Construction of Series of Degenerate Representations for Gsp(2) and Pgl(N)

CONSTRUCTION OF SERIES OF DEGENERATE REPRESENTATIONS FOR GSP(2) AND PGL(N)

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

Martin Bozhidarov Nikolov, M.Sc.

*****

The Ohio State University 2008

Dissertation Committee: Approved by Professor Yuval Flicker, Advisor

Professor James Cogdell Advisor Professor Wenzhi Luo Graduate Program in Mathematics

ABSTRACT

The relative trace formula is a tool for studying automorphic representations on symmetric spaces. In this work two relative trace formulas are studied: One for

GSp(2) relative to a form of SO(4) and one for PGL(n) relative to GL(n − 1).

The first trace formula is used to study the Saito-Kurokawa lifting of automorphic representations from PGL(2) to PGSp(2), thus characterising the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F ) associated to a quadratic extension E of the base field F , and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup.

The second trace formula is used to study the representations π of PGL(n) which have a nonzero linear form invariant under the Levi subgroup GL(n−1), and a certain nonvanishing degenerate Fourier coeﬃcient, over local and global ﬁelds. The result

0 is that these π are of the form I(1n−2, π ), namely normalizedly induced from the parabolic subgroup of type (n − 2, 2), trivial representation on the GL(n − 2)-factor and cuspidal π0 on the PGL(2)-factor. The analysis of the formula is intricate our main achievement is rigorously establishing the convergence of the formula. New in this case is that none of the representations of PGL(n) involved are cuspidal, they rather occur in the continuous spectrum, but discretely in our formula.

ii To Lore Thaler

iii ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor Professor Yuval Flicker for allowing me to work on this thesis, and for all of his time, encouragement, concern, and patience with me. I have appreciated the many conversations and discussions we have had, both mathematical and otherwise.

I would also like to thank all my committee members, my advisor Professor Yuval

Flicker, Professor James Cogdell, and Professor Wenzhi Luo, for agreeing to serve on my committee.

I would like to thank the Department of Mathematics for their support expressed through several teaching and research fellowships that I enjoyed.

iv VITA

2002-Present ...... Ph.D. Candidate, The Ohio State Univer- sity

2002 ...... M.Sc. Mathematics, Soﬁa University

2001-2002 ...... Researcher, Bulgarian Academy of Sci- ences, Institute of Mathematics and In- formatics,

2000-2001 ...... University of Electro Communications, Tokyo Japan

1997-2002 ...... Soﬁa University “St. Kliment Ohirdski”

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Number Theory

v TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

CHAPTER PAGE

1 Introduction ...... 1

2 Cusp Forms on GSp(2) with SO(4)-Periods ...... 10

2.1 Deﬁnitions and Notations ...... 10 2.2 The group ...... 20 2.3 Geometric side ...... 25 2.4 Asymptotes ...... 32 2.5 Matching ...... 39 2.6 Corresponding ...... 45 2.7 Spectral analysis ...... 51 2.8 Computing the integrals of truncated Eisenstein series . . 57 2.9 The contribution from the maximal parabolics ...... 64 2.10 The contribution from the minimal parabolic ...... 70 2.11 Comparison ...... 79 2.12 Converse ...... 89 2.13 Split cycles ...... 93 2.14 Invariance of Fourier coeﬃcients of cyclic automorphic forms 96

3 A Trace Formula for the Symmetric Space PGL(n)/ GL(n − 1) . . . . . 109

3.1 Notations ...... 109 3.2 Truncation ...... 115

vi 3.3 The Main Distribution ...... 126 3.4 Regularized Periods of Eisenstein series ...... 128 3.5 Periods of Truncated Eisenstein Series ...... 136 3.6 Limit Formula for J T (f) ...... 137 3.7 An application ...... 146

Bibliography ...... 154

vii CHAPTER 1

INTRODUCTION

The relative trace formula is a tool to study period integrals of automorphic forms and their relation to the principle of functoriality. For a cusp form φ on the group

G(A) of adèlepoints on a connected reductive group G over a number field F , these integrals are of the form Z φ(h)dh. S(F )\S(A) Here S is the group of fixed points of an involution of G over F . A representation

π of G(A) is called S-cyclic, or just cyclic if S is understood, if there is a φ in its representation space whose period integral is nonzero. It is expected that the cyclic representations are precisely those that are the functorial transfer from another group

G0. For more on the relative trace formula see [Lap1]. The present work consists of the study of to cases of the relative trace formula and their applications. Namely, the case ﬁrst case studies the pair G = GSp(2) and S = SO(4), the second deals with

G = GL(n) and S = GL(n − 1).

