz+r, D Thus, letting ( Φ D = det z+ z+ (r det D Q( rz+r, D (with GL r and D GL we have P(g = ( U ϕ E r, z+ (g + Φ(γg γ P r, To obtain a spectral decomposition of the Poincaré series P for GL r, we first recall from [Diaconu- Garrett 009] the spectral decomposition of Q for r =, and then form P r, Eisenstein series from the spectral fragments. s in [Diaconu-Garrett 009], a direct computation shows that the spectral expansion of the GL Poincaré series with constant term removed is Q(β, D = ( ϕ W F, ρ F L(β +, π F F F P GL (k + χ χ(d 4πiκ Re (s= ( P GL (k ϕ W E s,χ, L(β + s, χ L(β + s, χ L( s, χ d (β+s / E s,χ (D ds where F runs over an orthonormal basis of everywhere-spherical cuspforms, ρ F is the general GL analogue of the leading Fourier coefficient, π F is the cuspidal automorphic representation generated by F, W F, and W Es,χ, are the normalized spherical vectors in the corresponding archimedean Whittaker models, Λ(s, χ is the standard L function completed by adding the archimedean factors, and d is the differental idele. Thus, the individual spectral components of Φ are of the form ( ( Φ z+,ψ θ = (constant det z+ z+ (r det D Ψ(D (where θ K 0 D where Ψ is either a spherical GL cuspform or a spherical GL Eisenstein series, in either case with trivial central character. For Ψ a spherical GL cuspform F averaging over P r, k \G k produces a half-degenerate Eisenstein series E r, (g = z+,f,f (γ g Φ z+ γ P r, s in the appendix, the half-degenerate Eisenstein series E r, s,f has no poles in Re(s /. With s = (z + / this assures absence of poles in Re(z 0. The continuous spectrum part of Q produces degenerate Eisenstein series on G, as follows. With Ψ = E s,χ the usual spherical, trivial central character, Eisenstein series for GL, define an Eisenstein series E r, z+,es,χ (g = Φ z+,es,χ(γg γ P r, 6

17 s usual, for Re(s 0 and Re(z 0, this iterated formation of Eisenstein series is equal to a single-step Eisenstein series. That is, let Φ s,s,s 3,χ( 0 m θ = det s m s χ(m m 3 s3 χ(m 3 (for θ K, GL r 0 0 m 3 and E r,, s,s,s 3,χ(g = Φ s,s,s 3,χ(γg γ P r,, Taking s = z+, s = s (r (z+, and s 3 = s (r (z+, E r, z+,es,χ = E r,, z+,s (r (z+, s (r (z+,χ dding up these spectral components yields the spectral expansion of the Poincaré series. /// 4.3 Remark: Suitable archimedean data to give the Poincaré series a functional equation is best described in the context of the spectral expansion, and, due to the form of the spectral expansion, essentially reduces to GL. It is useful to describe the data via a differential equation, since this explains the outcome of the computation more transparently. Since each archimedean place affords its own opportunity for data choices, we simplify this aspect of the situation by taking groundfield k = Q. First, for G = GL (Q, let be the usual invariant Laplacian on the upper half-plane H, and consider the partial differential equations ( s(s ν u β s,ν = the distribution f 0 y β f ( y 0 dy 0 y ( ν Z and s, β C on H. Further, require that u β s,ν have the same equivariance as the target distribution, namely, u β s,ν(t z = t β u β s,ν(z (for t > 0 and z H Then u β s,ν(x + iy = y β ϕ β s,ν(x/y for a function ϕ β s,ν on R satisfying the corresponding differential equation (( + x + x( β x x + ( β(β s(s ν f = δ (with Dirac δ at 0 The generalized function δ is in the L Sobolev space on R with index ε for every ε > 0. By elliptic regularity, solutions f to this differential equation are in the local Sobolev space with index ν ε, and by Sobolev s lemma are locally at least C ν ε C ν. That is, by increasing ν solutions are made as differentiable as desired, and their Fourier transforms will have corresponding decay, giving convergence of the Poincaré series (for suitable s, β, as in [Diaconu-Garrett 009]. The spectral expansion of the GL Poincaré series P β s,ν formed with this archimedean data ϕ β s,ν is special case of the computation in [Diaconu-Garrett 009], recalled above, but in fact gives a much simpler outcome. For example, the cuspidal components are directly computed by unwinding, integrating by parts, and applying the characterization of ϕ β s,ν by the differential equation: P β s,ν, F = ρ F ( Λ(β +, F ( sf (s F s(s ν (where F = s F (s F where Λ(, F is the L-function completed with its gamma factors. Thus, P β s,ν = F ρ F ( Λ(β +, F F ( sf (s F s(s ν + (non-cuspidal 7

