aa r X i v : . [ m a t h . N T ] S e p SELF-ADJOINT BOUNDARY-VALUE PROBLEMS OFAUTOMORPHIC FORMS
ADIL ALI
Abstract.
We apply some ideas of Bombieri and Garrett to construct natu-ral self-adjoint operators on spaces of automorphic forms whose only possiblediscrete spectrum is λ s for s in a subset of on-line zeros of an L -function, ap-pearing as a compact period of cuspidal-data Eisenstein series on GL . Theseideas have their origins in results of Hejhal and Colin de Verdi´ere. In parallelwith the GL (2) case, the corresponding pair-correlation and triple-correlationresults limit the fraction of on-the-line zeros that can appear in this fashion. Contents
1. Introduction 12. Spectral Theory 33. Global Automorphic Sobolev Spaces 264. Pre-trace formula estimates on compact periods 265. Casimir Eigenvalue 306. Friedrichs self-adjoint extensions and complex conjugation maps 327. Moment bounds assumptions 348. Local automorphic Sobolev spaces 369. Main Theorem: Characterization and Sparsity of discrete spectrum 3810. L-function background 5111. Appendix I: Harmonic Analysis on GL Introduction
We apply the spectral theory of automorphic forms to the study of zeros of L -functions. A refined version of the spectral theory of automorphic forms plausiblyhas bearing on zeros of automorphic L -functions and other periods. This is power-fully illustrated by the following example, which is a much simpler analogue of ourpresent result. In 1977, H. Haas [Haas 1977] attempted to numerically computeeigenvalues λ of the invariant Laplacian∆ = y ( ∂ ∂x + ∂ ∂y )on SL ( Z ) \ H , parametrized as λ w = w ( w − w -values, intendingto solve the differential equation (∆ − λ w ) u = 0 Key words and phrases.
Automorphic Forms, Representation Theory.
H. Stark and D. Hejhal [Hejhal 1981] observed zeros of ζ and of an L -functionon the list. This suggested an approach to the Riemann Hypothesis, hoping thatzeros w of ζ would be in bijection with eigenvalues λ w = w ( w −
1) of ∆. Sincea suitable version of ∆ is a self-adjoint, non-positive operator, these eigenvalueswould necessarily be non-positive also, forcing either Re( w ) = or w ∈ [0 , − λ w ) u = δ afc ω allowing a multiple of an automorphic Dirac δ afc ω on the right hand side. Here ω isa cube root of unity, and δ afc ω ( f ) = f ( ω ) for an SL ( Z )-automorphic waveform f .However, since solutions u w of (∆ − λ ) u = δ afc ω are not genuine eigenfunctions ofthe Laplacian, this no longer implied non-positivity of the eigenvalues.The natural question was whether the Laplacian could be modified so as toexhibit a fundamental solution as a legitimate eigenfunction for the perturbed op-erator. That is, one would want a variant ∆ ′ for which(∆ ′ − λ w ) u w = 0 ⇐⇒ (∆ − λ w ) u w = C · δ afc ω Because of Y. Colin de Verdi`ere’s argument for meromorphic continuation of Eisen-stein series [CdV 1981], it was anticipated that ∆ ′ = ∆ Fr would be a fruitful choicefor the Friedrichs extension of a suitably chosen restriction. ∆ Fr is self-adjoint, andtherefore symmetric. This gave glimpses of progress toward the Riemann hypoth-esis.Friedrichs extensions have the desired properties and they played an essential rolein another story, namely Colin de Verdi`ere’s meromorphic continuation of Eisen-stein series, though there, the distribution that appeared was the evaluation ofconstant term at height y = a . There, the spaces of interest were the orthogonalcomplements L (Γ \ H ) a to the spaces of pseudo-Eisenstein series with test functiondata supported on [ a, ∞ ). ∆ a was ∆ with domain C ∞ c (Γ \ H ) and constant termvanishing above height y = a . ∆ Fr was the Friedrichs extension of ∆ a to a self-adjoint operator on L (Γ \ H ) a . In this way, a Friedrichs extension attached to thedistribution on Γ \ H given by T a ( f ) = ( c P f )( ia )has all eigenfunctions inside a +1-index global automorphic Sobolev space, definedas the completion of C ∞ c (Γ \ H ) with respect to the +1-Sobolev norm | f | H = h (1 − ∆) f, f i The Dirac δ on a two-dimensional manifold lies in a global Sobolev space H − − ǫ with index − − ǫ for all ǫ >
0, but not in H − , so by elliptic regularity, a funda-mental solution lies in the +1 − ǫ -Sobolev space. This implies that a fundamentalsolution could not be an eigenfunction for any Friedrichs extension of a restrictionof ∆ described by boundary conditions.The automorphic Dirac δ afc ω is an example of a period functional . Periods ofautomorphic forms have been studied extensively: after all, Mellin transforms ofcuspforms are noncompact periods. Hecke and Maass were aware of Eisensteinseries periods: in effect, Hecke treated finite sums over Heegner points attached to negative fundamental discriminants, and Maass treated compact geodesic periods ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 3 attached to positive fundamental discriminants. A simple example is given by E s ( i ) = ζ Q ( i ) ( s ) ζ Q (2 s )More generally, let ℓ a quadratic field extension of a global field k of characteristicnot 2. Let G = GL ( k ), and let H be a copy of ℓ × inside G . The period of anEisenstein series E s = P γ ∈ P k \ G k ϕ ( γg ) along H is defined by the compactly-supported integral period of E s along H = Z Z A H k \ H A E s Via Iwasawa-Tate integrals, Z Z A H k \ H A E s = ξ ℓ ( s ) ξ k (2 s )Noncompact periods have been studied extensively. Let G be a reductive groupover a number field F , and let H ⊂ G be a subgroup obtained as the fixed pointset of an involution θ . [Jacquet-Lapid-Rogowski 1997] studied the period integralΠ H ( ϕ ) = Z H ( F ) \ H ( A ) ϕ ( h ) dh The authors use a regularization procedure and a relative trace formula to obtainan Euler product for Π( E ), where E is an Eisenstein series.This paper examines the discrete spectrum of a Friedrichs extension e ∆ θ asso-ciated to a compactly-supported GL ( Z )-invariant distribution ˜ θ on G = GL (4),whose projection θ to the subspace of L ( GL ( Z ) \ GL ( R ) /O ( R )) spanned by 2 , f and f and the residue of thisEisenstein series, a Speh form. This distribution lies in the − w of the discrete spectrum λ w = w ( w − e ∆ θ interlace with the zeros of the constant term of the 2 , E Pf,f,s where f is a GL (2) cuspform. Such spacing is too regular to be compatiblewith the corresponding pair-correlation and triple-correlation conjectures, and thispowerfully constrains the number of zeros w of θE − w appearing in the discretespectrum of e ∆ θ . In particular, the discrete spectrum is presumably sparse.2. Spectral Theory
We follow [Langlands 1976], [MW 1990], [MW 1989], and [Garrett 2012]. Fix,once and for all, K ∞ = O ( R ), and K v = GL ( Z v ) for non-archimedean places v .Let z be the center of the enveloping algebra of G ∞ = GL ( R ). Definition 1.
Given a parabolic P in G = GL and a function f on Z A G k \ G A ,the constant term of f along P is c P f ( g ) = Z N k \ N A f ( ng ) dn where N is the unipotent radical of P . We will let k = Q throughout. An automorphic form is a cuspform if, for allparabolics P , the constant term along P is zero. This is the Gelfand condition(in the weak sense). Since the right G A -action commutes with taking constant ADIL ALI terms, the space of functions L ( Z A G k \ G A ) satisfying the Gelfand condition is G A -stable, and so is a sub-representation of L ( Z A G k \ G A ). We note that there arenon- K v -finite vectors in L ( Z A G k \ G A ). R. Godement, A. Selberg, I. Gelfand andI. I. Piatetski-Shapiro showed that integral operators attached to test functionson L ( Z A G k \ G A ) are compact. Specifically, for ϕ ∈ C ∞ c ( G A ) which is right K -invariant, the operator f → ϕ · f gives a compact operator from L ( Z A G k \ G A ) to itself. Here( ϕ · f )( y ) = Z Z A G k \ G A ϕ ( x ) · f ( yx ) dx By the spectral theorem for compact operators, this sub-representation decomposesinto a direct sum of irreducibles, each with finite multiplicity. The remainder of L is decomposed as follows.We classify non-cuspidal automorphic forms according to their cuspidal support,i.e. the smallest parabolics on which they have non-zero constant term. In GL (4)there are four associate classes of proper parabolic subgroups. There is P = GL , P , , , P , , , P , , , the maximal proper parabolic subgroups P , , P , and P , ,and the standard minimal parabolic subgroup P , , , . Definition 2.
A pseudo-Eisenstein series is a function of the form Ψ ϕ ( g ) = X γ ∈ P k \ G k ϕ ( γ · g ) where ϕ is a continuous function on Z A N A M k \ G A with cuspidal data on the Levicomponent. For example, given the 2 , ϕ φ,f ⊗ f ( (cid:18) A ∗ D (cid:19) ) = φ ( (cid:12)(cid:12)(cid:12)(cid:12) det A det D (cid:12)(cid:12)(cid:12)(cid:12) ) · f ( A ) · f ( D )where φ is a compactly-supported, smooth function on R and f and f are cusp-forms on GL with trivial central character. For the 3 , ϕ φ,f ⊗ f ( (cid:18) A ∗ d (cid:19) ) = φ ( (cid:12)(cid:12)(cid:12)(cid:12) det Ad (cid:12)(cid:12)(cid:12)(cid:12) ) · f ( A )where A ∈ GL and f is a cuspform on GL . For the 2 , , ϕ f,φ ,φ ( A b
00 0 c ) = f ( A ) · φ ( det Ab ) · φ ( det Ac )The 1 , , , Proposition 1.
In the following, abbreviate ϕ φ,f ⊗ f by ϕ . For any square-integrableautomorphic form f and any pseudo-Eisenstein series Ψ Pϕ , with P a parabolic sub-group h f, Ψ Pϕ i Z A G k \ G A = h c P f, ϕ i Z A N P A M Pk \ G A ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 5
Proof.
The proof involves a standard unwinding argument. Let N P and M P denotethe unipotent radical and Levi component of P , respectively. Observe that h f, Ψ Pϕ i Z A G k \ G A = Z Z A G k \ G A f ( g ) · Ψ Pϕ ( g ) dg = Z Z A G k \ G A f ( g )( X γ ∈ P k \ G k ϕ ( γ · g )) dg This is = Z Z A P k \ G A f ( g ) ϕ ( g ) dg = Z Z A N k M k \ G A f ( g ) ϕ ( g ) dg = Z Z A N A M k \ G A Z N k \ N A f ( ng ) ϕ ( ng ) dn dg = Z Z A N A M k \ G A ( Z N k \ N A f ( ng ) dn ) ϕ ( g ) dg = h c P f, ϕ i Z A N P A M Pk \ G A (cid:3) From this adjointness relation, we have the following
Corollary 1.
A square-integrable automorphic form is a cuspform if and only if itis orthogonal to all pseudo-Eisenstein series.
Since the critical issues arise at the archimedean place, we consider the real Liegroup. To this end, let G = P GL ( R ), Γ = P GL ( Z ). Definition 3.
The standard minimal parabolic B is defined as the subgroup B = P , , , of upper-triangular matrices, with standard Levi component A , unipotent radical N ,and Weyl group W , the latter represented by permutation matrices. Let A + be the image in G of positive diagonal matrices. Consider characters on B of the form χ = χ s : ( a ∗ ∗ ∗ a ∗ ∗ a ∗ a ) = | a | s · | a | s · | a | s · | a | s For the character to descend to
P GL n , necessarily s + s + s + s = 0. Definition 4.
The standard spherical vector is ϕ sph s ( pk ) = χ s ( p ) and the spherical Eisenstein series is E s ( g ) = X γ ∈ B ∩ Γ \ Γ ϕ sph s ( γ · g )The spherical Eisenstein series is convergent for Re( s ) ≫ s as in [Langlands 544, Appendix 1]. The function f → c B f ( g ) is left N ( B ∩ Γ)-invariant.
ADIL ALI
Recall that for ϕ ∈ C ∞ c ( N ( B ∩ Γ) \ G ) K ≈ C ∞ c ( A + ), letting h , i X be the pairingof distributions and test functions on a space X , the pseudo-Eisenstein series Ψ ϕ ( g )enters the adjunction relation h c B f, ϕ i N ( B ∩ Γ) \ G = h f, Ψ ϕ i Γ \ G That is, ϕ → Ψ ϕ is adjoint to f → c B f . Then c B f = 0 is equivalent to h f, Ψ ϕ i Γ \ G = 0for all ϕ . Proposition 2.
The pseudo-Eisenstein series Ψ ϕ admits a W -symmetric expan-sion as an integral of Eisenstein series. That is, Ψ ϕ = 1 | W | πi ) dim a Z ρ + i a ∗ E s · h Ψ ϕ , E ρ − s i Γ \ G ds Proof.
To decompose the pseudo-Eisenstein series Ψ ϕ as an integral of minimal-parabolic Eisenstein series, begin with Fourier transform on the Lie algebra a ≈ R n − of A + . Let h , i : a ∗ × a → R be the R -bilinear pairing of a with its R -lineardual a ∗ . For f ∈ C ∞ c ( a ), the Fourier transform is b f ( ξ ) = Z a e − i h x,ξ i f ( x ) dx Fourier inversion is f ( x ) = 1(2 π ) dim a Z a ∗ e i h x,ξ i b f ( ξ ) dξ Let exp : a → A + be the Lie algebra exponential, and log : A + → a the inverse.Given ϕ ∈ C ∞ c ( A + ), let f = ϕ ◦ exp be the corresponding function in C ∞ c ( a ). The(multiple) Mellin transform M ϕ of ϕ is the Fourier transform of f : M ϕ ( iξ ) = b f ( ξ )Mellin inversion is Fourier inversion in these coordinates: ϕ (exp x ) = f ( x ) = 1(2 π ) dim a Z a ∗ e i h ξ,x i b f ( ξ ) dξ = 1(2 π ) dim a Z a ∗ e i h ξ,x i M ϕ ( iξ ) dξ Extend the pairing h , i on a ∗ × a to a C -bilinear pairing on the complexification.Use the convention (exp) iξ = e i h ξ,x i = e h iξ,x i With a = exp x ∈ A + , Mellin inversion is ϕ ( a ) = 1(2 π ) dim a Z a ∗ a iξ M ϕ ( iξ ) dξ = 1(2 πi ) dim a Z i a ∗ a s M ϕ ( s ) ds With this notation, the Mellin transform itself is M ϕ ( s ) = Z A + a − s ϕ ( a ) da Since ϕ is a test function, its Fourier-Mellin transform is entire on a ∗ ⊗ R C . Thus,for any σ ∈ a ∗ , Mellin inversion can be written ϕ ( a ) = 1(2 πi ) dim a Z σ + i a ∗ a s M ϕ ( s ) ds ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 7
Identifying N ( B ∩ Γ) \ G/K ≈ A + , let g → a ( g ) be the function that picks out the A + component in an Iwasawa decomposition G = N A + K . For σ ∈ a + suitable forconvergence, the following rearrangement is legitimate,Ψ ϕ ( g ) = X γ ∈ ( B ∩ Γ) \ Γ ϕ ( a ( γ ◦ g )) = X γ ∈ B ∩ Γ \ Γ πi ) dim a Z σ + i a ∗ a ( γg ) s M ϕ ( s ) ds = 1(2 πi ) dim a Z σ + i a ∗ (cid:0) X γ ∈ B ∩ Γ \ Γ a ( γg ) s (cid:1) M ϕ ( s ) ds = 1(2 πi ) dim a Z σ + i a ∗ E s ( g ) M ϕ ( s ) ds This does express the pseudo-Eisenstein series as a superposition of Eisenstein se-ries, as desired. However, the coefficients M ϕ are not expressed in terms of Ψ ϕ itself. This is rectified as follows. Letting ρ denote the half-sum of positive roots, h f, E s i Γ \ G = Z Γ \ G f ( g ) E s ( g ) = Z B ∩ Γ \ G f ( g ) a ( g ) s dg = Z N ( B ∩ Γ) \ G Z N ∩ Γ \ N f ( ng ) a ( ng ) s dg = Z N ( B ∩ Γ) \ G c B f ( g ) a ( g ) s dg = Z A + c B f ( a ) a s daa ρ = Z A + c B f ( a ) a − (2 ρ − s ) da = M c B f (2 ρ − s )That is, with f = Ψ ϕ , h Ψ ϕ , E s i Γ \ G = M c B Ψ ϕ (2 ρ − s )On the other hand, a similar unwinding of the pseudo-Eisenstein series, and therecollection of the constant term c B E s , gives h Ψ ϕ , E s i Γ \ G = Z B ∩ Γ \ G ϕ ( g ) E s ( g ) dg = Z N ( B ∩ Γ) \ G Z N ∩ Γ \ N ϕ ( ng ) E s ( ng ) dg = Z N ( B ∩ Γ) \ G ϕ ( g ) c B E s ( g ) dg = Z A + ϕ ( a ) c B E s ( a ) daa ρ = Z A + ϕ ( a ) X w c w ( s ) a w · s daa ρ = X w c w ( s ) Z A + ϕ ( a ) a − (2 ρ − w · s ) da = X w c w ( s ) M ϕ (2 ρ − w · s )Combining these, M c B Ψ ϕ (2 ρ − s ) = h Ψ ϕ , E s i Γ \ G = X w c w ( s ) M ϕ (2 ρ − w · s )Replacing s by 2 ρ − s , noting that 2 ρ − w · (2 ρ − s ) = w · s , M c B Ψ ϕ ( s ) = X w c w (2 ρ − s ) M ϕ ( w · s )To convert the expressionΨ ϕ ( g ) = 1(2 πi ) dim a Z σ + i a ∗ E s ( g ) M ϕ ( s ) ds ADIL ALI into a W -symmetric expression, to obtain an expression in terms of c B Ψ ϕ , we mustuse the functional equations of E s . However, σ + i a ∗ is W -stable only for σ = ρ .Thus, the integral over σ + i a ∗ must be viewed as an iterated contour integral, andmoved to ρ + i a ∗ .Ψ ϕ = 1 | W | X w πi ) dim a ∗ Z ρ + i a ∗ E w · s M ϕ ( w · s ) ds = 1 | W | πi ) dim a Z ρ + i a ∗ E s (cid:0) X w c w ( s ) M ϕ ( w · s ) (cid:1) ds On ρ + i a ∗ , we have c w ( s ) = c w (2 ρ − s ). Therefore, X w c w ( s ) M ϕ ( w · s ) = X w c w (2 ρ − s ) M ϕ ( w · s ) = M c B Ψ ϕ ( s )This gives the desired spectral decomposition,Ψ ϕ = 1 | W | πi ) dim a Z ρ + i a ∗ E s · M Ψ ϕ ( s ) ds = 1 | W | πi ) dim a Z ρ + i a ∗ E s · h Ψ ϕ , E ρ − s i Γ \ G ds (cid:3) Proposition 3.