Chapter 2 concerns the determination of cusp forms on an ad´elegroup G(A) whose period − namely integral − over a closed subspace (“cycle”) arising from a subgroup

H(A), is nonzero. Such forms contribute to the cohomology of the symmetric space

G/H, and play a role in lifting automorphic forms to G(A) from another group

1 G0(A). Many advances in these studies so far have been made by means of the theory of the Weil representation [We1]; see Waldspurger [Wa1], Howe and Piatetski-Shapiro

[HPS], [PS1], Kudla-Rallis [KR], Oda [O]. This technique has the advantage − in addition to early maturity − of constructing cusp forms on G(A) directly from such forms on G0(A). Miraculously, the cusp forms on G(A) so obtained happen to have nonzero H(A)-periods. Our approach is based on a more naive and direct method, focusing more on the representation and its properties rather than on its particular realization. Thus we integrate both the spectral and the geometric expressions for the kernel Kf (x, y) of the convolution operator on the space of cusp form on G(A), over the cycle associated with

H(A). If both variables x and y are integrated over the cycle, one obtains a bi-period summation formula, involving the periods of the automorphic forms over the cycles

(in Jacquet [J1], and later in [FH], this is named a “relative trace” formula, although there are no traces in that formula). The case where G(A) is G0(A) × G0(A) and

H(A) is G0(A) embedded diagonally, coincides with that of the usual trace formula on G0(A); this case is also referred to as the “group case”. If the second variable is integrated over a unipotent radical of a parabolic subgroup against an additive character, the Fourier summation formula − involving Fourier co- eﬃcients of the automorphic forms − is obtained (see Jacquet [J2], where the formula is again named “relative trace” formula, and [F5], [F6] and [F4]). Only the cusp forms on G(A) with nonzero H(A)-periods survive on the spectral side. The geometric side is compared with the geometric side of an analogous summation formula on G0(A), for matching test functions on G(A) and G0(A). The resulting identity of spectral

2 sides can be used to establish lifting from G0(A) to G(A). In summary, both the bi-period and the Fourier summation formulae are special instances of the “relative trace” formula.

The derivation of the Fourier summation formula, and the characterization of the relevant orbital integrals, lead to deep chapters in global, and local, harmonic analysis, especially of symmetric spaces; cf., e.g., [OM], [BS]. The analytic problems thus raised might even be considered to be of greater importance than the motivating ﬁnal applications in representation theory. Conversely, these applications justify some of the work which has been done on symmetric spaces. One expects to derive identities of (bi-period or) Fourier-Period distributions intrinsically related to the (local) representation in question. These distributions are analogous to Harish-Chandra’s characters, which play a key role in studies of automorphic forms by means of the

Selberg trace formula. To fully harvest the (bi-period, or) Fourier-Period summation formulae, one would need an analogue of the orthogonality relations of characters, due to Harish-Chandra and Kazhdan [K], for these local distributions. The summation formula has been slow to evolve possibly since its application is based on panoply of techniques, substantially diﬀerent from each other. Yet it could be a source of inspiration in various branches of contemporary harmonic analysis.