18 summing over an orthonormal basis of cuspforms F. Granting convergence for ν sufficiently large and Re(s, Re(β large, the cuspidal part has a meromorphic continuation in s with poles at the values s F, as expected. Visibly, the cuspidal part of P β s,ν is invariant under s s, and in these coordinates the map β β maps F to F (whether or not F is self-contragredient. The leading term of the spectral expansion of P β s,ν, via Poisson summation, is a constant multiple Cs,ν β E β+ of the spherical Eisenstein series E β+. This happens regardless of the precise choice of archimedean data, simply due to the homogeneity we have required of the archimedean data throughout. Similarly, the continuous part of this Poincaré series on GL is 4πi Re (s e= ξ(β + s e ξ(β + s e E se ξ(s e ((s e (s e s(s ν ds e where ξ is the ζ-function completed with its gamma factor. In analogy with the cuspidal discussion, the product ξ(β + s e ξ(β + s e is invariant under β β, since ξ( z = ξ(z. The visual symmetry in s s is slightly deceiving, since the meromorphic continuation (in s through the critical line Re(s e = (over which the Eisenstein series is integrated introduces extra terms from residues at s e = s and s e = s. Indeed, parts of these extra terms cancel a pole in the leading term C β s,ν E β+ at β = 0. Despite this subtlety in the continuous spectrum, the special choice of archimedean data makes meromorphic continuation in s, β visible. In summary, for GL (Q, the special choice of archimedean data makes the cuspidal part of the Poincaré series have a visible meromorphic continuation, and satisfy obvious functional equations. The continuous part of the Poincaré series satisfies functional equations modulo explicit leftover terms. The ideal choice of archimedean data ϕ for the Poincaré series for GL r (Q is such that the averaged version of it, denoted ϕ in the proof above, restricts to the function ϕ β s,ν for GL just discussed: we want R r ϕ r 0 u 0 x du = ϕ r x = ϕ β s,ν(x (for x R It is not obvious that, given a reasonable (even function f on R, there is a rotationally symmetric function u on R r such that R r u(y + xe r dy = f(x (e i the standard basis for R r with R r sitting in the first r coordinates in R r. Fourier inversion clarifies this, as follows. Supposing the integral identity just above holds, integrate further in the (r th coordinate x, against e πiξx, to obtain û(ξe r = f(ξ (for ξ R where the Fourier transform on the left-hand side is on R r, on the right-hand side is on R. For u rotationally invariant, û is also rotationally invariant, and the latter equality can be rewritten as By Fourier inversion, u(x = û(ξ = f( ξ (for ξ R r R r e πi ξ,x f( ξ dξ (for x R r That is, given an even function f on R, the latter formula yields a rotationally invariant function on R r, whose averages along R r are the given f. This proves existence of an essentially unique ϕ yielding the prescribed ϕ β s,ν. Then the functional equation of the most-cuspidal part of the special-data Poincaré series on GL r is inherited from the functional equation of the cuspidal part of the special-data Poincaré series on GL. 8