The map f → ( s → h f, E s i ) is an inner-product-preserving mapfrom the Hilbert-space span of the pseudo-Eisenstein series to its image in L ( ρ + i a ) .Proof. Let f ∈ C ∞ c (Γ \ G ), ϕ ∈ C ∞ c ( N \ G ), and assume Ψ ϕ is orthogonal to residuesof E s above ρ . Using the expression for Ψ ϕ in terms of Eisenstein series, h Ψ ϕ , f i = h | W | πi ) dim a Z ρ + i a ∗ h Ψ ϕ , E ρ − s i · E s ds, f i = 1 | W | πi ) dim a Z ρ + i a ∗ h Ψ ϕ , E ρ − s i · h E s , f i ds (cid:3) The map Ψ ϕ → h Ψ ϕ , E ρ − s i with s = ρ + it and t ∈ a ∗ , produces functions u ( t ) = h Ψ ϕ , E ρ − it i satisfying u ( wt ) = h Ψ ϕ , E ρ − w · s i = h Ψ ϕ , E w · (2 ρ − s ) i = h Ψ ϕ , E ρ − s c w (2 ρ − s ) i = c w ( s ) · u ( t ) for all w ∈ W since c w (2 ρ − s ) = c w ( s ) = 1 c w ( s )on ρ + i a ∗ . Proposition 4.
Any u ∈ L ( ρ + i a ∗ ) satisfying u ( wt ) = c w ( s ) · u ( t ) for all w ∈ W is in the image. ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 9
Proof.
First, for compactly-supported u meeting this condition, we claimΨ u = 1 | W | πi ) dim a Z ρ + i a ∗ u ( t ) · E ρ + it dt = 0It suffices to show c B Ψ u is not 0. With s = ρ + it , the relation implies u ( t ) E ρ − s is invariant by W . Let C = { t ∈ a ∗ : h t, α i > α > } be the positive Weyl chamber in a ∗ , where h , i is the Killing form transported to a ∗ by duality. ThenΨ u = 1 | W | πi ) dim a Z ρ + i a ∗ u ( t ) · E s dt = 1(2 πi ) dim a Z ρ + iC u ( t ) · E s dt Since u ( tw ) = u ( t ) · c w ( ρ + it ), the constant term of Ψ u is c B Ψ u = 1(2 πi ) dim a Z ρ + i a ∗ u ( t ) · a s dt This Fourier transform does not vanish for non-vanishing u . (cid:3) Given G = GL ( R ), Γ = GL ( Z ), and K = O ( R ), it is necessary to invokethe complete spectral decomposition of L (Γ \ G/K ), that cuspforms and cuspidaldata Eisenstein series attached to non-minimal parabolic Eisenstein series attachedto non-minimal parabolics, and their L residues, as well as the minimal-parabolicpseudo-Eisenstein series, span L (Γ \ G/K ). And we must demonstrate the or-thogonality of integrals of minimal-parabolic Eisenstein series to all other spectralcomponents.We now decompose the pseudo-Eisenstein series with cuspidal data. We carrythis out for the 3 , , , , P = P , . We decompose P , and P , pseudo-Eisenstein series with cuspidal support. The data for a P pseudo-Eisenstein seriesis smooth, compactly-supported, and left Z A M Pk N P A -invariant. For now, we assumethat the data is spherical, i.e. right K -invariant. This means that the function isdetermined by its behavior on Z A M Pk \ M P A . In contrast to the minimal paraboliccase, this is not a product of copies of GL , so we can not simply use the GL spectral theory (Mellin inversion) to accomplish the decomposition. Instead, thisquotient is isomorphic to GL ( k ) \ GL ( A ), so we will use the spectral theory for GL . If η is the data for a P , pseudo-Eisenstein series Ψ η , we can write η as atensor product η = f ⊗ µ on Z GL ( A ) GL ( k ) \ GL ( A ) · Z GL ( k ) \ Z GL ( A ) Saying that the data is cuspidal means that f is a cusp form. Similarly, the data ϕ = ϕ F,s for a P , -Eisenstein series is the tensor product of a GL cusp form F and a character χ s = | . | s on GL . We show that Ψ f,η is the superposition ofEisenstein series E F,s where F ranges over an orthonormal basis of cusp forms and s is on the critical line. Proposition 5.
The pseudo-Eisenstein series Ψ f,η admits a spectral decomposition Ψ f,η = X F Z s h Ψ f,η , E F,s i · E F,s ds where the sum is over spherical cuspforms F on GL ( k ) \ GL ( A ) .Proof. Using the spectral expansions of f and η , η = f ⊗ η = (cid:0) X cfms F h f, F i (cid:1) · (cid:0) Z s h µ, χ s i · χ s ds (cid:1) = X cfms F Z s h η f,µ , ϕ F,s i · ϕ F,s ds So the pseudo-Eisenstein series can be re-expressed as a superposition of Eisensteinseries Ψ f,η ( g ) = X γ ∈ P k \ G k η f,µ ( γg )= X γ ∈ P k \ G k X cfms F Z s h η f,µ , ϕ F,s i · ϕ F,s ( γg ) ds = X cfms F Z s h η f,µ , ϕ F,s i X γ ∈ P k \ G k ϕ F,s ( γg ) ds = X cfms F Z s h η f,µ , ϕ F,s i · E F,s ds The coefficient h η, ϕ i GL is the same as the pairing h Ψ η , E ϕ i GL , since h Ψ η , E ϕ i = h c P (Ψ η ) , ϕ i = h η, ϕ i So the spectral decomposition isΨ f,η = X cfms F Z s h Ψ f,η , E F,s i · E F,s ds (cid:3) It now remains to show that pseudo-Eisenstein series for the associate parabolic, Q = P , can also be decomposed into superpositions of P -Eisenstein series. No-tice that in the decomposition above, when we decomposed P -pseudo-Eisensteinseries into genuine P -Eisenstein series, we did not use the functional equation tofold up the integral, as in the case of minimal parabolic pseudo-Eisenstein series.For maximal parabolic Eisenstein series, the functional equation does not relatethe Eisenstein series to itself, but rather the Eisenstein series of the associate para-bolic. We will use this functional equation to obtain the decomposition of associateparabolic pseudo-Eisenstein series. The functional equation is E QF,s = b F,s · E PF, − s where b F,s is a meromorphic function that appears in the computation of the con-stant term along P of the Q -Eisenstein series. Proposition 6.
The pseudo-Eisenstein series Ψ Qf,µ admits a spectral decomposition Ψ Qf,µ = X F Z s h Ψ Qf,µ , E
PF, − s i · | b F, − s | · E PF, − s where F ranges over an orthonormal basis of cuspforms. ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 11
Proof.
We consider a Q -pseudo-Eisenstein series Ψ Qf,µ with cuspidal data. By thesame arguments used above to obtain the decomposition of P -pseudo-Eisensteinseries, we can decompose Ψ Qf,µ into a superposition of Q -Eisenstein series,Ψ Qf,µ ( g ) = X cfms F Z s h η f,µ , ϕ F,s i · E QF,s ( g )Now using the functional equation,Ψ Qf,µ ( g ) = X cfms F Z s h Ψ Qf,µ , b
F,s · E PF, − s i · b F,s · E PF, − s = X cfms F Z s h Ψ Qf,µ , E
PF, − s i · | b F,s | · E PF, − s giving the proposition. (cid:3) So we have a decomposition of Q -pseudo-Eisenstein series (with cuspidal data)into a P -Eisenstein series (with cuspidal data). In order to use the functionalequation we did have to move some contours, but in this case there are no poles,so we did not pick up any residues. Likewise, if η is the data for a P , , pseudo-Eisenstein series Ψ η , we can write η as a tensor product η = f ⊗ µ ⊗ µ on Z GL ( A ) \ Z GL ( A ) × Z GL ( A ) × Z GL ( A ) Similarly, the data ϕ = ϕ F,s ,s for a P , , -Eisenstein series is the tensor productof a GL cuspform and characters χ s and χ s on GL . We show that Ψ f,µ isthe superposition of Eisenstein series E F,s ,s where F ranges over an orthonormalbasis of cusp forms and s and s are on the vertical line. Proposition 7.
The , , pseudo-Eisenstein series Ψ f,µ ,µ admits a spectral ex-pansion Ψ f,µ ,µ = X F Z s Z s h η f,µ ,µ , ϕ F,s ,s i · E F,s ,s where F ranges over an orthonormal basis of cuspforms.Proof. Using the spectral expansions of f and µ , η = f ⊗ µ ⊗ µ = (cid:0) X cfms F h f, F i· F (cid:1) · (cid:0) Z s h µ , χ s i· χ s ds (cid:1) · (cid:0) Z s h µ , χ s i· χ s ds (cid:1) = X cfms F Z s Z s h η f,µ ,µ , ϕ F,s ,s i · ϕ F,s ,s ds ds
22 ADIL ALI
Therefore, the pseudo-Eisenstein series can be re-expressed as a (double) superpo-sition of Eisenstein series.Ψ f,µ ,µ = X γ ∈ P k \ G k η f,µ ,µ ( γg )= X γ ∈ P k \ G k X cfms F Z s Z s h η f,µ ,µ , ϕ F,s ,s i · ϕ F,s ,s ( γg ) ds ds = X cfms F Z s Z s h η f,µ ,µ , ϕ F,s ,s i X γ ∈ P k \ G k ϕ F,s ,s ( γg ) ds ds = X cfms F Z s Z s h η f,µ ,µ , ϕ F,s ,s i · E F,s ,s ( g ) (cid:3) Finally, if η is the data for a P , pseudo-Eisenstein series Ψ η , we can write η f,g,µ = f ⊗ g ⊗ µ on Z GL ( A ) /Z GL ( A ) × Z GL ( A )where f and g are cuspforms, and µ is a compactly-supported smooth function on GL (1). Similarly, the data ϕ = ϕ f ,f ,s for a P , -Eisenstein series is the tensorproduct of GL (2) cuspforms f and f and a character χ s . Proposition 8.
The , pseudo-Eisenstein series Ψ η has a spectral expansion interms of , Eisenstein series Ψ η = X F ,F Z s h η f,g,µ , ϕ F ,F ,s i E F ,F ,s ds where F and F are cuspforms on GL (2) .Proof. Writing η = f ⊗ g ⊗ µ = (cid:0) X cfms F h f, F i · F (cid:1)(cid:0) X cfms F h g, F i · F (cid:1) · (cid:0) Z s h µ, χ s i · χ s (cid:1) = X cfms F ,F Z s h η f,g,µ , ϕ F ,F ,s i · ϕ F ,F ,s ds As before, the corresponding pseudo-Eisenstein series will unwindΨ η = X γ ∈ P k \ G k η f,g,µ ( γg ) = X cfms F ,F Z s h η f,g,µ , ϕ F ,F ,s i · E F ,F ,s ds (cid:3) Recall the construction of 2 , φ ∈ C ∞ c ( R ) and let f be a spherical cuspform on GL with trivial central character. Let ϕ ( (cid:18) A B D (cid:19) ) = φ ( (cid:12)(cid:12)(cid:12) det A det D (cid:12)(cid:12)(cid:12) ) · f ( A ) · f ( D ) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 13 extending by right K -invariance to be made spherical. Define the P , pseudo-Eisenstein series by Ψ ϕ ( g ) = X γ ∈ P k \ G k ϕ ( γg )We recall the construction of 2 , , f be a sphericalcuspform on GL ( k ) \ GL ( A ), and let φ , φ ∈ C ∞ c ( R ). Let ϕ f,φ ,φ ( A b
00 0 c ) = f ( A ) · φ ( det Ab ) · φ ( det Ac )The 2 , , ϕ = X γ ∈ P k \ G k ϕ f,φ ,φ ( γg ) Proposition 9.
The pseudo-Eisenstein series Ψ , ϕ is orthogonal to all other pseudo-Eisenstein series in Sob (+1) .Proof. Recall by [MW p.100] that h Ψ , ϕ , Ψ , , ψ i L = 0Let us now check that they’re also orthogonal in the +1-Sobolev space. Notethat h Ψ , ϕ , Ψ , , ψ i +1 = h Ψ , ϕ , Ψ , , ψ i L + h ∆Ψ , ϕ , Ψ , , ψ i L Since the first summand is zero, it suffices to prove that the second is zero. To thisend, we rewrite the Casimir operatorΩ = Ω + Ω + Ω + Ω where Ω = 12 H , + E , E , + E , E , and Ω = 12 H , + E , E , + E , E , while Ω = 14 H , , , We let Ω be the remaining terms appearing in the expression of Casimir. Weprove that application of Ω to Ψ ϕ produces another function in the span of 2 , , ϕ orthogonal to all other non-associate pseudo-Eisenstein series. We will provethat when restricted to G/K , Ω acts as the SL -Laplacian on the cuspform f , Ω acts as the SL -Laplacian on f , while Ω acts as a second derivative on the testfunction. Indeed, let Ω = 12 H , + E , E , + E , E , where H , = diag(1 , − , ,
0) and E i,j is the matrix with 1 in the ij th position and0’s elsewhere. We check how H , acts on smooth functions on ϕ . Let A = (cid:18) a bc d (cid:19) D = (cid:18) f gh i (cid:19) Observe that H , · ϕ ( (cid:18) A ∗ D (cid:19) ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ( a b c d f g h i · e t e − t )This is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ( ae t be t ce t de t f g h i ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ ( (cid:12)(cid:12)(cid:12) det A det D (cid:12)(cid:12)(cid:12) ) · f ( (cid:18) ae t be − t ce t de − t (cid:19) ) · f ( D )Use Iwasawa coordinates on the upper left hand GL (2) block of the Levi component,namely n x = x m y = √ y √ y As in the discussion for SL ( R ),( H , f )( n x m y ) = 2 y ∂∂y f ( n x m y )Therefore, letting ∆ be Ω restricted to G/K , we see that the effect of ∆ on thecuspform f is just ∆ ( f ) = y ( ∂ ∂x + ∂ ∂y ) f = λ f · f Therefore, ∆ ( ϕ φ,f,f ) = ϕ φ,λ f f,f = λ f · ϕ φ,f,f A similar argument which uses H , , E , and E , as the standard basis in thelower right 2 × the restriction of Ω to smooth functionson G/K , ∆ ( ϕ φ,f,f ) = ϕ φ,f,f = λ f ϕ φ,λ f f,f It remains to check the effect of Ω = H , , , . Observe that H , , , ϕ ( a b c d f g h i ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ( a b c d f g h i · e t e t e − t
00 0 0 e − t )Yet this is just = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ( ae t be t ce t de t f e − t ge − t he − t ie − t )Which gives= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ ( e t det Ae − t det D ) · f ( (cid:18) ae t be t ce t de t (cid:19) ) · f ( (cid:18) f e − t ge − t he − t ie − t (cid:19) ) = 2 · φ ′ · f ( A ) · f ( D ) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 15 since both f and f have trivial central character. Therefore, the effect of H , , , as a differential operator on ϕ φ,f,f is14 H , , , · ϕ φ,f,f = ϕ φ ′′ ,f,f That is, ∆ ϕ φ,f,f = ϕ φ ′′ ,f,f Together the effect of the three differential operators is(∆ + ∆ + ∆ ) ϕ φ,f,f = ϕ ( λ f + λ f ) φ + φ ′′ ,ff Therefore, (∆ + ∆ + ∆ )(Ψ ϕ φ,f,f ) = Ψ ϕ ( λf + λf ) φ + φ ′′ ,f,f The operator ∆ acts by 0 on the vector ϕ φ,f,f . Therefore,∆Ψ ϕ φ,f,f = Ψ ϕ ( λf + λf ) φ + φ ′′ ,f,f The function Ψ ϕ ( λf + λf ) φ + φ ′′ ,f,f is another 2 , λ f + λ f ) φ + φ ′′ is another functionin C ∞ c ( R ), so [MW, p.100] applies again to give h Ψ ϕ ( λf + λf ) φ + φ ′′ ,f,f , Ψ , , ψ i L = 0Therefore, h ∆Ψ ϕ,f,f , Ψ , , ψ i L = 0proving that the pseudo-Eisenstein series are orthogonal in the +1-index Sobolevspace. An inductive argument shows that they are orthogonal in every Sobolevspace.An analogous argument shows that 2 , , , , (cid:3) We turn our attention to the 3 , Proposition 10. , pseudo-Eisenstein series are orthogonal to all other (non-associate) pseudo-Eisenstein series in Sob (+1) .Proof. We review the construction of 3 , f be a spherical cuspform on GL ( k ) \ GL ( A ) and φ ∈ C ∞ c ( R ). Consider the vector ϕ f,φ ( (cid:18) A ∗ d (cid:19) ) = f ( A ) · φ ( det Ad )Working in GL consider the element H = ∈ gl ( R ) We determine the effect of H as a differential operator on ϕ f,φ . To this end, let n x x x = x x
00 1 x
00 0 1 00 0 0 1 m y y y y = y y y
00 0 0 y Then H · ϕ f,φ ( n x x x m y y y y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( n x x x m y y y y e t )This is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( n x x x m y e t y y y ) = y ∂∂y ϕ f,φ ( n x x x m y y y y )Therefore, H · ϕ f,φ ( n x x x m y y y y ) = y ∂∂y ϕ f,φ ( n x x x m y y y y )The effect of H and H is computed similarly. That is H · ϕ f,φ ( n x x x m y y y y ) = y ∂∂y ϕ f,φ ( n x x x m y y y y )while H · ϕ f,φ ( n x x x m y y y y ) = y ∂∂y ϕ f,φ ( n x x x m y y y y )With notation as before, we determine the effect of E , as a differential operator.Observe that E , · ϕ f,φ ( n x x x m y y y y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( n x x x m y y y y t )This is just ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( n x + y tx x m y y y y ) = y ∂∂x ϕ f,φ ( n x x x m y y y y )Therefore, the effect of E , is y ∂∂x , and E differentiates only the cuspform f .Similar arguments show that the effect of E , as a differential operator is E , → y ∂∂x and E , → y ∂∂x Observe that E , , E , , and E , act by 0 on ϕ f,φ . We prove this for E , , theargument being identical for E , and E , . Note E , · ϕ f,φ ( n x x x m y y y y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( n x x x m y y y y t ) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 17
This is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( x x ∗ x
00 0 1 00 0 0 1 · y y y
00 0 0 y ) = 0Let H = Then H · ϕ f,φ ( x x
00 1 x
00 0 1 00 0 0 1 · y y y
00 0 0 y )Which is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ( x x
00 1 x
00 0 1 00 0 0 1 · y y y
00 0 0 y t ) = ϕ f,φ ′ It is clear to see that this is the only element of the Lie algebra differentiating thetest function datum. If { X i } is a basis of gl ( R ) and { X i } is the dual basis relativeto the trace pairing, define an element Ω ∈ U g byΩ = X i X i X ∗ i Let Ω be the element of Z gl given byΩ = 12 H + 12 H + 12 H + E , E , + E , E , + E , + E , As shown above, this element differentiates the cuspidal-data, and does not interactwith the test function datum. Since Ω ∈ Z gl , it acts by a scalar λ f on theirreducible unramified principal series generated by f . Then,Ω = Ω + H + Ω where Ω = Ω − Ω − H . Since Ω interacts with neither the cuspidal data nor thetest function data, its effect as a differential operator on ϕ f,φ will be 0. Note thatΩ · ϕ f,φ = ϕ λ f f,φ , while H · ϕ f,φ = ϕ f,φ ′ . Therefore,Ω ϕ f,φ = ϕ f, ( λ f φ + φ ′ ) producing another 3 , , , , , , , (cid:3) Finally, we consider 2 , , X , X , . . . , X n is abasis for gl ( R ), with dual basis X ∗ , X ∗ , . . . , X ∗ n relative to the trace pairing. LetΩ = P i X i · X ∗ i ∈ Z g , and let ∆ be Ω descended to G/K . We will show thatapplication of ∆ to a 2 , , f and test functions φ , φ produces another 2 , , will prove that 2 , , , , f be a spherical cuspform on GL ( k ) \ GL ( A ),and let φ , φ ∈ C ∞ c ( R ). Let ϕ f,φ ,φ ( A b
00 0 c ) = f ( A ) · φ ( det Ab ) · φ ( det Ac )The 2 , , ϕ = X γ ∈ P k \ G k ϕ f,φ ,φ ( γg ) Proposition 11.