In chapter 2 we study the case of automorphic forms on G = GSp(2) and the cycle

H = SO(4, E/F ) associated to a quadratic separable extension E of the global base

ﬁeld F . More precisely, G is the algebraic group of g ∈ GL(4) with gJ tg = λJ, λ =

0 w 0 1 × λ(g) ∈ GL(1),J = ( −w 0 ) , w = ( 1 0 ); A is the ring of ad´eles of F . We ﬁx θ ∈ F which is not a square in F , put θ = ( 0 1 ) and Θ = θ1 0 , and let H be the centralizer 1 θ 0 0 θ1

3 of Θ in G. Denote by Z(F ),Z(A) the center of G(F ) and G(A) respectively. Also

IX x y consider the unipotent radical N = {n = ( 0 I ); X = ( z x )} of the Siegel parabolic subgroup P of type (2,2) of G, a complex valued nontrivial character ψ of the additive

0 1 group A/F , and the character ψθ(n) = ψ(trTX) = ψ(z − θy),T = ( −θ 0 ). Our main global achievement in this work is to advance the theory of the Fourier summation formula, namely develop such a formula by expanding geometrically and spectrally the integral of the kernel Kf (n, h) of the standard convolution operator r(f) (for a test function f) on the space of automorphic forms. In fact we multiply

Kf (n, h) by ψθ(n), and integrate over n ∈ N(F )\N(A) and h ∈ Z(A)H(F )\H(A). The Fourier summation formula is recorded in Proposition 18. On the spectral side we use the mixed truncation following [JLR]. On the geometric side we obtain a new type of orbital integrals of the form R R f (nγh)ψ (n)dndh. Nv Hv v θ Our main local achievement is in characterizing the functions of γ thus obtained, by studying their asymptotic behavior; see Proposition 8. This study involves integration over a certain quadric in the aﬃne 5-space. We are led to Fourier analysis with respect to quadratic forms, involving Weil’s factor γψ. Underlying our computations is the stationary phase method, where we use the Morse Lemma. We discover that the asymptotic behavior of these Fourier orbital integrals is compatible with that of analogous Fourier orbital integrals obtained in the analysis of the Fourier summa-

∗ 0 1 ∗ tion formula for N\PGL(2)/A, A = ( 0 1 ) ,N = ( 0 1 ); see Proposition 12 (due to Jacquet [J2]). We relate these Fourier orbital integrals on PGSp(2) and PGL(2),

4 proving the existence of matching, in Corollary 13 and compare the summation formula for PGSp(2) (Proposition 18) with that of PGL(2) (Proposition 16, [F5], [J2]) in Proposition 35.

In Proposition 17 we record the statement that naturally related spherical functions on PGSp(2) and PGL(2) have matching Fourier orbital integrals. The case of the unit elements in the Hecke algebras is proven in Proposition 14. The general case was proposed as a conjecture in an early draft of this work, and it has later been proved in Zinoviev’s OSU thesis [Z], using the case of the unit elements. It would be interesting to ﬁnd an alternative proof of this “fundamental lemma”, possibly based on a “symmetric space” analogue of the regular functions technique of [F3], which might reduce the spherical case to that of smooth test functions, or to that of the unit element in the Hecke algebra, which are analyzed here.

As an application of our summation formula and study of orbital integrals we recover a result of Piatetski-Shapiro [PS1], which − together with the work of Weissauer [Wr1]

− in fact motivated our study. Let ρv be an admissible representation of PGL(2,Fv), and ζ a complex number. Write I(ρv, ζ) for the Gv = PGSp(2,Fv)-module on the space of φ : Gv → ρv which satisfy

3 A ∗ −1 ζ+ 2 φ 0 λw tA−1w g = |λ det A| ρv(A)(φ(g))

× 1 for A ∈ PGL(2,Fv), g ∈ Gv and λ ∈ Fv . Write J(ρv, 2 ) for the Langlands’ quotient 1 (see [BW]) of I(ρv, 2 ) (it is unramiﬁed if so is ρv, and nontempered if ρv is unitarizable). Proposition 36 asserts that if ρ is a cuspidal representation of PGL(2, A) with

1 × × L( 2 , ρ⊗χθ) 6= 0 (χθ is the quadratic character of A /F associated with the quadratic

5 √ 1 extension E = F ( θ) of F ) and L( 2 , ρ) = 0, then there exists a cuspidal represen- 1 tation π of PGSp(2, A) with πv ' J(ρv, 2 ) for almost all v. This π is H(A)-cyclic, R namely the period ΠH (Φ) = Φ(h)dh over the cycle Z( )H(F )\H( ) is Z(A)H(F )\H(A) A A R nonzero for some Φ ∈ π, and is θ-generic, namely Wψ (Φ) = Φ(n)ψθ(n)dn θ N(F )\N(A) is nonzero for some (possibly other) Φ ∈ π. More precise local results could be obtained from our global theory had we had orthogonality relations for our Fourier-

Period distributions, analogous to those of Kazhdan [K] in the case of characters.