19 5. ppendix: half-degenerate Eisenstein series Take q >, and let f be a cuspform on GL q (, in the strong sense that f is in L (GL q (k\gl q (, and f meets the Gelfand-Fomin-Graev conditions N k \N f(ng dn = 0 (for almost all g and f generates an irreducible representation of GL q (k v locally at all places v of k. For a Schwartz function Φ on q r and Hecke character χ, let ϕ(g = ϕ χ,f,φ (g = χ(det g q f(h χ(det h r Φ(h [0 q (r q q ] g dh GL q( This function ϕ has the same central character as f. It is left invariant by the adele points of the unipotent radical ( r q N = { } (unipotent radical of P = P r q,q r The function ϕ is left invariant under the k-rational points M k of the standard Levi component of P, ( a M = { : a GL d r q, d GL r } To understand the normalization, observe that ξ(χ r, f, Φ(0, = ϕ( = GL q( f(h χ(det h r Φ(h [0 q (r q q ] dh is a zeta integral as in [Godement-Jacquet 97] for the standard L function attached to the cuspform f. Thus, the Eisenstein series formed from ϕ includes this zeta integral as a factor, so write ξ(χ r, f, Φ(0, Eχ,f,Φ(g P = ϕ(γ g (convergent for Re(χ γ P k \GL r(k The meromorphic continuation follows by Poisson summation: = χ(det g q γ P k \GL r(k = χ(det g q GL q(k\gl q( ξ(χ r, f, Φ(0, E P χ,f,φ(g GL q(k\gl q( f(h χ(det h r f(h χ(det h r α GL q(k y k q r, full rank Φ(h [0 α] g dh Φ(h y g dh The Gelfand-Fomin-Graev condition on f fits the full-rank constraint. nticipating that we can drop the rank condition suggests that we define Θ Φ (h, g = Φ(h y g y k q r s in [Godement-Jacquet 97], the non-full-rank terms integrate to 0: 5. Proposition: For f a cuspform, less-than-full-rank terms integrate to 0, that is, f(h χ(det h r Φ(h y g dh = 0 GL q(k\gl q( y k q r, rank <q 9

20 Proof: Since this is asserted for arbitrary Schwartz functions Φ, we can take g =. By linear algebra, given y 0 k q r of rank l, there is α GL q (k such that α y 0 = ( yl r 0 (q l r (with l-by-r block y l r of rank l Thus, without loss of generality fix y 0 of the latter shape. Let Y be the orbit of y 0 under left multiplication by the rational points of the parabolic ( P l,q l l-by-l = { } GL 0 (q l-by-(q l q This is some set of matrices of the same shape as y 0. Then the subsum over GL q (k y 0 is GL q(k\gl q( f(h χ(det h r Φ(h y dh = f(h χ(det h r Φ(h y dh y GL P l,q l q(k y 0 k \GL q( y Y Let N and M be the unipotent radical and standard Levi component of P l,q l, ( ( l l-by-l 0 N = M = 0 q l 0 (q l-by-(q l Then the integral can be rewritten as an iterated integral = = f(h χ(det h r N k M k \GL q( y Y N M k \GL q( y Y N M k \GL q( y Y Φ(h y dh f(nh χ(det nh r Φ((nh y dn dh N k \N ( χ(det h r Φ(h y f(nh dn dh N k \N since all fragments but f(nh in the integrand are left invariant by N. The inner integral of f(nh is 0, by the Gelfand-Fomin-Graev condition, so the whole is 0. /// Let ι denote the transpose-inverse involution. Poisson summation gives = det(h ι r det g ι q Θ Φ (h, g = y k q r Φ(h y g y k q r Φ((h ι y g ι = det(h ι r det g ι q Θ Φ(h ι, g ι s with Θ Φ, the lower-rank summands in Θ Φ integrate to 0 against cuspforms. Thus, letting we have = χ(det g q GL + q = {h GL q ( : det h } GL q = {h GL q ( : det h } ξ(χ r, f, Φ(0, E P χ,f,φ(g = χ(det g q GL q(k\gl + q GL q(k\gl q( f(h χ(det h r Θ Φ (h, g dh + χ(det g q = χ(det g q GL q(k\gl + q 0 f(h χ(det h r Θ Φ (h, g dh GL q(k\gl q f(h χ(det h r Θ Φ (h, g dh f(h χ(det h r Θ Φ (h, g dh

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