The , , pseudo-Eisenstein series Ψ ϕ is orthogonal to all other(non-associate) pseudo-Eisenstein series in Sob (+1) .Proof. We consider basis elements of the Lie algebra gl ( R ). Let E ij be as before.Let H i be the matrix with 1 on the i th diagonal entry and 0’s elsewhere. We considerthe effect of the H i ’s as differential operators on ϕ f,φ ,φ . It will be convenient touse an Iwasawa decomposition on the GL block in the upper left hand corner. Wewill be considering right K -invariant functions, so ϕ is determined by its effect on n x m y y where n x = x and m y y = y y We calculate H ’s effect on ϕ f,φ ,φ ( n x m y y ). Note that H · ϕ ( n x m y y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ ( n x m y e t y ) = y ∂∂y ϕ ( n x m y y )Similarly, H · ϕ ( n x m y y ) = y ∂∂y ϕ ( n x m y y )Therefore, H and H differentiate the cuspform f , and leave the functions φ and φ as they are. As before, E , · ϕ ( n x m y y ) = y ∂∂x ϕ ( n x m y y )Let us consider the effect of H as a differential operator on ϕ . Observe that H · ϕ f,φ ,φ ( n x m y y y y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( n x m y y y e t y )This is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( A ) φ ( det Ay e − t ) φ ( det Ay ) = − f ( A ) · φ ′ ( det Ay ) φ ( det Ay )Therefore, H · ϕ f,φ ,φ ( n x m y y y y ) = ϕ f, − φ ′ ,φ Similarly, H · ϕ f,φ ,φ ( n x m y y y y ) = ϕ f,φ , − φ ′ ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 19
Observe that E , acts as 0 on ϕ f,φ ,φ . Indeed, E , · ϕ f,φ ,φ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b
00 0 0 c t
00 1 0 00 0 1 00 0 0 1 )This is just ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b
00 0 0 c ) = 0The effect of E , is computed similarly. Observe E , · ϕ f,φ ,φ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b
00 0 0 c t )Which is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b
00 0 0 c ) = 0The elements E , , E , , E , and E , also act as 0. To see that E , acts by 0,note E , · ϕ f,φ ,φ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b
00 0 0 c t )Which is ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ f,φ ,φ ( z z z z b bt c ) = 0 = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( A ) · φ ( det Ab ) φ ( det Ac ) = 0Likewise, E , acts by 0 as a differential operator. The terms which contributenon-trivially to the effect of the P GL ( R )-Laplacian are( H + H + E , E , + E , E , ) + H + H the parenthetical expression acts by a scalar λ f on the cuspform f . That is,( H + H + E , E , + E , E , ) ϕ f,φ ,φ = ϕ λ f f,φ ,φ since H + H + E , E , + E , E , is the Laplacian on P GL ( R ). The remainingtwo terms in expression act as follows: H ϕ f,φ ,φ = ϕ f, φ ′′ ,φ Therefore,( H + H + E , E , + E , E , + H + H ) ϕ f,φ ,φ = ϕ λ f f,φ ,φ + ϕ f, φ ′′ ,φ + ϕ f,φ , φ ′′ Therefore, with ∆ the
P GL ( R )-Laplacian,∆Ψ ϕ = Ψ ϕ λf,φ ,φ + Ψ ϕ f, φ ′′ ,φ + Ψ ϕ f,φ , φ ′′ is again in the vector space spanned by 2 , , L , as claimed. (cid:3) We review Maass-Selberg relations and the theory of the constant term for GL ,as in [Harish-Chandra, p.75], [MW, p.100-101] and [Garrett 2011a]. Let P = P , be the standard, maximal parabolic subgroup P , = (cid:18) GL ∗ GL (cid:19) with unipotent radical N P and standard Levi component M P . The parabolic P isself-associate. Let f be an everywhere spherical cuspform on GL ( k ) \ GL ( A ) withtrivial central character and let ϕ be the vector ϕ ( nmk ) = ϕ s,f ( nmk ) = | det m | s | det m | − s · f ( m ) · f ( m )where m = (cid:18) m ∗ m (cid:19) with m , m in GL , so that m is in the standard Levi component M of the parabolicsubgroup P , n ∈ N its unipotent radical, k ∈ K , and | · | is the idele norm. Definition 5.
The spherical Eisenstein series is E Ps,f ( g ) = E s,f ( g ) = X γ ∈ P k \ G k ϕ Ps,f ( γ · g ) for Re ( s ) ≫ s ) sufficiently large, this series converges absolutely and uniformly oncompacta. We define truncation operators. For a standard maximal proper para-bolic P = P , as above, for g = nmk with m = (cid:18) m ∗ m (cid:19) as above, n ∈ N P and k ∈ O (4) define the spherical function h P ( g ) = h P ( pk ) = | det m | | det m | = δ P ( nm ) = δ P ( m )where δ P is the modular function on P . For fixed large real T , the T -tail of the P -constant term of a left N Pk -invariant function Fc TP F ( g ) = (cid:26) c P F ( g ) : h P ( g ) ≥ T h P ( g ) ≤ T Definition 6.
The truncation operator is Λ T E Pϕ = E Pϕ − E P ( c TP E Pϕ ) where E P ( ϕ )( g ) = X γ ∈ P Z \ Γ ϕ ( γg )These are square-integrable, by the theory of the constant term([MW, pp.18-40],[Harish-Chandra]). The Maass-Selberg relations describe their inner product asfollows. The inner product h Λ T E Pϕ , Λ T E Pψ i ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 21 of truncations Λ T E Pϕ and Λ T E Pψ of two Eisenstein series E Pϕ and E Pψ attached tocuspidal-data ϕ , ψ on maximal proper parabolics P = P , is given as follows.The term c s refers to the quotient of Rankin-Selberg L-functions appearing in theconstant term c P E Pϕ . That is, c s = L (2 s − , π ⊗ π ′ ) L (2 s, π ⊗ π ′ )as in [Langlands 544,Section 4] where π is locally everywhere an unramified principalseries isomorphic to the representation generated by the cuspform f locally. Proposition 12.
Maass-Selberg relations h Λ T E Pg , Λ T E Pg i = h g , g i T s + r − s + r − h g , g w i c g r T s +(1 − r ) − s + (1 − r ) − h g w , g i c g s T (1 − s )+ r − (1 − s ) + r − h g w , g w i c g s c g r T (1 − s )+(1 − r ) − (1 − s ) + (1 − r ) − P in GL n , on thehalf-plane Re( s ) ≥ the only possible poles are on the real line, and only occur if h f, f w i 6 = 0. In that case, any pole is simple, and the residue is square-integrable.In particular, taking f = f o × f o h Res s o E Pϕ , Res s o E Pϕ i = h f o , f o i · Res s o c ϕs as in [Harish-Chandra,p.75]. The group GL gives the first instance of non-constant,noncuspidal contribution to the discrete spectrum; the residues of the Eisensteinseries at its poles give Speh forms. Recall ([Langlands 544] Section 4, though heuses a different normalization), that the constant term is equal to (cid:12)(cid:12) det A det D (cid:12)(cid:12) s · f ( A ) · f ( D ) + (cid:12)(cid:12) det A det D (cid:12)(cid:12) − s · Λ(2 s − , π ⊗ π ′ )Λ(2 s, π ⊗ π ′ ) · f ( A ) · f ( D )The L -function appearing in the numerator necessarily has a residue at the uniquepole in the right half-plane. This residue of the Eisenstein series at this pole is theSpeh form [Jacquet] attached to a GL (2) cuspform f , and is in L .We now compute the 2 , , f and f . Let P = P , be the self-associate standard parabolic in G = GL with Levi component GL × GL . Let f and f be spherical cuspforms on GL ( k ) \ GL ( A ). Define the spherical vector ϕ Ps,f ,f ( (cid:18) A ∗ D (cid:19) ) = (cid:12)(cid:12) det A det D (cid:12)(cid:12) s · f ( A ) · f ( D )and then extending to G A by right K v -invariance and Z v -invariance everywherelocally. Define cuspidal-data Eisenstein series for Re( s ) ≫ E Ps,f ,f ( g ) = X γ ∈ P k \ G k ϕ Ps,f ,f ( γg ) Proposition 13.
The P -constant term of the P -Eisenstein series E Ps,f ,f ( g ) isgiven by c P E Ps,f ,f ( g ) = (cid:12)(cid:12) det A det D (cid:12)(cid:12) s · f ( A ) · f ( D )+ (cid:12)(cid:12) det A det D (cid:12)(cid:12) − s · f ( A ) · f ( D ) · L ( π ⊗ π , s − L ( π ⊗ π , s ) where π is the G A -representation generated by f and π is the G A -representationgenerated by f .Proof. The constant term of E s,f ,f along P is given by c P E Ps,f ,f ( g ) = Z N k \ N A E Ps,f ,f ( ng ) dn = X ξ ∈ P k \ G k /N k Z ξ − P k ξ ∩ N k \ N A ϕ s,f ,f ( ξγng ) dn The double coset space P \ G/N surjects to W P \ W/W P which has three doublecoset representatives, two of which give a nonzero contribution. The identity cosetcontributes a volume, which we will compute later. The nontrivial representativeis ξ = σ σ σ σ . Observe that ξ · P k · ξ − ∩ N k = { } so that c P E Ps,f ,f ( g ) = Z N k \ N A ϕ s,f ,f ( ng ) dn + Z N A ϕ s,f ,f ( ξng ) dn To compute the contribution of the integral Z N A ϕ s,f ,f ( ξng ) dn we must re-express the Eisenstein series representation-theoretically. To this end,let π f = ⊗ π f ,v be the representation of G A generated by f and let π f = ⊗ π f ,v be the G A -representation generated by f . For places v outside a finite set S , fixisomorphisms j v : Ind χ f ,v → π f ,v and l v : Ind χ f ,v → π f ,v Their tensor product j v ⊗ l v is a representation of the Levi M = GL ⊗ GL . Extendrepresentations of Levi components trivially to parabolics. A π f -valued Eisensteinseries is formed by a convergent sum E Pϕ = X γ ∈ P k \ G k ϕ ◦ γ Let T = ⊗ v T v : ϕ → R N A ϕ ( ξng ) dn . We have a chain of intertwinings ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 23 O v ∈ S Ind G v P v (cid:0) ( π f ,v ) ⊗ π f ,v ) ν sP v (cid:1) ⊗ O v / ∈ S Ind G v B v (cid:0) ( χ f ,v ⊗ χ f ,v ) ν s,s,s, − sB v (cid:1) l iterated induction O Ind G v P v ( π f ,v ⊗ π f ,v ) v sP v ⊗ O Ind G v P v (cid:0) Ind P v B v ( χ f ,v ⊗ χ f ,v ) ν s,s,s, − sB v (cid:1) l ⊗ (cid:0) ⊗ Ind
GvPv ( j v ⊗ l v ) (cid:1)O Ind G v P v ( π f ,v ⊗ π f ,v ) ν sP v ⊗ O Ind G v P v (cid:0) ( π f ,v ⊗ π f ,v ) ν sP v (cid:1) l T = ⊗ T v O Ind G v P v (cid:0) ( π f ,v ⊗ π f ,v ) ν − sP v (cid:1) ⊗ O Ind G v P v (cid:0) ( π f ,v ⊗ π f ,v ν − sP v (cid:1) l ⊗ (cid:0) N Ind
GvPv ( j − v ⊗ l − v ) (cid:1)O Ind G v P v (cid:0) ( π f ,v ⊗ π f ,v ) ν − sP v (cid:1) ⊗ O Ind G v P v (cid:0) Ind P v B v ( χ f ,v ⊗ χ f ,v ) ν − s,s − ,s − ,s − B v (cid:1) l iterated induction O Ind G v P v (cid:0) ( π f ,v ⊗ π f ,v ) ν − sP v ⊗ (cid:1) ⊗ O Ind G v B v ( χ f ,v ⊗ χ f ,v ) ν − s,s − ,s − ,s − B v The advantage of this set-up is that for v outside the finite set S, the minimalparabolic unramified principal series has a canonical spherical vector, namely thatspherical vector taking value 1 at 1 ∈ G v . Therefore the isomorphism T v can becompletely determined by computing its effect on the canonical spherical vector.The intertwinings T v among minimal-parabolic principal series can be factored ascompositions of similar intertwining operators attached to reflections correspond-ing to positive simple roots, each of which is completely determined by its effecton the canonical spherical vector in the unramified principal series. The simplereflection intertwinings’ effect on the normalized spherical functions reduce to GL computations.Thus, with simple reflections σ = σ = σ = and with corresponding root subgroups N σ = x N σ = y
00 0 1 00 0 0 1 N σ = z The simple-reflection intertwinings S σ f ( g ) = Z N σ f ( σ ng ) dn S σ f ( g ) = Z N σ f ( σ ng ) dnS σ f ( g ) = Z N σ f ( σ ng ) dn are instrumental because we wish to compute the effect of S σ ◦ S σ ◦ S σ ◦ S σ on the normalized spherical vector in the unramified minimal-parabolic principalseries I ( s , s , s , s ). Furthermore, S στ = S σ ◦ S τ Therefore, we must understand the effect of the individual S σ i ’s. Recall that S σ : I ( s , s , s , s ) → I ( s , s + 1 , s − , s )Similarly, S σ : I ( s , s , s , s ) → I ( s + 1 , s − , s , s )and S σ : I ( s , s , s , s ) → I ( s , s , s + 1 , s − f ∈ I ( s , s , s , s ) is mapped by S σ to amultiple of the normalized spherical function in I ( s +1 , s − , s , s ). The constantis S σ f (1) = Z f ( σ x ) dx = Z f ( x ) dx Using the Iwasawa decomposition for GL ( k v ), we show that this calculation re-duces to a GL calculation. Indeed, there is (cid:18) a bc d (cid:19) in the maximal compact of GL ( k v ) such that (cid:18) x (cid:19) (cid:18) a bc d (cid:19) = (cid:18) ∗ ∗ ∗ (cid:19) Therefore, x a b c d = ∗ ∗ ∗ From this, it follows that the constant S σ f (1) with S σ : I ( s , s , s , s ) → I ( s + 1 , s − , s , s )is the same as the constant in the intertwining from I ( s , s ) → I ( s + 1 , s −
1) of GL principal series, namely ϕ ( (cid:18) x (cid:19) ) dx ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 25 where ϕ is the normalized spherical vector in the GL principal series. A similarargument applies to the other intertwining operators attached to other simple re-flections. We recall the GL computation below. At absolutely unramified finiteplaces, (cid:18) x (cid:19) ∈ K v = GL ( σ v ) for x ≤
1. For x > (cid:18) x (cid:19) (cid:18) − x x (cid:19) = (cid:18) x x (cid:19) (cid:18) −
11 0 (cid:19)
Thus, with local parameter ω and residue field cardinality q , since the measure of { x ∈ k v : | x | = q r } is ( q − q r − , we see that Z k v ϕ ( (cid:18) x (cid:19) ) dx = Z | x |≤ dx + Z | x | > ϕ ( (cid:18) x
10 1 (cid:19) ) dx This is 1 + (1 − q ) X r ≥ q r (1 − s + s ) = ζ v ( s − s − ζ v ( s − s )with the Iwasawa-Tate unramified local zeta integral ζ v ( s ).Using this GL reduction, we see that S σ : I ( s , s , s , s ) → I ( s , s + 1 , s − , s )and maps the normalized spherical vector in I ( s , s , s , s ) to ζ v ( s − s − ζ v ( s − s )times the normalized spherical function in I ( s , s + 1 , s − , s ). Then S σ : I ( s , s + 1 , s − , s ) → I ( s + 2 , s − , s − , s )and sends the normalized spherical function in I ( s , s + 1 , s − , s ) to ζ v ( s − s − ζ v ( s − s − I ( s + 2 , s − , s − , s ). Then S σ maps the normalized spherical vector in I ( s + 2 , s − , s − , s ) to ζ v ( s − s − ζ v ( s − s − I ( s + 2 , s − , s + 1 , s − S σ : I ( s + 2 , s − , s + 1 , s − → I ( s + 2 , s + 2 , s − , s −
2) and sends thenormalized spherical function in I ( s + 2 , s − , s + 1 , s −
2) to ζ v ( s − s − ζ v ( s − s − I ( s +2 , s +2 , s − , s − S σ ◦ S σ ◦ S σ ◦ S σ maps the normalized spherical vector in I ( s , s , s , s ) to ζ v ( s − s − ζ v ( s − s ) · ζ v ( s − s − ζ v ( s − s − · ζ v ( s − s − ζ v ( s − s − · ζ v ( s − s − ζ v ( s − s − I ( s + 2 , s + 2 , s − , s − For ( s , s , s , s ) = ( s + s f , s − s f , − s + s f , − s − s f ) we get ζ v ( s − s f − ( − s + s f ) − ζ v ( s − s f − ( − s + s f )) · ζ v ( s + s f − ( − s + s f ) − ζ v ( s + s f − ( − s + s f ) − · ζ v ( s − s f − ( − s − s f ) − ζ v ( s − s f − ( s − s f ) − · ζ v ( s + s f − ( − s − s f ) − ζ v ( s + s f − ( − s − s f ) − L ( π ⊗ π , s − L ( π ⊗ π , s ) (cid:3) Global Automorphic Sobolev Spaces
We recall basic ideas about global automorphic Sobolev spaces. For example,see Decelles [2011b], [Grubb], and [Garrett 2010]. Consider the group G = GL (4)defined over a number field k . At each place v , let K v be the standard maximalcompact subgroup of the v -adic points G v of G . That is, K v = GL ( O v ) for nonar-chimedean places v where O v denotes the local ring of integers, and K v = O ( R )for v real and K = U ( n ) for v complex. Consider the space C ∞ c ( Z A G k \ G A , ω )where ω is a trivial central character. We define positive index global archimedeanspherical automorphic Sobolev spaces as right K ∞ -invariant subspaces of comple-tions of C ∞ c ( Z A G k \ G A , ω ) with respect to a topology induced by norms associatedto the Casimir operator Ω. The operator Ω acts on the archimedean component f ∈ C ∞ c ( Z A G k \ G A , ω ) by taking derivatives in the archimedean component. Thenorm | . | ℓ on C ∞ c ( Z A G k \ G A , ω ) K is | f | ℓ = h (1 − Ω) ℓ f, f i where h , i gives the norm on L ( Z A G k \ G A , ω ), induces a topology on the space C ∞ c ( Z A G k \ G A , ω ) K . Definition 7.