Conversely, Proposition 37 asserts (but only in the function ﬁeld case) that given an H(A)-cyclic θ-generic discrete series representation π of PGSp(2, A), either there 1 1 exists a cuspidal PGL(2, A)-module ρ with L( 2 , ρ ⊗ χθ) 6= 0 and πv ' J(ρv, 2 ) for 1 almost all v, or πv ' J(χθ,v ◦det, 2 ) for almost all v (here χθ ◦det is a one-dimensional, residual, discrete series representation of PGL(2, A) deﬁned by the quadratic character χθ). A brief discussion of the description of packets of such representations of

PGSp(2, A) is given in the beginning of Section 2.12, see [F9].

In Section 2.13 we explain why there are no cusp forms on G(A) with periods by

ε 0 1 0 the split cycle SO(4) = H0 = ZG (( 0 −ε )) , ε = ( 0 −1 ). For this reason we consider only SO(4, E/F )-periods. In Section 2.14 we employ another form of the Fourier summation formula to study invariance of Fourier coeﬃcients of cusp forms under the action of a certain stabilizer subgroup, recovering part of [PS2].

Under the isomorphism of PGSp(2) with SO(3, 2), the image of H0 is the split group SO(2, 2), and that of H is SO(3, 1; E/F ), the special orthogonal group associated with the sum of the hyperbolic form xy and the norm form z2 − θt2. Our techniques apply also with G = PGSp(1,D) ' SO(4, 1), where D is a quaternion algebra, and H

6 is again SO(3, 1; E/F ), if the ﬁeld E embeds in D. It would be interesting to study this situation, and its relation to the present work. Further, here we consider cyclic cusp forms with nonzero Fourier coeﬃcients with respect to a generic character of the unipotent radical of the Siegel parabolic subgroup. There are no generic (with respect to a maximal unipotent subgroup) cyclic cusp forms on GSp(2). It would be interesting to consider also cyclic cusp forms generic with respect to the Heisenberg unipotent subgroup. Naturally it would be interesting to consider cusp forms on

SO(n) cyclic with respect to SO(n − 1), for a suitable choice of inner forms of these groups. The present work is a ﬁrst step in this direction.

In Chapter 3 we consider a variation of the relative trace formula initiated in [F5], in the case where G = GL(n) and S = GL(n − 1) × GL(1). In this case there are no cuspidal cyclic representations, and we are led to introducing periods of induced representations by means of regularization of analogous integrals. The deﬁnition of the regularized integrals is given in section 5, where their main properties are described. We derive an analogue of Arthur’s spectral expansion, which combined with the geometric side developed in [F5] constitutes our trace formula.

h 0 Embed S in G via (h, a) 7→ ( 0 a ) for h ∈ GL(n − 1) and a ∈ GL(1). Denote by N 1 p z the group of matrices of the form n = 0 I tq , where z is a scalar, I the identity 0 0 1 (n − 2) × (n − 2) matrix and p and q are row vectors. Here tq denotes the transpose of q. Let ψe be a complex valued nontrivial additive character of A trivial on F .

Denote by ψ the character of N(A) trivial on N(F ) deﬁned by ψ(n) = ψe(p1 + qn−2) for n ∈ N(A).

7 ∞ 1 For a test function f ∈ Cc (G(A) ) let

X −1 Kf (x, y) = f(x γy) γ∈G(F ) be the kernel of the right regular representation of G(A)1 on L2(G(F )\G(A)1). Our main goal is the computation of the ﬁne spectral expansion of the distribution

Z Z f 7→ J(f) = Kf (n, h)ψ(n)dndh. 1 N(F )\N(A) S(F )\S(A)∩G(A) This is recorded in Theorem 59. To derive our formula we use a modiﬁcation of

Arthur’s truncation, as in [JLR]. We truncate the kernel Kf (n, h) with respect to the second variable, noting that for a suﬃciently positive truncation parameter the truncated kernel is equal to the original kernel. As in [JLR] we compute, in sections 5 and 6, the period integrals of the truncated Eisenstein series. Using Arthur’s (G, M)- families we compute in section 7 the ﬁne expansion of the distribution J(f).

In more detail, the constant term φP of an automorphic form φ has a decomposition

l X P φP (namk) = αi(HP (a))ψi (amk). i=1

P Here αi are polynomials and ψi are level P automorphic forms satisfying

P hλi+ρP ,HP (a)i P ψi (ag) = e ψi (g),

∗ for some λi ∈ aP . R ∗ The regularized integral 1 φ(h)dh is deﬁned to be S(F )\S(A)∩G(A) ZZ Z # X X T,P P hλi,Xi (Λ (ψi ))(mk)dkdm αi(X)e τb(X − T )dX. P ⊂G i

8 The #-integral is deﬁned through analytic continuation in [JLR]. Thus in section 5 we compute the regularized integrals of cuspidal Eisenstein series. Namely

Z ∗ E(h, φ, λ)dh = J(ω, φ, λ) 1 S(F )\S(A)∩G(A) where ! X Z Z J(ω, φ, λ) = ehλ+ρP ,HP (ηh)i φ(mηh)dm dh. 1 η Sη(A)\S(A) Mη(F )\Mη(A)∩G(A)

In section 6 we compute the (convergent) integrals of truncated Eisenstein series in terms of the regularized integrals of their constant terms. In section 7 we use the combinatorics of Arthur’s (G, M)-families to compute the ﬁne spectral expansion of the distribution J(f). As noted above, the equality of the spectral expression of J(f) and the geometric expression of J(f), obtained in [F5], constitutes our trace formula.

For applications, in section 8 we change our setting from that of S(A) ∩ G(A)1 inside

G(A)1 (where G = GL(n) and S = GL(n − 1) × GL(1)), which is used by Arthur, to that of S = GL(n − 1) in G = PGL(n), which is more suitable for our applications.

Thus using this rephrased absolutely convergent expression and the results of [F5] and [F8] we describe the set of automorphic representations of PGL(n, A) which are S-cyclic. More precisely given a cuspidal representation π0 of PGL(2, A), let

0 π = I(n−2,2)(1n−2 × π ) be the induced representation of PGL(n, A). Then π is S- cyclic and every S-cyclic automorphic representation of PGL(n, A) is of this form.

9 CHAPTER 2

CUSP FORMS ON GSP(2) WITH SO(4)-PERIODS

2.1 Deﬁnitions and Notations

Let F be a number ﬁeld and let G be the group GSp(2), the algebraic group deﬁned over F by

t 0 w 0 w G = g ∈ GL(4) | g ( −w 0 ) g = λ(g)( −w 0 ) , λ(g) ∈ GL(1) ,

t 0 1 × ×2 where g denotes the transpose of g and w is ( 1 0 ). Fix θ ∈ F − F and put Θ = θ1 0 , where θ = ( 0 1 ) . Let H = Z (Θ) be the centralizer of Θ in G. 0 θ1 1 θ 0 G Let B be the upper triangular Borel subgroup. It has a Levi decomposition B =

MBNB. Here MB is the diagonal subgroup and NB the unipotent upper triangular subgroup. With this choice of a minimal parabolic subgroup the standard parabolic subgroups are the Borel subgroup B, the Siegel parabolic subgroup P1 = M1N1 and

IX x y the Heisenberg parabolic subgroup P2 = M2N2. Here N1 = {n1 = ( 0 I ); X = ( z x )} , A 0 λ M1 = 0 λwtA−1w where A ∈ GL(2), and M2 = diag(a, A, a ); detA = λ , N2 = n u(x) yu( z ) o y ; u(x) = ( 1 x ) . 0 u(−x) 0 1 Fix a nontrivial additive character ψ0 of A/F . Let ε be diag(1, −1). Deﬁne the 0 character ψ of N1(A)/N1(F ) by ψ(n) = ψ tr(εθ1X) .