The completion H ℓ ( Z A G k \ G A , ω ) is the ℓ -th global automorphicSobolev space. H ℓ ( Z A G k \ G A , ω )is a Hilbert space with respect to this topology. Definition 8.
For ℓ > , the Sobolev space H − ℓ ( Z A G k \ G A , ω ) is the Hilbert spacedual of H ℓ ( Z A G k \ G A , ω ) . Since the space of test functions is a dense subspace of H ℓ ( Z A G k \ G A , ω ) with ℓ >
0, dualizing gives an inclusion of H − ℓ ( Z A G k \ G A , ω ) into the space of distributions.The adjoints of the dense inclusions H ℓ → H ℓ − are inclusions H − ℓ +1 ( Z A G k \ G A , ω ) → H − ℓ ( Z A G k \ G A , ω )4. Pre-trace formula estimates on compact periods
We give a standard argument. See, for example, [Iwaniec] and [Garrett 2010].Set k = Q throughout. Let Θ be a k -subgroup of G . Let [Θ] = ( Z A ∩ Θ)Θ k \ Θ A and [ G ] = Z A G k \ G A /K ∞ . For smooth f on Z A G k \ G A , define the [Θ] x -period of f to be f Θ ,x = Z [Θ] f ( hx ) dh ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 27
Similarly, with φ an automorphic form on Θ k \ Θ A , the [Θ] , x, φ -period of f is h f, φ i Θ = Z [Θ] φ ( h ) · f ( hx ) dh For finite places v , fix a compact open subgroup K v of G v such that at almost allplaces, K v is the standard maximal compact subgroup of G v , and let K fin = Q v K v .Let K = K ∞ · K fin . Proposition 14.
The distribution given by integration along a compact quotient Θ k \ Θ A lies in H − s ( Z A G k \ G A ) for all s > dim ( G ∞ /K ∞ ) − dim (Θ ∞ /K Θ ∞ )2 Proof.
Consider smooth f on Z A G k \ G A generating unramified principal series atarchimedean places. The usual action of compactly-supported measures η on suit-able f on G k \ G A /K ∞ is given by( η · f )( x ) = Z G A η ( g ) f ( xg ) dg The Θ k \ Θ A x -period of η · f admits a useful rearrangement( η · f ) Z A Θ k \ Θ A x = Z Z A Θ k \ Θ A ( η · f )( hx ) dh = Z [Θ] Z G A η ( g ) f ( hxg ) dg dh = Z [Θ] Z G A η ( x − h − g ) dg dh = Z [Θ] Z [ G ] X γ ∈ G k η ( x − h − γg ) f ( g ) dg dh = Z [ G ] f ( g ) Z [Θ] X γ ∈ G k η ( x − h − γg ) dh dg Denote the inner sum and integral by q ( g ) = q Θ ,x ( g ). For η a left and right K fin -invariant measure, for f a spherical vector in a copy of a principal series, η · f will be K fin -invariant. Since the spherical vector in an irreducible representationis unique (up to scalar), η · f = λ f ( η ) · f for some constant λ f ( η ). Let η ∞ be thecharacteristic function of a shrinking ball B ǫ in G ∞ /K ∞ of geodesic radius ǫ > v , let η v be the characteristic function of K v . The ball B ǫ has v -adic components in K v for almost all v , and archimedean component lyingwithin a ball of radius ǫ . Identify B ǫ with its pre-image B ǫ · K v in G v . Here, wemake use of a G ∞ -invariant metric d ( x, y ) = ν ( x − y )on G ∞ /K ∞ where ν ( g ) = log sup( | g | , | g − | )Here | · | is the operator norm on the group G v given by | T | = sup u ≤ || T u || Let η = ⊗ v η v . The action of such η changes the period by the eigenvalue. To seethis, observe that( η v · f )( x ) = Z K v η v ( k ) f ( gk ) dk = Z K v f ( g ) dk = vol( K v ) · f ( g )Also, η ∞ · f will be a spherical vector. Since the spherical vector is unique up to aconstant multiple, η ∞ · f = λ ∞ · f for some scalar λ ∞ . Therefore,( η · f ) Θ ,x = λ f ( η ) · vol( K fin ) · f Θ ,x An upper bound for the L ( Z A G k \ G A , ω ) norm of q , and a lower bound for λ f ( η )contingent on restrictions on the spectral parameter of f , yield, by Bessel’s inequal-ity, an upper bound for a sum-and-integral of periods h f, φ i Θ ,x as follows. Estimatethe L norm of q : Z [ G ] | q ( g ) | = Z [ G ] Z [Θ] Z [Θ] X γ ∈ G k X γ ∈ G k η ( x − h − γg ) η ( x − h − γ g ) dg dh dg = Z G A Z [Θ] Z [Θ] X γ ∈ G k η ( x − h − γg ) η ( x − h − g ) dh dh dg With C a large enough compact subset of Θ A to surject to [Θ] = ( Z A ∩ Θ)Θ k \ Θ A , Z [ G ] | q ( g ) | ≤ Z G A Z C Z C X γ ∈ G k | η | ( x − h − γg ) | η | ( x − h − g ) dh dh dg The setΦ = Φ
H,x,η = { γ ∈ G k : η ( x − h − γg ) η ( x − h − g ) = 0 for some h, h ∈ C and g ∈ G A } = { γ ∈ G k : γ ∈ CxB ǫ g − , g ∈ CxB ǫ } ⊂ G k ∩ CxB ǫ · ( CxB ǫ ) − } the last set in the sequence above is the intersection of a closed, discrete set with acompact set, so is finite, and can only shrink as ǫ → + .For K a compact open subgroup in the finite adele part G of G A , a ball ofarchimedean radius ǫ is the product B ǫ × K . Here B ǫ is the inverse image in G ∞ of the geodesic ball of radius ǫ in G ∞ /K ∞ . For each γ ∈ Φ, for each h ∈ C , η ( x − h − γg ) = 0 only for g in a ball in X = G A /K ∞ of radius ǫ , with volumedominated by ǫ dim X . Thus, Z G k \ G A | q ( g ) | dg ≪ Z C ǫ dim X +dim Y dh ≪ ǫ dim X +dim Y By automorphic Plancherel, with η as above, X cfm F | λ F ( η ) | · |h η · F, φ i| + . . . ≪ ǫ dim X +dim Y Next,we give a bound on the spectral data to give a non-trivial lower bound for λ f ( η ). Left and right K -invariant η necessarily gives η · f = λ f ( η ) · f , since up toscalars f is the unique spherical vector in the irreducible representation f generates.This is an intrinsic representation-theoretic relation, because an isomorphism of ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 29 principal series sends a spherical vector in the first representation to a constantmultiple of the spherical vector in the second representation. That is, if ϕ : V → W is an isomorphism of representations, and f ∈ V and f ∈ W are the uniquespherical vectors, then ϕ ( f ) = c · f for a constant c . To see this, observe that k · ϕ ( f ) = ϕ ( k · f ) = ϕ ( f )Therefore, ϕ ( f ) is indeed invariant under the K -action, so is the spherical vectorin the representation V . Then a calculation gives λ f ( η ) · ϕ ( f ) = η · ϕ ( f ) = η · ( c · f ) = c · λ f ( η ) = λ f ( η ) · c · f = λ f ( η ) · ϕ ( f )so that λ f = λ f , as claimed. Therefore, the eigenvalue λ f ( η ) can be computed inthe usual model of the principal series at an archimedean place, as η · ϕ s (1) = λ f ( η )for ϕ s the normalized spherical vector for s ∈ a ∗ ⊗ R C , and ϕ (1) = 1. Thus, λ f ( η ) = ( η · ϕ s (1)) = Z G R η ( g ) · ϕ s ( g ) dg = Z B ǫ ϕ s ( g ) dg Let P + be the connected component of the identity in the standard minimal para-bolic. The Jacobian of the map P + × K → G R is non-vanishing at 1, and ϕ (1) = 1,so a suitable bound of ǫ on the spectral parameter s ∈ a ∗ ⊗ R C will keep ϕ s ( g ) near1 on B ǫ . In the example of GL n ( R ) with ϕ s the usual spherical vector, bounds ofthe form | s j | ≪ ǫ assure that Re ϕ s ( g ) ≥ on B ǫ , which prevents cancellation inthe real part of ϕ s ( g ) for g ∈ B ǫ , so | λ f ( η ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ǫ ϕ s ( g ) dg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≫ Z B ǫ Re ϕ s ( g ) dg ≫ Z B ǫ dg ≫ ǫ dim X Combining the upper bound on | q | L with its lower bound on eigenvalues t F ( t F − T = ǫ ,( ǫ dim X ) × X cfm F | t F |≤ T | F Θ k \ Θ A x | + . . . ≪ ǫ dim X +dim Y so X cfm F: | t F |≪ T | F Θ k \ Θ A x | + . . . ≪ T dim X − dim Y Similarly, X cfm F: | t F |≪ T |h η · F, φ i| + . . . ≪ T dim X − dim Y (cid:3) Casimir Eigenvalue
Let G = SL ( R ) and I ( s , s , s , s ) a minimal-parabolic principal series. Let g = sl be the Lie algebra of G . For i = j , let E i,j be the matrix with 1 in the ( i, j )-thposition and 0 elsewhere. Let H i,j be the matrix with 1 in the ( i, i )-th positionand − j, j )-th position. Observe that H i,i +1 span the Cartan subalgebra h and the E i,j for i = j span the rest of the Lie algebra. Assume without loss ofgenerality that i < j . We have the bracket relations[ E i,j , E j,i ] = H i,j As before, the Casimir element is given byΩ = 12 H , + 12 H , + 12 H , + ( X j,i E i,j E i,j + E i,j E j,i )Rearranging, this givesΩ = 12 H , + 12 H , + 12 H , + ( X i,j E j,i E i,j + H i,j )The lie algebra g acts on C ∞ ( G ) by X · f ( g ) = ddt | t =0 f ( ge tX )The product E j,i E i,j act by 0, so Casimir is simplyΩ = ( 12 H , − H , ) + ( 12 H , − H , ) + ( 12 H , − H , ) + H , + H , + H , Proposition 15.
The Casimir operator acts on I ( s , s , s , s ) by the scalar
12 ( s − s ) − ( s − s ) + 14 ( s + s − s − s ) − ( s − s ) + 12 ( s − s ) − ( s − s ) − ( s − s ) − ( s − s ) − ( s − s ) Proof.
Let us see how H , acts on I ( s , s , s , s ). Note that e tH , = e t e − t Therefore ddt | t =0 f ( e t e − t ) = ddt | t =0 χ ( e t e − t )= ddt | t =0 e ts · e − ts ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 31
This is just ( s − s ). Likewise, we see that H i,j will act on I s by s i − s j . Therefore,the Casimir operator will act by12 ( s − s ) − ( s − s ) + 12 ( s − s ) − ( s − s ) + 12 ( s − s ) − ( s − s ) + ( s − s )+ ( s − s ) + ( s − s )Let G = GL and I ( s , s , s , s ) a minimal-parabolic principal series. Let g = gl be the Lie algebra of G . For i = j , let E ij be the matrix with 1 in the ( i, j )-thposition and 0 elsewhere. Let H ij be the matrix with 1 in the ( i, i )-th position and − j, j )-th position and let H = diag(1 , , − , − H i,i +1 span the Cartan subalgebra h and the E ij for i = j span the rest of the Lie algebra.Assume without loss of generality that i < j . We have the bracket relations[ E ij , E ji ] = H ij As before, the Casimir element is given byΩ = 12 H + 14 H + 12 H + ( X ji E ij E ji + E ji E ij )Rearranging, this givesΩ = 12 H + 14 H + 12 H + ( X ij E ij E ji − H ij )The Lie algebra g acts on C ∞ ( G ) by X · f ( g ) = ddt | t =0 f ( ge tX )The product E ij E ji act by 0, so Casimir is simplyΩ = ( 12 H − H , ) + ( 14 H − H ) + ( 12 H − H ) − H − H − H As an example computation, let us see how H acts on I ( s , s , s , s ). Note that e tH = e t e − t Therefore ddt | t =0 f ( e t e − t ) = ddt | t =0 χ ( e t e − t )= ddt | t =0 e ts · e − ts This is just ( s − s ). Likewise, we see that H ij will act on I s by s i − s j . Therefore,the Casimir operator will act by12 ( s − s ) − ( s − s ) + 14 ( s + s − s − s ) − ( s − s ) + 12 ( s − s ) − ( s − s ) − ( s − s ) − ( s − s ) − ( s − s ) (cid:3) Letting s = s + s f , s = − s + s f , s = s − s f , s = − s − s f , we see that( s − s ) = 2 s , ( s − s ) = − s + 2 s f , ( s − s ) = 2 s , ( s − s ) = 2 s + 2 s f ,( s − s ) = 2 s f , ( s − s ) = 2 s f , and finally ( s + s − s − s ) = 4 s f . Putting allthis into the above expression for Casimir’s action gives that Casimir acts by λ s,f = 4 s + 4 s f − s f − s Observe that λ s,f − λ w,f = 4( s ( s − − w ( w − Friedrichs self-adjoint extensions and complex conjugation maps
We review the result due to Friedrichs that a densely-defined, symmetric, semi-bounded operator admits a canonical self-adjoint extension with a useful charac-terization. We follow [Grubb], [Garrett 2011c], [Friedrichs 1935a] and [Friedrichs1935b].Let T be a densely defined, symmetric, unbounded operator on a Hilbert space V ,with domain D . Assume further, that T is semi-bounded from below in the sensethat || u || ≤ h u, T u i for all u ∈ D. Let h x, y i = h T x, y i on D. Let V be the completion of D with respect to the newinner product. The operator T remains symmetric for h , i . That is, h T x, y i = h x, T y i for x, y ∈ D . By Riesz-Fischer, for y ∈ V , the continuous linear functional f ( x ) = h x, y i can be written f ( x ) = h x, y ′ i for a unique y ′ ∈ V . Set T − y = y ′ That is, the inverse T − of the Friedrichs extension T Fr of T is an everywhere-defined map T − : V → V continuous for the h , i topology on V , characterized by h T x, T − y i = h x, y i We will prove that, given θ ∈ V − and T θ = T | ker θ , the Friedrichs extension ˜ T θ hasthe feature that ˜ T θ u = f for u ∈ V , f ∈ V ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 33 exactly when T θ u = f + c · θ for some c ∈ C Define a conjugation map on V to be a complex-conjugate-linear automorphism j : V → V with h jx, jy i = h y, x i and j = 1. A conjugation map is equivalent to acomplex-linear isomorphism Λ : V → V ∗ of V with its complex-linear dual, via Riesz-Fischer, byΛ( y )( x ) = h x, jy i = h y, jx i Assume j stabilizes D and that T ( jx ) = jT x for x ∈ D . Then j respects h , i : h jx, jy i = h y, T x i = h y, x i for x, y ∈ D . Also, j commutes with T Fr : h x, T − jy i = h x, jy i = h y, jx i = h T − y, jx i = h x, jT − y i for x ∈ V and y ∈ V . Let V − be the complex-linear dual of V . We have V ⊂ V ⊂ V − . By design, T : D → V ⊂ V − is continuous when V has the subspace topology from V − : | T y | − = sup | x | ≤ | Λ( T y )( x ) | = sup |h x, jT y i| = |h x, T jy i| ≤ sup | x |·| y | = | y | by Cauchy-Schwarz-Bunyakowsky. Thus the map T : D → V extends by continuityto an everywhere-defined, continuous map T : V → V − by ( T y )( x ) = h x, jy i Further, T : V → V − agrees with T Fr : D → V on the domain D = BV of T Fr ,since ( T y )( x ) = h x, jy i = h T x, jy i = h T x, T − T Fr jy i = h T − T x, T Fr jy i which is = h x, T Fr jy i = Λ( T Fr y )( x ) for x ∈ D and y ∈ D This follows since T Fr extends T , and noting the density of D in V .The following were presented as heuristics in [CdV 1982/1983] and treated moreformally by Garrett in [Garrett 2011a]. We give complete proofs. Theorem 1.
The domain of T Fr is D = { u ∈ V : T u ∈ V } .Proof. T u = f ∈ V implies that h x, ju i = ( T u )( x ) = Λ( T u )( x ) = Λ( f )( x ) = h x, jf i for all x ∈ V By the characterization of the Friedrichs extension, T Fr ( ju ) = jf . Since T Fr com-mutes with j , we have T Fr u = f . (cid:3) Extend the complex conjugation j to V − by ( jλ )( x ) = λ ( jx ) for x ∈ V , andwrite h x, θ i V × V − = ( jθ )( x ) = θ ( jx ) (for x ∈ V and θ ∈ V − )For θ ∈ V − , θ ⊥ = { x ∈ V : h x, θ i V × V − = 0 } is a closed co-dimension-one subspace of V in the h , i -topology. Assume θ / ∈ V .This implies density of θ ⊥ in V in the h , i -topology. Theorem 2.
The Friedrichs extension T θ = ( T | θ ⊥ ) Fr of the restriction T | θ ⊥ of T to D ∩ θ ⊥ has the property that T θ u = f for u ∈ V and f ∈ V exactly when T u = f + cθ for some c ∈ C . Letting D be the domain of T Fr , the domain of T θ isdomain T θ = { x ∈ V : h x, θ i V × V − = 0 , T x ∈ V + C · θ } Proof. T u = f + c · θ is equivalent to h x, ju i = T ( u )( x ) = ( f + c · θ )( x ) = h x, jf i (for all x ∈ θ ⊥ ) . This gives h x, ju i = h x, jf i . The topology on θ ⊥ is the restriction of the h , i -topology of V , while θ ⊥ is dense in V in the h , i -topology. Thus, ju = T − θ jf bythe characterization of the Friedrichs extension of T θ ⊥ . Then u = T − θ f , since j commutes with T . (cid:3) Given an everywhere-defined map ˜ T − : V → V , characterized by h T x, ˜ T − y i = h x, y i (for x ∈ D, y ∈ V )we review the proof that given θ ∈ V − and T θ = T | ker θ , the Friedrichs extension˜ T θ has the feature that ˜ T θ u = f for u ∈ V , f ∈ V exactly when T θ u = f + c · θ for some c ∈ C Observe that T θ u = f + c · θ is equivalent to h x, u i = h x, T u i = h x, f + c · θ i V × V − = h x, f i V × V − ⇐⇒ ˜ T θ u = f where the second equality follows from restricting in the first argument and extend-ing in the second. 7. Moment bounds assumptions
We will need to assume a moment bound to know that the projected distributionis in the desired Sobolev space. This assumption is far weaker than Lindelof, buthighly non-trivial.
Proposition 16.
For a degree n L -function L ( s ) with suitable analytic continuationand functional equation, a second-moment bound T Z | L ( 12 + it ) | dt ≪ T A ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 35 implies a pointwise bound L ( σ o + it, f ) ≪ σ o ,ǫ (1 + | t | ) A + ǫ ( for every ǫ > Proof.
The proof of this is a standard argument, as follows. Fix σ o > . For0 < t o ∈ R , let s o = σ o + it o . Let R be a rectangle in C with vertices ± iT and2 ± iT for T > t o . By Cauchy’s Theorem L ( s o , f ) = 12 πi Z R e ( s − s o ) s − s o · L ( s, f ) ds Since the L-function has polynomial vertical growth, we can push the top andbottom of R to ∞ , giving L ( s o ) = 12 π ∞ Z −∞ e (( − σ o )+ i ( t − t o )) ( − σ o ) + i ( t − t o ) · L ( 12 + it ) dt + O (1)The part of the integral where | t − t o | ≥ t o is visibly ≪ n,σ o e − t o : | e (( − σ o + i ( t − t o )) ) | = e ( − σ o ) − ( t − t o ) ≪ σ o e − t o · e − ( t − to )22 ≪ e − t o for | t − t o | ≥ t o . Squaring the convexity bound for L ( + it ) gives | L ( 12 + it ) | ≪ | t | n + ǫ (for all ǫ > ∞ Z t o e (( − σ o + i ( t − t o )) ) ( − σ o ) + i ( t − t o ) · L ( 12 + it ) dt ≪ σ o e − t o ∞ Z t o e − ( t − to )22 · t n + ǫ ≪ ǫ e − t o The other half of the tail, where t <
0, is estimated similarly. For 0 < t < t o , usethe assumed moment estimate and the trivial estimate e (( − σ o + i ( t − t o )) ) ( − σ o ) + i ( t − t o ) ≪ σ o e ( − σ o ) − ( t − t o ) ≪ σ o t o Z e (( − σ o + i ( t − t o )) ) ( − σ o ) + i ( t − t o ) · L ( 12 + it ) dt ≪ σ o t o Z | L ( 12 + it ) | dt ≪ t Ao Thus, L ( s o ) = 12 π ∞ Z −∞ e ( + it − s o ) + it − s o · L ( 12 + it, f ) dt + O (1) ≪ n,σ o t Ao Then a standard convexity argument [Lang, p.263] gives the asserted | t o | A + ǫ on σ o = for all ǫ > (cid:3) Local automorphic Sobolev spaces
A notion of local automorphic Sobolev spaces H s lafc defined in terms of globalautomorphic Sobolev spaces H s gafc is necessary to discuss the meromorphic contin-uation of solutions u = u w to differential equations (∆ − λ w ) u = θ for compactly-supported automorphic distributions θ . We want a continuous embedding of globalautomorphic Sobolev spaces into local automorphic Sobolev spaces. This will followimmediately from the description, below. Second, compactly-supported distribu-tions θ ∈ H − s gafc should extend to continuous linear functionals in H − s lafc . A convenientcorollary is that such θ moves inside integrals appearing in a spectral decomposi-tion/synthesis of automorphic forms lying in global automorphic Sobolev spaces.Finally, we want automorphic test functions to be dense in the local automorphicSobolev spaces.The necessity of the introduction of larger spaces than global automorphic Sobolevspaces is apparent already in the simplest situations. On Γ \ H , with Γ = SL ( Z ),when θ ∈ H − − ǫ gafc is an automorphic Dirac δ afc at z ∈ Γ \ H , the spectral expansionin Re( w ) > for a solution u w to that differential equation yields u w ∈ H − ǫ gafc , butthe meromorphic continuation to Re( w ) = and then to Re( w ) < includes anEisenstein series E w which lies in no global automorphic Sobolev space. That E w lies in local automorphic Sobolev space H ∞ lafc is immediate from the smoothness of E w and the definition of the local spaces, below.We describe local automorphic Sobolev spaces. Given a global automorphicSobolev norm | . | s , the corresponding local automorphic Sobolev norms, indexed byautomorphic test functions ϕ , are given by f → | f | s,ϕ = | ϕ · f | s for f smooth automorphic Definition 9.
The s -th local automorphic Sobolev space is given by H s lafc ( X ) = quasi-completion of C ∞ c ( X ) with respect to these semi-norms By definition, C ∞ c ( X ) is dense in H s lafc ( X ). Continuity of the embedding of theglobal automorphic Sobolev spaces into the local uses integration by parts. TheLie algebra g admits a decomposition g = k ⊕ s where k is the Lie algebra of themaximal compact subgroup K and s is the algebra of symmetric matrices. Choosean orthonormal basis { x i } for s with respect to the Killing form h , i . Define thegradient ∇ = X i X x i ⊗ x i where X x i is the differential operator given by X x i f ( g ) = ∂∂t | t =0 f ( g · e tx i ). Observethat in the universal enveloping algebra ∇ f · ∇ F = ( X i X x i f ⊗ x i ) · ( X j X x j F ⊗ x j ) = X i X x i f · X x j F where the product is the Killing form on s . Proposition 17.
For f, F ∈ C ∞ c (Γ \ G ) , we have the integration-by-parts formula Z Γ \ G ( − ∆ f ) F = Z Γ \ G ∇ f · ∇ F ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 37
Proof.
Letting X = Γ \ G , consider the integral Z X ∂∂t f ( g · e tx i ) ∂∂t F ( g · e tx i ) dg Let u = ∂∂t f ( g · e tx i ) and dv = ∂∂t F ( ge tx i ) dg . Then du = ∂ ∂t ∂t f ( g · e t x i · e t x i ),while v = F ( g ). Then, using the compact support of f and its derivatives, we get Z X ∂∂t f ( g · e t x i ) ∂∂t F ( g · e t x i ) dg = Z X − ∂ ∂t ∂t f ( g · e t x i e t x i ) F ( g ) dg Taking limits as t and t approach 0 gives the integration-by-parts formula Z X X x i f · X x i F = Z X ( − X x i f ) · F and Z X ( − ∆ f ) · F = Z X ∇ f · ∇ F (cid:3) Now we can compare the local automorphic Sobolev +1-norm to the globalautomorphic Sobolev +1-norm as follows:
Proposition 18.
Every local automorphic Sobolev +1 -norm is dominated by theglobal automorphic Sobolev +1 -norm.Proof. | f | H = | ϕf | H = Z X (1 − ∆)( ϕf ) ϕf = Z X ∇ ( ϕf ) · ∇ ( ϕf ) + Z X ϕf · ϕf This is Z X ( f ∇ ϕ + ϕ ∇ f ) · ( f ∇ ϕ + ϕ ∇ f ) + | ϕf | L = Z X f ||∇ ϕ || + Z X ( f ϕ ∇ f · ∇ ϕ + ϕf ∇ f ∇ ϕ ) + | ϕf | L The first and last summands are dominated by ( C + C ) | f | L where C = sup k ϕ k and C = sup k∇ ϕ k . For the middle term, we use Cauchy-Schwarz and a constantbigger than 2 · k ϕ k · k∇ ϕ k ( f ϕ ∇ f · ∇ ϕ + ϕf ∇ f ∇ ϕ ) ≤ Z X ϕ | f |k∇ f kk∇ ϕ k ≪ Z X | f |k∇ f k≤ ( Z X | f | ) ( Z X k∇ f k ) = | f | L · ( Z M − ∆ f · f ) ≤ | f | L · ( Z X (1 − ∆) f · f ) = | f | L · | f | H ≤ | f | H That is, with an implied constant independent of f , | ϕf | H ≪ | f | H (cid:3) Proposition 19.
There is a continuous map H gafc → H lafc Proof.
The previous result proves continuity of H → H ,ϕ for every automorphictest function ϕ . Since H is the projective limit of the H ,ϕ over all automorphictest functions ϕ , the universal property of the projective limit guarantees that theremust be a continuous map H → H . (cid:3) Main Theorem: Characterization and Sparsity of discretespectrum
Recall the construction of 2 , φ ∈ C ∞ c ( R ) and let f be a spherical cuspform on GL ( k ) \ GL ( A ) with trivial central character. Let ϕ ( (cid:18) A B D (cid:19) ) = φ ( (cid:12)(cid:12)(cid:12) det A det D (cid:12)(cid:12)(cid:12) ) · f ( A ) · f ( D )extending by right K -invariance to be made spherical. Define the P , pseudo-Eisenstein series by Ψ ϕ ( g ) = X γ ∈ P k \ G k ϕ ( γg )Given g = (cid:18) A b D (cid:19) , let h = h ( g ) = | det A det D | be the height of g . The spectraldecomposition for θ in a global automorphic Sobolev space H − s is e θ = X F cfm GL h e θ, F i · F + h e θ, ih , i + X F cfm GL h e θ, Υ F i · Υ F + X F ,F cfm GL Z + i ∞ − i ∞ h e θ, E , F ,F ,s i · E , F ,F ,s ds + X F cfm GL Z + i ∞ − i ∞ h e θ, E , F ,s i · E , F ,s ds + X F cfm GL Z ρ + i ∞ ρ − i ∞ h e θ, E , , F ,λ i · E , , F ,λ dλ + Z ρ + i a ∗ min h e θ, E λ i · E λ dλ where F and F ′ are cuspforms on GL (2) and the Υ F ’s are Speh forms. We areinterested in the subspace V of L ( Z A G k \ G A ) spanned by 2 , f and f , where f is everywhere locally spherical. Let D a,f be the subspace of V consisting of the L -closure of the span of 2 , f and f with test function ϕ supportedon h ( g ) < a and whose constant terms have support on h ( g ) < a .Let ∆ a be ∆ restricted to D a,f , and let e ∆ a be the Friedrichs extension of ∆ a to aself-adjoint (unbounded) operator. By construction, the domain of e ∆ a is containedin a Sobolev space Φ +1 a , defined as the completion of D a,f with respect to the +1-Sobolev norm h f, f i = h (1 − ∆) f, f i L . We recall [M-W,141-143], and [Garrett2014] the Theorem 3.
The inclusion Φ a → Φ a , from Φ a with its finer topology, is compact,so that the space Φ a decomposes discretely. ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 39
Indeed, let L η be the subspace of L (PGL \ PGL ( R ) / O ( R )) with all constantterms vanishing above given fixed heights, specified by a real-valued function η onsimple positive roots described below. By its construction, the resolvent of theFriedrichs extension maps continuously from L to the automorphic Sobolev space H = H (PGL ( Z ) \ PGL ( R ) / O ( R )) with its finer topology. Letting H η = H ∩ L η with the topology of H , it suffices to show that the injection H η → L η is compact. To prove this compactness, we show that the image of the unit ball of H η is totally bounded in L η .Let A be the standard maximal torus consisting of diagonal elements of GL , Z the center of G , and K = O ( R ). Let A + be the subgroup of A R with positivediagonal entries, and let Z + = Z R ∩ A + . A standard choice of positive simple rootsis Φ = { α i ( a ) = a i a i +1 i = 1 , . . . , r − } where a is the matrix a = a a a
00 0 0 a Let N min be the unipotent radical of the standard minimal parabolic P min con-sisting of upper-triangular elements of G . For g ∈ G R , let g = n g a g k g be thecorresponding Iwasawa decomposition with respect to P min . From basic reductiontheory, the quotient Z R G Z \ G R is covered by the Siegel set S = N min Z \ N min R · Z + \ A +0 · K = Z + N min Z (cid:15)(cid:8) g ∈ G : α ( a g ) ≥ √ , for all α ∈ Φ (cid:9) Further, there is an absolute constraint so that Z S | f | ≪ Z Z R G Z \ G R | f | for all f . For a non-negative real-valued function η on the set of simple roots, let X αη = { g ∈ S : α ( a g ) ≥ η ( α ) } for α ∈ Φ. Let C η = { g ∈ S : α ( a g ) ≤ η ( α ) for all α ∈ Φ } This is a compact set, and S = C η ∪ [ α ∈ Φ X αη For α ∈ Φ, let P α be the standard maximal proper parabolic whose unipotentradical N α has Lie algebra n α including the α th root space. That is, for α ( a ) = a i a i +1 ,the Levi component M α of P α is GL i × GL − i . As before, let ( c P f )( g ) denote theconstant term along a parabolic P of a function f on G Z \ G R . For P = P α , write c α = c P . For a non-negative real-valued function η on the set of simple roots, the space of square-integrable functions with constant terms vanishing above heights η is L η = { f ∈ L ( Z R G Z \ G R /K ) : c α f ( g ) = 0 for α ( a g ) ≥ η ( α ) , for all α ∈ Φ } Vanishing is meant in a distributional sense. The global automorphic Sobolev space H is the completion of C ∞ c ( Z R G Z \ G R ) K with respect to the H Sobolev norm | f | H = Z Z R G Z \ G R (1 − ∆) f · f where ∆ is the invariant Laplacian descended from the Casimir operator Ω. Put H η = H ∩ L η . Proposition 20.
The Friedrichs self-adjoint extension e ∆ η of the restriction of thesymmetric operator ∆ to test functions in L η has compact resolvent, and thus haspurely discrete spectrumProof. Let A +0 = { a ∈ A : α ( a ) ≥ √
32 : for all α ∈ Φ } We grant ourselves that we can control smooth cut-off functions:
Lemma 1.
Fix a positive simple roots α . Given µ ≥ η ( α ) + 1 , there are smoothfunctions ϕ αµ for α ∈ Φ and ϕ µ such that: all these functions are real-valued, takingvalues between and , ϕ is supported in C µ +1 , and ϕ α µ is supported in X αµ , and ϕ µ + P α ϕ αµ = 1 . Further, there is a bound C uniform in µ ≥ η ( α ) + 1 , such that | f · ϕ µ | H ≤ C · | f | H , and | f · ϕ αµ | H ≤ C · | f | H for all µ ≥ η ( α ) + 1 . Then the key point is
Claim 1.
For α ∈ Φ , lim µ →∞ (cid:18) sup | f | L | f | H (cid:19) = 0 where the supremum is taken over f ∈ H η and support ( f ) ⊂ X αµ . Temporarily grant the claim. To prove total boundedness of H η → L η , given ǫ >
0, take µ ≥ η ( α ) + 1 for all α ∈ Φ, large enough so that f · ϕ αµ | L < ǫ , for all f ∈ H η , with | f | H ≤
1. This covers the images { f · ϕ αµ : f ∈ H η } with α ∈ Φ withcardΦ open balls in L of radius ǫ . The remaining part { f · ϕ µ : f ∈ H η } consistsof smooth functions supported on the compact C µ . The latter can be covered byfinitely-many coordinate patches φ i : U i → R d . Take smooth cut-off functions ϕ for this covering. The functions ( f · ϕ i ) ◦ φ − i on R d have support strictly insidea Euclidean box, whose opposite faces can be identified to form a flat d-torus T d .The flat Laplacian and the Laplacian inherited from G admit uniform comparisonon each φ ( U i ) , so the H ( T d )-norm of ( f · ϕ ) ◦ φ − i is uniformly bounded by the H -norm. The classical Rellich lemma asserts compactness of H ( T d ) → L ( T d )By restriction, this gives the compactness of each H · ϕ i → L . A finite sum ofcompact maps is compact, so H · ϕ µ → L is compact. In particular, the image ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 41 of the unit ball from H admits a cover by finitely-many ǫ -balls for any ǫ > ǫ -balls with the card(Φ) balls covers the image of H η in L η by finitely-many ǫ -balls, proving that H η → L is compact.It remains to prove the claim. Fix α = α i ∈ Φ, and f ∈ H η with support inside X αmu for µ ≫ η ( α ). Let N = N α , P = P α , and let M = M α be the standard Levicomponent of P . Use exponential coordinates n x = (cid:18) i x − i (cid:19) In effect, the coordinate x is in the Lie algebra n of N R . Let Λ ⊂ n be the latticewhich exponentiates to N Z . Give η the natural inner product h , i invariant underthe (Adjoint) action of M R ∩ K that makes root spaces mutually orthogonal. Fixa nontrivial character ψ on R / Z . We have the Fourier expansion f ( n x m ) = X ξ ∈ Λ ′ ψ h x, ξ i ˆ f ξ ( m )with n ∈ N R , m ∈ M R , and Λ ′ is the dual lattice to Λ in n with respect to h , i , andˆ f ξ ( m ) = Z n \ Λ ˆ ψ h x, ξ i f ( n x m ) dx Let ∆ n be the flat Laplacian on n associated to the inner product h , i normalizedso that ∆ n ψ h x, ξ i = −h ξ, ξ i · ψ h x, ξ i Let U = M ∩ N min . Abbreviating A u = Ad u , | f | L ≤ Z S | f | = Z Z + \ A +0 R U Z \ U R Z A − u Λ \ n | f ( un x a ) | dx du daδ ( a )with Haar measures dx , du , da , and where δ is the modular function of P R . Usingthe Fourier expansion, f ( un x a ) = f ( un x u − · ua ) = X ξ ∈ λ ′ ψ h A u x, ξ i · ˆ f ξ ( ua )= X ξ ∈ Λ ′ ψ h x, A ∗ u ξ i · ˆ f ξ ( ua )Then − ∆ n f ( un x a ) = X ξ ∈ Λ ′ h A ∗ u ξ, A ∗ u ξ i · ψ h x, A ∗ u ξ i · ˆ f ξ ( ua )The compact quotient U Z \ U R has a compact set R of representatives in U R , so thereis a uniform lower bound for 0 = ξ ∈ Λ ′ :0 < b ≤ inf u ∈ R inf = ξ ∈ Λ ′ h A ∗ u ξ, A ∗ u ξ i By Plancherel applied to the Fourier expansion in x , using the hypothesis thatˆ f = 0 in X αµ , Z A − µ Λ \ n | f ( un x a ) | dx = Z A − u Λ \ n | f ( un x u − · ua ) | dx = X ξ ∈ Λ ′ | ˆ f ξ ( ua ) | ≤ b − X ξ ∈ Λ ′ h A ∗ u ξ, A ∗ u ξ i · | ˆ f ξ ( ua ) | = X ξ ∈ Λ ′ − c ∆ n f ξ ( ua ) · ˆ f ( ua ) = Z u − Λ u \ n − ∆ n f ( un x u − · ua ) · f ( un x u − · ua ) dx = Z A − u Λ \ n − ∆ n f ( un x a ) · f ( un x a ) dx Thus, for f with ˆ f (0) = 0 on α ( g ) ≥ η , | f | L ≪ Z Z + \ A +0 Z U Z \ U R Z A − u Λ \ n − ∆ n f ( un x a ) · f ( un x a ) dx du daδ ( a )Next, we compare ∆ n to the invariant Laplacian ∆. Let g be the Lie algebra of G R , with non-degenerate invariant pairing h u, v i = trace( uv )The Cartan involution v → v θ has +1 eigenspace the Lie algebra k of K , and − s , the space of symmetric matrices.Let Φ N be the set of positive roots β whose root space g β appears in n . For each β ∈ Φ N , take x β ∈ g β such that x β + x θβ ∈ s , x β − x θβ ∈ k , and h x β , x θβ = 1: for β ( a ) = a i a j with i < j , x β has a single non-zero entry, at the ij th place. LetΩ ′ = X β ∈ Φ N ( x β x θβ + x θβ x β )Let Ω ′′ ∈ U g be the Casimir element for the Lie algebra m of M R , normalized sothat Casimir for g is the sum Ω = Ω ′ + Ω ′′ . We rewrite Ω ′ to fit the Iwasawacoordinates: for each β , x β x θβ + x θβ x β = 2 x β x θβ +[ x θβ , x β ] = 2 x β − x β ( x β − x θβ )+[ x θβ , x β ] ∈ x β +[ x θβ , x β ]+ k Therefore, Ω ′ = X β ∈ Φ N x β + [ x θβ , x β ] modulo k The commutators [ x θβ , x β ] ∈ m . In the coordinates un x a with U g acting on theright, x β ∈ n is acted on by a before translating x , by un x a · e tx β = un x · e tβ ( a ) · x β · a = un x + β ( a ) x β a That is, x β acts by β ( a ) · ∂∂x β .For two symmetric operators S, T on a not-necessarily-complete inner productspace V , write S ≤ T when h Sv, v i ≤ h
T v, v i for all v ∈ V . We say that a symmetric operator T is non-negative when 0 ≤ T .Since a ∈ A +0 , there is an absolute constant so that α ( a ) ≥ µ implies β ( a ) ≫ µ .Thus, − ∆ n = − X β ∈ Φ N ∂ ∂x β ≪ µ · − X β ∈ Φ N x β on C ∞ c ( X αµ ) K with the L inner product. We claim that − X β ∈ Φ N [ x θβ , x β ] − Ω ′′ ≥ ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 43 on C ∞ c ( X αµ ) K . From this, it would follow that − ∆ n ≪ µ · − X β ∈ Φ N x β ≤ µ · − X β ∈ Φ N x β − X β ∈ Φ N [ x θβ , x β ] − Ω ′′ = 1 µ · ( − ∆)Then, for f ∈ H η with support in X αµ we would have | f | L ≪ Z S − ∆ n f · f ≪ µ Z S − ∆ f · f ≪ µ Z Z R G Z \ G R − ∆ f · f ≪ µ · | f | H Taking µ large makes this small. Since we can do the smooth cutting-off to affectthe H norm only up to a uniform constant, this would complete the proof of totalboundedness of the image in L of the unit ball from H η .To prove the claimed nonnegativity of T = − P β ∈ Φ N [ x θβ , x β ] − Ω ′′ , exploit theFourier expansion along N and the fact that x ∈ n does not appear in T : notingthat the order of coordinates n x u differs from that above, Z Z + \ A +0 Z U Z \ U R Z Λ \ n T f ( n x ua ) f ( n x ua ) dx du daδ ( a )= Z Z + \ A +0 Z U Z \ U R Z Λ \ n T X ξ ψ h x, ξ i ˆ f ( ua ) X ξ ′ ψ h x, ξ ′ i ˆ f ( ua ) dx du daδ ( a )Only the diagonal summands survive the integration in x ∈ n , and the exponentialscancel, so this is Z Z + \ A +0 Z U Z \ U R X ξ T ˆ f ξ ( ua ) · ˆ f ( ua ) du daδ ( a )Let F ξ be a left- N R -invariant function taking the same values as ˆ f ξ on U R A + K ,defined by F ξ ( n x uak ) = ˆ f ξ ( uak )for n x ∈ N , u ∈ U , a ∈ A + , k ∈ K . Since T does not involve n and since F ξ is left N R -invariant, T ˆ f ξ ( ua ) = T F ξ ( n x ua ) = − ∆ F ξ ( n x ua )and then Z Z + \ A +0 Z U Z \ U R X ξ T ˆ f ( ua ) · ˆ f ξ ( ua ) du daδ ( a ) = Z Z + \ A +0 Z U Z \ U R X ξ − ∆ F ξ ( ua ) · F ξ ( ua ) du daδ ( a )The individual summands are not left- U Z -invariant. Since ˆ f ξ ( γg ) = ˆ f A ∗ γ ξ ( g ) for γ normalizing n , we can group ξ ∈ Λ ′ by U Z orbits to obtain U Z subsums and thenunwind. Pick a representative ω for each orbit [ ω ], and let U ω be the isotropysubgroup of ω in U Z , so Z U Z \ U R X ξ − ∆ F ξ ( ua ) · F ξ ( ua ) du = X [ ω ] Z U Z \ U R X ξ ∈ [ ω ] − ∆ F ξ ( ua ) · F ξ ( ua ) du = X [ ω ] Z U Z \ U R X γ ∈ U ω \ U Z − ∆ F A ∗ γ ω ( ua ) · F A ∗ γ ω ( ua ) du = X ω Z U ω \ U R − ∆ F ω ( ua ) · F ω ( ua ) du Then Z Z + \ A +0 Z U Z \ U R X ξ − ∆ F ξ ( ua ) · F ξ ( ua ) du = X ω Z Z + \ A +0 Z U ω \ U R − ∆ F ω ( ua ) · F ω ( ua ) du daδ ( a )Since − ∆ is a non-negative operator on functions on every quotient Z + N R U ω \ G R /K of G R /K , each double integral is non-negative, proving that T is non-negative.This completes the proof that H η → L η is compact, and thus, that the Friedrichsextension of the restriction of ∆ to test functions in L η has purely discrete spectrum. (cid:3) Since the pseudo-Eisenstein series appearing in the spectral decomposition areorthogonal to all other automorphic forms appearing in the spectral expansion inevery Sobolev space, we can speak of the projection θ of the period distribution e θ to the subspace V of L ( Z A G k \ G A ). That is, θ = h e θ, Υ f i · Υ f + 14 πi Z + i ∞ − i ∞ h e θ, E f,f,s i · E f,f,s where h , i is the pairing of distributions with functions. To check θ is well-defined,we must check that, for every square-integrable automorphic form f not in the L -span of 2 , h θ, f i = 0To this end, let us check it for 3 , f ,φ with cuspidaldata f and test function data φ . Then h θ, Ψ f ,φ i = * h e θ, Υ f i · Υ f + h e θ, Ψ , f,f,φ i · Ψ , f,f,φ , Ψ , f ,φ + This is * h e θ, Υ f i · Υ f , Ψ , f ,φ + + * h e θ, Ψ , f,f,φ i · Ψ , f,f,φ , Ψ , f ,φ + = 0The Speh form Υ f is a ∆-eigenfunction. Furthermore, it is orthogonal to 3 , L . Indeed, using the adjunction relation, h Υ f , Ψ , ϕ f ,φ i = h c , Υ f , ϕ f ,φ i Since the 3 , f is zero, the above is zero. There-fore, the Speh form Υ f is orthogonal to 3 , , , h θ, Ψ f ,φ i = * h e θ, Υ f i · Υ f + h e θ, Ψ , f,f,φ i · Ψ , f,f,φ , Ψ , f ,φ + = 0We now prove that for a 2 , , ϕ f ,φ ,φ with cuspidaldata f and test functions φ and φ , that h θ, Ψ ϕ f ,φ ,φ i = 0As before, this is just * h e θ, Υ f i · Υ f , Ψ , , ϕ f ,φ ,φ + + * h e θ, Ψ , f,f,φ i · Ψ , f,f,φ , Ψ , , f ,φ ,φ + ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 45
The second term is zero, because the pseudo-Eisenstein series are orthogonal. Thefirst term gives zero. Indeed h Υ f , Ψ , , ϕ f ,φ ,φ i = h c , , Υ f , ϕ f ,φ ,φ i = 0since the 2 , , f is zero.Let ∆ θ be ∆ with domain ker θ ∩ V . We will show that parameters for thediscrete spectrum λ s,f = s f ( s f −
2) + s ( s −
1) (if any) of the Friedrichs extension e ∆ θ are contained in the zero-set of the L -function appearing in the period.To legitimize applying the distribution θ to cuspidal-data Eisenstein series E f,f,s requires discussion of local automorphic Sobolev spaces. Recall that θ is in the − − E f,f,s is in the +1 local automorphic Sobolev space, we can apply θ to it. Theorem 4.
For Re ( w ) = , if the equation (∆ − λ w,f ) u = θ has a solution u ∈ V , then θE f,f,w = 0 . Conversely, if θE f,f,w = 0 for Re ( w ) = , thenthere is a solution to that equation in V , and the solution is unique with spectralexpansion u = θ (Υ f ) · Υ f ( λ Υ f − λ w ) + 14 πi Z ( ) θE f,f, − s λ s,f − λ w,f · E f,f,s ds convergent in V +1 Proof.
The condition θ ∈ V − is that Z R | θE f,f, − s | t dt < ∞ Thus, u ∈ V +1 , and u has a spectral expansion of the form u = A f · Υ f + 14 πi Z ( ) A s · E f,f, − s ds with t → A + it in L ( R ). The distribution θ has spectral expansion in V − , θ = θ (Υ f ) · Υ f + 14 πi Z ( ) θE f,f, − s · E f,f,s ds We describe the vector-valued weak integrals of [Gelfand 1936] and [Pettis 1938]and summarize the key results. We follow [Bourbaki 1963].
Definition 10.
For
X, µ a measure space and V a locally convex, quasi-completetopological vector space, a Gelfand-Pettis (or weak) integral is a vector-valued inte-gral C c ( X, V ) → V denoted f → I f such that for all α ∈ V ∗ , we have α ( I f ) = Z X α ◦ f dµ where the latter is the usual scalar-valued Lebesgue integral. Proposition 21.
Hilbert, Banach, Frechet, and LF spaces together with their weakduals are locally convex, quasi-complete topological vector spaces.
Proposition 22.
Gelfand-Pettis integrals exist and are unique.
Proposition 23.
Any continuous linear operator between locally convex, quasi-complete topological vector spaces T : V → W commutes with the Gelfand-Pettisintegral: T ( I f ) = I T f
Note that E f,f,s lies in a local automorphic Sobolev space. By the Gelfand-Pettistheory, if T : V → W is a continuous linear map of locally convex topological vectorspaces, where convex hulls of compact sets in V have compact closures and if f isa continuous, compactly-supported V -valued function on a finite measure space X ,then the W -valued function T ◦ f has a Gelfand-Pettis integral, and T (cid:18)Z X f (cid:19) = Z X T ◦ f Let V = H ( X ). Note that V is a locally convex, quasi-complete topologicalvector space since it is the completion of C ∞ c ( X ) with respect to a family of semi-norms. Given a compactly-supported distribution θ ∈ H − ( X ), θ extends to acontinuous linear functional θ ∈ H − ( X ), by section 7. Since θ is a continuousmapping θ : H − ( X ) → C , given a continuous, compactly-supported H ( X )-valued function f , θ Z X f = Z X θ ◦ f Gelfand-Pettis theory allows us to move θ inside the integral. Thus( λ Υ f − λ w ) A f = θ (Υ f )and ( λ s,f − λ w,f ) · A s = θE f,f, − s The latter equality holds at least in the sense of locally integrable functions. Letting w = + iτ , by Cauchy-Schwarz-Bunyakowsky, for any ǫ > τ + ǫ Z τ − ǫ | θE f,f − it | dt = τ + ǫ Z τ − ǫ | ( λ + it,f − λ + iτ,f ) A + it | dt Using s = + it and rewriting the difference of eigenvalues gives us equality of theabove with Z τ + ǫτ − ǫ | ( t − τ )( t − τ ) A + it | dt ≤ Z τ + ǫτ − ǫ | t − τ | dt · Z τ + ǫτ − ǫ | ( t − i + τ ) A + it | dt ≪ ǫ The function t → θE f,f, + it is continuous, in fact s → θE f,f,s is meromorphic, since θ is compactly supported (see [Grothendieck 1954] and [Gar-rett 2011 e]), so θE f,f, − w = 0Conversely, when θE − w = 0, the function t → θE f,f, − it ( λ + it − λ w ) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 47 is continuous and square-integrable, assuring H -convergence of the integral u = θ (Υ f ) · Υ f λ Υ f − λ w,f + 14 πi Z ( ) θE f,f, − s · E f,f,s ( λ s,f − λ w,f ) ds this spectral expansion produces a solution of the differential equation. Any solutionin V +1 admits such an expansion, and the coefficients are uniquely determined,giving uniqueness. (cid:3) Let X a = { A, D ∈ GL : | det A det D | = a } . Let H be the subgroup of GL × GL consisting of pairs ( B, C ) so that | det B · det C | = 1. The group H acts simplytransitively on X a , so X a has an H -invariant measure. Fix GL cuspforms f and f and define η a F = Z Z R H k \ X a c P ( F ( a )) · f ( A ) · f ( D ) dx Proposition 24.
Take Re ( w ) = . For a ≫ such that the support of e θ is below h = a , the constant term c P u of a solution u ∈ V +1 to (∆ − λ w,f ) u = θ vanishesfor height h ≥ a .Proof. Let η a,f ⊗ f be the functional above. This functional is in H − − ǫ for all ǫ >
0. Thus, for u ∈ H +1 , η a,f ⊗ f u = η a,f ⊗ f θ (Υ f ) · Υ f ( λ Υ f − λ w ) h , i + 14 πi Z ( ) θE f,f, − s λ s − λ w · E f,f,s ds ! We can break up the integral into two tails and a truncated finite part. Thetruncated finite part is a continuous, compactly-supported integral of functions ina local automorphic Sobolev space, so Gelfand-Pettis theory allows us to movecompactly-supported distributions inside the integral. The tails are spectral ex-pansions of functions in H +1 , and since H +1 embeds into a local automorphicSobolev space, the Gelfand-Pettis theory applies there also, allowing us to movethe distribution inside the integral. θ (Υ f ) · η a,f ⊗ f (Υ f )( λ Υ f − λ w,f ) + 14 πi Z ( ) θE f,f, − s · η a,f ⊗ f E f,f,s λ s,f − λ w,f ds This is θ η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) + 14 πi Z ( ) η a,f ⊗ f E f,f,s ( λ s,f − λ w,f ) · E f,f, − s ds ! which is θ η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) + 14 πi Z ( ) C ( a − s + c − s a s )( λ s,f − λ w,f ) · E f,f, − s ds ! where C = Z Z R H k \ X a f ( A ) · f ( D ) · f ( A ) · f ( D ) dx Since θ has compact support below h = a , the last integral need be evaluated onlyfor h ≤ a . Using the functional equation c − s E f,f,s = E f,f, − s we see Z ( ) c − s a s ( λ s,f − λ w,f ) · E f,f,s ds = Z ( ) a − s ( λ s,f − λ w,f ) · E f,f,s ds by changing variables. Thus, for g with h ( g ) ≤ a , the integral can be evaluated byresidues of vector-valued holomorphic functions as in [Grothendieck] and [Garrett2011 e]. θ η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) + 14 πi Z ( ) C ( a − s + c − s a s )( λ s,f − λ w,f ) · E f,f, − s ds ! = θ η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) + 12 πi Z ( ) C ( a − s )( λ s,f − λ w,f ) · E f,f,s ds ! Consider the integral Z ( ) a − s θE f,f,s ( λ s,f − λ w,f ) ds With s = α + iT , consider a rectangle with vertices ± iT and T ± iT . Let γ bethe line segment from + iT to T + iT . Let γ be the line segment from T + iT to T − iT , and let γ be the line segment from T − iT to − iT . We invoke ourassumed subconvexity bound θE f,f,s ≪ | s | − ǫ . Then we get an estimate (cid:12)(cid:12) Z γ a − s · θE f,f,s λ s,f − λ w,f ds (cid:12)(cid:12) ≪ a − s · | s | − ǫ | λ s,f − λ w,f | · ( T −
12 )since γ has length T − . Then, a − s · | s | − ǫ | λ s,f − λ w,f | · ( T −
12 ) ≤ a − s · | s | − ǫ | λ s,f − λ w,f | · ( | s | −
12 ) → T → ∞ , since the denominator is a degree 2 polynomial in s , while the numeratoris a polynomial of degree 2 − ǫ . Likewise, for the curve γ , we get an estimate (cid:12)(cid:12) Z γ a − s · θE f,f,s λ s,f − λ w,f ds (cid:12)(cid:12) ≪ a − s · | s | − ǫ | λ s,f − λ w,f | · (2 T )since γ has length 2 T . Then, a − s · | s | − ǫ | λ s,f − λ w,f | · ( T −
12 ) ≤ a − s · | s | − ǫ | λ s,f − λ w,f | · (2 | s | ) → T → ∞ , since the denominator is a degree 2 polynomial in s , while the numeratoris a polynomial of degree 2 − ǫ . A similar argument shows that the integrals along γ and γ go to 0 as T →
0. Therefore, the original integral Z ( ) a − s θE f,f,s ( λ s,f − λ w,f ) ds = − πi (sum of residues in the right half-plane) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 49
This implies12 πi Z ( ) a − s · C · θE f,f,s ( λ s,f − λ w,f ) ds = − (sum of residues in the right half-plane)The Eisenstein series E f,f,s has a simple pole at s = 1 ([MW] and [Garrett 2011f]), with residue η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f )Therefore θE f,f,s has residue at s = 1 given by θ (cid:18) η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) (cid:19) Thus,12 πi Z ( ) a − s · C · θE f,f,s ( λ s,f − λ w,f ) ds = − θ (cid:18) η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) (cid:19) + a − w − w · C · θE f,f, − w Returning to the original equation, θ η a,f ⊗ f (Υ f ) · Υ f ( λ Υ f − λ w,f ) + 12 πi Z ( ) C ( a − s )( λ s,f − λ w,f ) · θE f,f,s ds ! = a − w − w · C · θE − w,f,f Since θE − w,f,f = 0, we are done. (cid:3) Recall that Φ a decomposes discretely, with (square-integrable) eigenfunctionsconsisting of truncated Eisenstein series ∧ a E s j ,f,f of Eisenstein series for s j suchthat a s · f ( A ) · f ( D ) + a − s · c s · f ( A ) · f ( D ) = 0where ( A, D ) ∈ X a , and finitely-many other eigenfunctions. In fact, these trun-cations are in H − ǫ for every ǫ >
0, since they are solutions to the differentialequation (∆ − λ w,f ) u = η a,f ⊗ f . There are finitely-many other eigenfunctions inaddition to these truncated Eisenstein series.Let S denote the operator S = 1 − ˜∆ a with dense domain in Φ +1 a as before. Then S is an unbounded, symmetric, densely-defined operator. We have the continuousinjections Φ +1 a → Φ a → Φ − a Then S extends by continuity to S : Φ a → Φ − a . Since we have the naturalinclusion j : Φ a → H +1 taking adjoints produces an inclusion j ∗ : H − → Φ − a Let j ∗ θ denote the image of θ under this mapping.Then we can solve the differentialequation ( S − λ w ) u = j ∗ θ because j ∗ θ ∈ Φ − a . Proposition 25.
Take a ≫ such that the (compact) support of θ is below height a . If necessary, adjust a so that θE s j = 0 for any s j such that a s j · f ( A ) · f ( D ) + a − s j · c s j · f ( A ) · f ( D ) = 0 where ( A, D ) ∈ X a . For w not among the s j , the equation ( S − λ w,f ) v = j ∗ θ has aunique solution v w ∈ V ∩ Φ a , this solution lies in H +1 , and has spectral expansion v w = X j θE f,f, − s j λ s j ,f − λ w,f · ∧ a E f,f,s j || ∧ a E f,f,s j || Proof.
As before, any solution is in H +1 , since θ ∈ H − . The solution v ∈ V ∩ Φ a has an expansion in terms of the orthogonal bases ∧ a E s j ,f,f , v w = X j A j ∧ a E s j ,f,f || ∧ a E s j ,f,f || convergent in H +1 Thus, j ∗ θ = ( S − λ w,f ) v w = X j ( λ s j ,f − λ w,f ) A j ∧ a E f,f,s j || ∧ a E f,f,s j || Indeed, since the compact support of e θ is below h = a , the projection θ to V is inthe H − completion of V ∩ Φ a . Therefore, the expansion of j ∗ θ in terms of truncatedEisenstein series must be j ∗ θ = X j θE f,f,s j · ∧ a E f,f,s j || ∧ a E f,f,s j || noting that θE f,f,s j = θ ∧ a E f,f,s j . Thus, the coefficients A j are uniquely deter-mined, also giving uniqueness. (cid:3) Proposition 26.
Solutions w to the equation θv w = 0 all lie on ( + i R ) ∪ [0 , ,and there is exactly one such between each pair s j , s j +1 of adjacent solutions of (cid:12)(cid:12) det A det D (cid:12)(cid:12) s + (cid:12)(cid:12) det A det D (cid:12)(cid:12) − s · Λ(2 s − , π ⊗ π ′ )Λ(2 s, π ⊗ π ′ ) = 0 . Proof.
Using the expansion of v w in H +1 in terms of the truncated Eisenstein series,and that of θ ∈ H − in those terms, θv w = X j | θE − s j ,f,f | ( λ s j ,f − λ w,f ) · k ∧ a E s j ,f,f k Since every λ s j ,f is real, for λ w,f / ∈ R , the imaginary part of θv w is easily seen tobe nonzero, thus θv w = 0. Thus, any solution lies in ( + i R ) ∪ R . For λ w > w ∈ ( + i R ) ∪ [0 , w ) = with λ s j +1 ,f < λ w,f < λ s j ,f . Note that θv w ∈ R for such w . For w on the vertical line segment between s j and s j +1 , all summands butthe j th and ( j + 1) th are bounded. As w → s j , 0 < λ s j ,f − λ w,f → + and λ s j +1 ,f − λ w,f is bounded. As w → s j +1 , 0 > λ s j +1 ,f − λ w → − and λ s j − λ w isbounded. Since w → v w is a holomorphic H +1 -valued function, θv w is continuous. ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 51
By the intermediate value theorem, there is at least one w between s j and s j +1 with θv w = 0.To see that there is at most one w giving θv w = 0 between each adjacent pair s j , s j +1 again use holomorphy of w → v w , and take the derivative in w : ∂∂w θv w = X j | θE − s j ,f | · (2 w − λ s j ,f − λ w,f ) · k ∧ a E s j ,f k Everything is positive real except the purely imaginary 2 w −
1, because, in fact, theheight a was adjusted so that no θE − s j ,f vanishes. That is, away from poles, thederivative is non-vanishing, so all zeros are simple. Returning to the proof of thetheorem: suppose u ∈ V such that ( S − λ w ) u = j ∗ θ with Re( w ) = . For u to bean eigenfunction for e ∆ θ requires θu = 0 by the nature of the Friedrichs extension.From above, η a u vanishes above a height a depending on the compact support of˜ θ . Thus, u ∈ V ∩ Φ a , so u must be the solution v w expressed as a linear combinationof truncated Eisenstein series, and θv w = 0. Since there is at most one w giving θv w = 0 between any two adjacent roots s j of (cid:12)(cid:12) det A det D (cid:12)(cid:12) s + (cid:12)(cid:12) det A det D (cid:12)(cid:12) − s · Λ(2 s − , π ⊗ π ′ )Λ(2 s, π ⊗ π ′ ) = 0giving the constraint. (cid:3) L-function background
Recall that the 2 , , f and f at height h = a is a s + c s a − s where c s = Λ(2(1 − s ) , f ⊗ f )Λ(2 s, f ⊗ f )A standard argument principle computation shows that the number of zeros of a s + c s a − s with imaginary parts between 0 and T > N ( T ) = Tπ log( T πe + T log a + O (log T ))All zeros of a s + c s a − s are on Re( s ) = for a ≥
1. Recall ([Iwaniec-Kowalski,p.115]) thatlog L (1 + iu, f ⊗ f ) − log L (1 + it, f ⊗ f ) = O ( log t log log t ) · ( u − t )for u ≥ t . Lemma 2.
The gaps between consecutive zeros of a s + c s a − s at height greaterthan or equal to T are π log T + O ( 1 log log T ) Proof.
The condition for the vanishing of a s + c s a − s can be rewritten asΛ(2 s, f ⊗ f )Λ(2(1 − s ) , f ⊗ f ) = − s, f ⊗ f ) = π − s · Γ( s + µ − ν s − µ + ν s − µ − ν s + µ + ν · L ( s, f ⊗ f )where µ is the parameter for the principal series I µ generated by f , while ν is theparameter for the principal series generated by f . Therefore, with s on the criticalline, we have − it + µ − ν )Γ( it − µ + ν )Γ( it − µ − ν )Γ( it + µ + ν )Γ( − it + µ − ν )Γ( − it − µ + ν )Γ( − it − µ − ν )Γ( − it + µ + ν ) π − it L (1 + 2 it, f ⊗ f ) L (1 − it, f ⊗ f )All the factors on the right-hand side are of absolute value 1. The count of zerosas t = Im( s ) moves from 0 to T is the number of times the right-hand side assumesthe value −
1. Regularity is entailed by upper and lower bounds for the derivativeof the logarithm of that right-hand side, for large t . Observe that ddt Im log Γ( a + it )Γ( a − it ) = 2 ddt Im log Γ( a + it )From the Stirling asymptotic,log Γ( s ) = ( s −
12 )log s − s + 12 log 2 π + O δ ( 1 s )in Re( s ) ≥ δ >
0. From this, we havelog Γ( a + it ) = it log ( a + it ) − ( a + it ) + 12 log 2 π + O δ ( 1 a + it )= it (cid:0) i ( π + O ( 1 t )) + log t + O ( 1 t ) (cid:1) − ( a + it ) + 12 π log 2 π + O δ ( 1 a + it )Therefore, Im log Γ( a + it ) = t log t − t + O ( 1 t )Consider, for 0 < δ ≪ t ,Im log Γ( a + i ( t + δ )) − Im log Γ( a + it ) = (cid:0) ( t + δ )log ( t + δ ) − ( t + δ ) (cid:1) − ( t log t − t )+ O ( 1 t )Which is = δ log t − ( t + δ ) δt − δ + O δ ( 1 t ) = δ log t − δ + O δ ( 1 t )In particular, for 0 < δ ≤ t ,Im log Γ( a + i ( t + δ )) − Im log Γ( a + it ) = δ log t + O ( 1log t )Let f ( t ) = Γ( it + µ − ν )Γ( it − µ + ν )Γ( it − µ − ν )Γ( it + µ + ν )Γ( − it + µ − ν )Γ( − it − µ + ν )Γ( − it − µ − ν )Γ( − it + µ + ν )Then using the calculation above,Im log f ( t + δ ) − Im log f ( t ) = 4 δ log t + O ( 1log t ) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 53
The result on L (1 + it, f ⊗ f ) quoted above giveslog L (1 + 2 i ( t + δ ) , f ⊗ f ) − log L (1 + 2 it, f ⊗ f ) = O ( log t log log t )Therefore,Im log Λ(1 + 2 i ( t + δ ) , f ⊗ f ) − Im log Λ(1 + 2 it, f ⊗ f ) = 4 δ log t + O ( log t log log t ) · δ The presence of the 4 being due to the four factors of Γ appearing. Thus, if t givesa 0 of the constant term, the next t ′ = t + δ giving a zero of the constant term mustbe such that 4 δ log t + O ( log t log log t ) · δ ≥ π On the other hand, when that inequality is satisfied, then the unit circle will havebeen traversed, and a zero of the constant term occurs. (cid:3)
Since periods of automorphic forms produce L -functions, it is anticipated that θE s will produce a self-adjoint, degree 4 L -function, with a corresponding pair-correlation conjecture. That is, given ǫ >
0, there are many pairs of zeros of θE s within ǫ of each other. The previous section exhibits the zeros w of θE s asparamaters of the discrete spectrum of e ∆ θ . Since parameters of the discrete spectrainterlace with the zeros s j of a s + c s a − s , and these are regularly spaced by theargument above, the discrete spectrum is presumably sparse.11. Appendix I: Harmonic Analysis on GL Given a parabolic P in G = GL and a function f on Z A G k \ G A , the constantterm of f along P is c P f ( g ) = Z N k \ N A f ( ng ) dn where N is the unipotent radical of P . An automorphic form satisfies the Gelfandcondition if, for all maximal parabolics P , the constant term along P is zero. Ifsuch a function is also z -finite, and K -finite, it is called a cuspform. Since theright G A -action commutes with taking constant terms, the space of functions meet-ing Gelfand’s condition is G A -stable, so is a sub-representation of L ( Z A G k \ G A ).Godement, Selberg, and Piatetski-Shapiro showed that integral operators on thisspace are compact. Specifically, for ϕ ∈ C ∞ c ( G ), the operator f → ϕ · f gives acompact operator from L ( Z A G k \ G A ) to itself. Here,( ϕ · f )( y ) = Z Z A G k \ G A ϕ ( x ) · f ( yx ) dx By the spectral theorem for compact operators, this sub-representation decomposesinto a direct sum of irreducibles, each appearing with finite multiplicity. To decom-pose the remainder of L demands an understanding of the continuous spectrum,consisting of pseudo-Eisenstein series. We classify non-cuspidal automorphic formsaccording to their cuspidal support, the smallest parabolic on which they have anonzero constant term. In GL , there are three conjugacy classes of proper par-abolic subgroups. We will consider the standard parabolic subgroups P = GL , P , and P , the maximal parabolics, and P , , the minimal parabolic. Given the 2 , ϕ by ϕ ( (cid:18) A ∗ d (cid:19) ) = ϕ φ ,f ( (cid:18) A ∗ d (cid:19) ) = φ ( det Ad ) · f ( A )where f is a GL -cuspform and φ is a compactly-supported smooth function. Thepseudo-Eisenstein series attached to ϕ is the functionΨ , ϕ ( g ) = X P k \ G k ϕ ( γg )Likewise, given the 2 , ψ by ψ ( (cid:18) a ∗ D (cid:19) ) = ψ φ ,f ( (cid:18) a ∗ D (cid:19) ) = φ ( a det D ) · f ( D )again φ is a compactly-supported smooth function and f is a cuspform on GL .Finally, given the 1 , , ψ by ψ ( a ∗ ∗ b ∗ c ) = ψ g ,g ( a ∗ ∗ b ∗ c ) = g ( ab ) · g ( bc )where g and g are compactly-supported smooth functions. Then,Ψ ψ ( g ) = X γ ∈ P k \ G k ψ ( γ · g )Next, we exhibit the spaces spanned by non-associate pseudo-Eisenstein series asthe orthogonal complement to L cuspforms. Proposition 27.
For any square-integrable automorphic form f and any pseudo-Eisenstein series Ψ Pϕ , with P a parabolic subgroup h f, Ψ Pϕ i Z A G k \ G A = h c P f, ϕ i Z A N P A M Pk \ G A Proof.
The proof involves a standard unwinding argument. Observe that h f, Ψ Pϕ i Z A G k \ G A = Z Z A G k \ G A f ( g ) · Ψ Pϕ ( g ) dg = Z Z A G k \ G A f ( g )( X γ ∈ P k \ G k ϕ ( γ · g )) dg This is = Z Z A P k \ G A f ( g ) ϕ ( g ) dg = Z Z A N k M k \ G A f ( g ) ϕ ( g ) dg = Z Z A N A M k \ G A Z N k \ N A f ( ng ) ϕ ( ng ) dn dg = Z Z A N A M k \ G A ( Z N k \ N A f ( ng ) dn ) ϕ ( g ) dg = h c P f, ϕ i Z A N P A M Pk \ G A (cid:3) ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 55
The space spanned by P , pseudo-Eisenstein series is the same as the spacespanned by P , pseudo-Eisenstein series. More generally, pseudo-Eisenstein se-ries of associate parabolics span the same space. The L decomposition is that L ( Z A G k \ G A ) decomposes as the direct sum of cuspforms together with the spacesspanned by the minimal parabolic pseudo-Eisesntein series and 2 , GL case, we will decompose the pseudo-Eisenstein series into genuine Eisenstein series. There are several kinds of Eisensteinseries in GL . For a parabolic P , the P -Eisenstein series is E λ = X γ ∈ P k \ G k f λ ( γg )where f λ is a spherical vector in a representation λ of M P , extended to a P -representation by left N -invariance, and induced up to G . One of the chief in-gredients in the spectral decomposition for GL pseudo-Eisenstein series was thatthe Levi component was a product of copies of GL , allowing us to reduce to thespectral theory for GL . Unfortunately, this is no longer true for non-minimal para-bolic pseudo-Eisenstein series, because the Levi component contains a copy of GL .Therefore, we will first decompose the minimal parabolic pseudo-Eisenstein series.To this end, we need the functional equation of the Eisenstein series. Because of theincrease in dimension, there is more than one functional equation. The symmetriesof the Eisenstein series can be described in terms of the action of the Weyl group W on the standard maximal torus A , on its Lie algebra a , and the dual space i a ∗ .We describe the constant term and the functional equations of the Eisenstein seriesand use them in the spectral decomposition. For GL n the standard maximal torus A is the product of m copies of GL , and representations of A are products of rep-resentations of GL ; in the unramified case, these representations are just y → y s i ,for complex s i . The Weyl group W can be identified with the group of permutationmatrices in GL n . It acts on A by permuting the copies of GL , and it acts on thedual in the canonical way, permuting the s i , in the unramified case. We give apreliminary sketch of the constant term and functional equation of the Eisensteinseries, with details to be filled in later. The constant term of the Eisenstein series(along the minimal parabolic) has the form c P ( E λ ) = X w ∈ W c w ( λ ) · w λ where w λ is the image of λ under the action of w and c w ( λ ) is a constant dependingon w and λ with c ( λ ) = 1. The Eisenstein series has functional equations c w ( λ ) · E λ = E w λ for all w ∈ W We start the decomposition of Ψ ϕ by using the spectral expansion of its data ϕ . Recall that ϕ is left N A -invariant, so it is essentially a function on the Levicomponent, which is a product of copies of k × \ J . By Fujisaki’s lemma, this isthe product of a ray with a compact abelian group. We assume that the compactabelian group is trivial. So spectrally decomposing ϕ is a higher-dimensional versionof Mellin inversion. ϕ = Z h ϕ, λ i · λ dλ Winding up gives Ψ ϕ ( g ) = Z i a ∗ h ϕ, λ i · E λ ( g ) dλ In order for this decomposition to be valid, the parameters of λ must have Re( s i ) ≫
1. However, in order to use the symmetries of the functional equations, we needthe parameters to be on the critical line. In moving the contours, we pick up someresidues, which are mercifully constants. Breaking up the dual space according toWeyl chambers and changing variables,Ψ ϕ ( g ) − (residues) = X w ∈ W Z h ϕ, w λ i · E w λ ( g ) dλ Now using the functional equations,Ψ ϕ ( g ) − (residues) = X w ∈ W Z (1) h ϕ, w λ i · c w ( λ ) · E λ ( g ) dλ This is Z (1) X w ∈ W h ϕ, c w ( λ ) w λ i · E λ ( g ) dλ We recognize the constant term of the Eisenstein series and apply the adjointnessrelation X w ∈ W h ϕ, c w ( λ ) w λ i = h ϕ, c P E λ i = h Ψ ϕ , E λ i So we have Ψ ϕ ( g ) = X (1) h Ψ ϕ , E λ i · E λ ( g ) dλ + residuesOur next goal is to show that the remaining automorphic forms, namely thosewith cuspidal support P , or P , can be written as superpositions of genuine P , Eisenstein series. To do this it suffices to decompose P , and P , pseudo-Eisenstein series with cuspidal support. Let P = P , and Q = P , . We start bylooking more carefully at pseudo-Eisenstein series with cuspidal data. The data fora P pseudo-Eisenstein series is smooth, compactly-supported, and left Z A M Pk N P A -invariant. Assume the data is spherical. Then the function is determined by itsbehavior on Z A M Pk \ M P A . In contrast to the minimal parabolic case, this is nota product of copies of GL , so we can not use the GL spectral theory of Mellininversion to establish the decomposition. Instead the quotient is isomorphic to GL ( k ) \ G A , so we will use the spectral theory for GL . If η is the data for a P , pseudo-Eisenstein series Ψ η , we can write η as a tensor product η = f × ν on Z GL ( A ) GL ( k ) \ GL ( A ) · Z GL ( k ) \ Z GL ( A ) Saying that the data is cuspidal means that f is a cuspform. Similarly the data ϕ = ϕ F,s for a P , -Eisenstein series is the tensor product of a GL cusp form F and a character λ s = | . | s on GL . We show that Ψ f,ν is the superposition ofEisenstein series E F,s where F ranges over an orthonormal basis of cuspforms and s is on a vertical line.Using the spectral expansions of f and ν , η = f ⊗ ν = (cid:0) X cfms F h f, F i · F (cid:1) · (cid:0) Z s h ν, λ s i · λ s ds (cid:1) = X cfms F Z s h η f,ν , ϕ F,s i · ϕ F,s ds ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 57
So the pseudo-Eisenstein series can be re-expressed as a superposition of Eisensteinseries Ψ f,ν ( g ) = X cfms F Z s h η f,ν , ϕ F,s i · E F,s ( g ) ds In fact the coefficient h η, ϕ i GL is the same as the pairing h Ψ η , E ϕ i GL , since h Ψ η , E ϕ i = h c P (Ψ η ) , ϕ i = h η, ν i So the spectral expansion isΨ f,ν = X cfms F Z s h Ψ f,ν , E F,s i · E F,s ( g ) ds So far, we have not had to shift the line of integration to the critical line + i R .It now remains to show that pseudo-Eisenstein series for the associate parabolic Q = P , can also be decomposed into superpositions of P -Eisenstein series. Formaximal parabolic pseudo-Eisenstein series, the functional equation does not relatethe Eisenstein series to itself but rather to the Eisenstein series of the associateparabolic. We will use this functional equation to obtain the decomposition ofassociate parabolic pseudo-Eisenstein series. The functional equation is E QF,s = b F,s · E PF, − s where b f,s is a meromorphic function that appears in the computation of the con-stant term along P of the Q -Eisenstein series.We consider a Q -pseudo-Eisenstein series Ψ Qf,ν with cuspidal data. By the samearguments used above to obtain the decomposition of P -pseudo-Eisenstein series,we can decompose Ψ Pf,ν into a superposition of Q -Eisenstein seriesΨ Qf,ν ( g ) = X cfms F Z s h η f,ν , ϕ F,s i · E QF,s ( g )Now using the functional equation,Ψ Qf,ν ( g ) = X cfms F Z s h Ψ Qf,ν , b
F,s · E PF, − s i · b F,s · E PF, − s = X cfms F Z s h Ψ Qf,ν , E
PF, − s i · | b F,s | · E PF, − s So we have a decomposition of Q -pseudo-Eisenstein series (with cuspidal data) into P -Eisenstein series (with cuspidal data). In order to use the functional equationwe moved some contours, but there are no poles, so no residues are acquired.We have described the spectral decomposition of L ( Z A G k \ G A ) as the directsum/integral of irreducibles. Any automorphic form ξ can be written as ξ = X GL cfms f h ξ, f i· f + X GL cfms F Z s h ξ, E , F,s i· E , F,s + Z (1) h ξ, E , , λ i· E , , λ dλ + h ξ, ih , i This converges in L . Bibliography [Bourbaki 1963] N. Bourbaki,
Topological Vector Spaces, ch. 1-5 , Springer-Verlag,1987.[CdV 1982,83] Y. Colin de Verdi`ere,
Pseudo-laplaciens, I, II,
Ann. Inst. Fourier(Grenoble) (1982) no. 3, 275-286, no. 2, 87–113.[CdV 1981] Y. Colin de Verdi`ere, Une nouvelle demonstration du prolongementmeromorphe series d’Eisenstein , C. R. Acad. Sci. Paris Ser. I Math. (1981),no. 7, 361-363.[DeCelles 2011a] A. DeCelles,
Fundamental solution for (∆ − λ z ) ν on a symmetricspace G/K , arXiv:1104.4313 [math.RT].[DeCelles 2011b] A. DeCelles,
Automorphic partial differential equations and spec-tral theory with applications to number theory , Ph.D thesis, University of Minnesota,2011.[DeCelles 2012] A. DeCelles
An exact formula relating lattice points in symmetricspaces to the automorphic spectrum , Illinois J. Math. (2012), 805-823.[Fadeev 1967] L. D. Faddeev, Expansion in eigenfunctions of the Laplace operatoron the fundamental domain of a discrete group on the Lobacevskii plane , TrudyMoskov. math 0-ba , 323–350 (1967).[Faddeev-Pavlov 1972] L. Faddeev, B. S. Pavlov, Scattering theory and automorphicfunctions,
Seminar Steklov Math. Inst (1972), 161–193.[Feigon-Lapid-Offen 2012] B. Feigon, E. Lapid, O. Offen, On representations distin-guished by unitary groups , Publ. Math. Inst. Hautes Etudes Sci. (2012), 185-323.[Garrett 2010] P. Garrett,
Examples in automorphic spectral theory
Colin de Verdi`ere’s meromorphic continuation of Eisen-stein series
Pseudo-cuspforms, pseudo-Laplaciens
Unbounded operators, Friedrichs’ extension theorem
Vector-Valued Integrals
ELF-ADJOINT BOUNDARY-VALUE PROBLEMS OF AUTOMORPHIC FORMS 59 [Garrett 2011 e] P. Garrett
Holomorphic vector-valued functions
Slightly non-trivial example of Maass-Selberg relations
Most continuous automorphic spectrum for GL n Discrete decomposition of pseudo-cuspforms on GL n Sur un lemme de la theorie des espaces lineaires ,Comm. Inst. Sci. Math de Kharkoff, no. 4, (1936),35-40.[Grothendieck 1952], A. Grothendieck, Sur cetains espaces de fonctions holomor-phes I, II, III , J. Reine Angew. Math. (1953), 35-64 and 77-95.[Grothendieck 1955], A. Grothendieck,
Produits tensoriels topologiques et espacesnucleaires , Mem. Am. Math. Soc. , 1955.[Grubb 2009] G. Grubb, Distributions and operators , Springer-Verlag, 2009.[Haas 1977] H. Haas,
Numerische Berechnung der Eigenwerte der Differentialgle-ichung y ∆ u + λu = 0 fur ein unendliches Gebiet im R , Diplomarbeit, Universit¨atHeidelberg (1977) 155.pp.[Harish-Chandra 1968] Harish–Chandra, Automorphic Forms on semi-simple LieGroups , Lecture Notes in Mathematics, no. 62, Springer-Verlag, Berlin, Heidel-berg, New York, 1968.[Hejhal 1981] D. Hejhal,
Some observations concerning eigenvalues of the Lapla-cian and Dirichlet L-series in Recent Progress in Analytic Number Theory , ed. H.Halberstam and C. Hooley, vol. 2, Academic Press, NY, 1981, 95–110.[Hejhal 1976] D. Hejhal
The Selberg trace formula for SL ( R ) I , Lecture Notes InMath. , Springer-Verlag, Berlin, 1976.[Hejhal 1983] D. Hejhal The Selberg trace formula for SL ( R ) II , Lecture Notes InMath. , Springer-Verlag, Berlin, 1983.[Hejhal 1990] D. Hejhal On a result of G. P´olya concerning the Riemann ξ -function ,J. d’Analyse Mathematique 55 (1990), pp. 59-95.[Hejhal 1994] D. Hejhal On the triple correlation of zeros of the zeta function
In-ternat. Math. Res. Notices 7, pp. 293-302, 1994. [Jacquet 1983] H. Jacquet,
On the residual spectrum of GL ( n ), in Lie Group Rep-resentations, II , Lecture notes in Math. 1041, Springer-Verlag, 1983.[Jacquet-Lapid-Rogowski 1999] H. Jacquet, E. Lapid, J. Rogowski,
Periods of au-tomorphic forms , J. Amer. Math. Soc. (1999), no. 1, 173-240.[Lang 1970] S. Lang, Algebraic number theory , Addison-Wesley, 1970.[Lapid-Offen 2007] E. Lapid, O. Offen,
Compact unitary periods , Compos. Math.
On the Functional Equations satisfied by Eisen-stein series , Lecture Notes in Mathematics no. 544, Springer-Verlag, New York,1976.[Lax-Phillips 1976] P. Lax, R. Phillips,
Scattering theory for automorphic functions ,Annals of Math. Studies, Princeton, 1976.[Maass 1949] H. Maass,
Uber eine neue Art von nichtanalytischen automorphenFunktionen , Math. Ann. (1949), 141–183.[Moeglin–Waldspurger 1989] C. Moeglin, J. L. Waldspurger,
Le spectre residuel de GL ( n ), with appendix Poles des fonctions L de pairs pour GL ( n ), Ann. Sci. EcoleNorm. Sup. (1989), 605–674.[Moeglin–Waldspurger 1995] C. Moeglin, J. L. Waldspurger, Spectral decomposi-tions and Eisenstein series , Cambridge Univ. Press, Cambridge, 1995.[Pettis] B. J. Pettis,
On integration in vectorspaces , Trans. AMS , 1938, 277-304.[P´olya] G. P´olya, Bemerkung ¨uber die Integraldarstellung der Riemannschen ξ -Funktion Acta Math. (1926), 305-317.[Rudin 1991] W. Rudin, Functional Analysis , second edition, McGraw-Hill, 1991.[Rudnick-Sarnak 1994] Z. Rudnick, P. Sarnak,
The n-level correlations of zeros ofthe zeta function , C.R. Acad. Sci. Paris , 1027-1032, 1994.[Shahidi 2010] F. Shahidi,
Eisenstein series and automorphic L-functions , AMSColloquium Publ, , AMS, 2010. E-mail address : [email protected]@umn.edu