10 Recall the following deﬁnitions (for a general connected reductive group G). Fix a maximal compact subgroup K of G(A) satisfying (see [MW], I.1.4)

1. G(A) = B(A)K;

2. K∩M(A) is a maximal compact group of M(A) for all Levi subgroups M ⊂ G;

3. P (A) ∩ K = (MP (A) ∩ K)(NP (A) ∩ K).

Let P be a standard parabolic subgroup of G. Let MP be its unique Levi subgroup containing MB and NP its unipotent radical. Let TP be the maximal split torus in the center of MP . Let X∗(TP ) = Hom(Gm,TP ) be the group of co-characters of TP .

∗ Let X (TP ) = Hom(TP , Gm) be the group of characters. Set

eaP = X∗(TP ) ⊗ R and d(P ) = dim(eaP ).

For any two parabolic subgroups Q ⊂ P , there are canonical injection eaP ,→ eaQ and surjection eaQ eaP , hence a canonical decomposition

P eaQ = aQ ⊕ eaP .

Since eaG is a summand of eaP for any P , set aP = eaP /eaG . Then we have the decomposition

P aQ = aQ ⊕ aP .

∗ Write aP = HomR(aP , R) for the dual space (over R) of aP . Let g be the Lie algebra of G and Ad : G → Aut(g) the adjoint representation of G on its Lie algebra g. Then Ad(TB) is a group of diagonalizable commuting

11 (T ) endomorphisms g → g. Hence they are simultaneously diagonalizable. Set gα =

∗ (T ) {X ∈ g|Ad(t)(X) = α(t)X} for α ∈ X (TB). If the space gα is not zero, α is called a weight. Hence we have

(T ) (T ) g = g0 ⊕ ⊕α∈Φgα .

Only ﬁnitely many nonzero α appear. These α are called roots of G with respect to

TB. The ﬁnite set of roots is denoted by Φ.

∗ There is a natural pairing h, i : X (TB) × X∗(TB) → Z, deﬁned as follows: For

∗ α ∈ X (TB) and β ∈ X∗(TB) the composition α ◦ β : GL(1) → GL(1) gives a character of GL(1). Since X∗(GL(1)) ' Z there is an n ∈ Z such that α ◦ β(t) = tn. Deﬁne hα, βi = n.

∨ For each root α ∈ Φ the coroot α ∈ X∗(TB) is the unique homomorphism GL(1) →

∨ TB satisfying hα, α i = 2. A choice of a minimal parabolic subgroup determines an ordering of the set of roots

Φ. A root α is called simple if α > 0 and it cannot be written as α = β1 + β2 with βi both positive.

∗ Let ∆ and ∆b be the subsets of simple roots and simple weights in aG, respectively.

∨ ∨ We write ∆ for the basis of aG dual to ∆,b and ∆b for the basis of aG dual to ∆.

Thus, ∆∨ is the set of coroots and ∆b ∨ is the set of coweights.

For every P , let ∆P ⊂ aG be the set of nontrivial restrictions of elements of ∆ to

∨ ∨ aP . For each α ∈ ∆P , let α be the projection of β to aP , where β is the root in

∨ ∨ ∆ whose restriction to aP is α. Set ∆P = {α : α ∈ ∆P }. Then we may also deﬁne

∨ their dual bases. Namely, we denote the dual basis of ∆P by ∆b P and the dual basis

∨ of ∆P by ∆b P .

12 P If Q ⊂ P , we write ∆Q to denote the subset of α ∈ ∆Q appearing in the action of

TQ on the unipotent radical of Q ∩ MP . Then aP is the subspace of aQ annihilated

P P ∨ ∨ P ∨ P P by ∆Q. Let (∆Q) = {α : α ∈ ∆Q}. We then deﬁne (∆b )Q and ∆b Q to be the basis

P P ∨ dual to ∆Q and (∆Q) , respectively.

P P Given two parabolic subgroups Q ⊂ P denote by χQ and χbQ respectively the characteristic functions of the sets

P {H ∈ aB : hα, Hi > 0 for all α ∈ ∆Q} and

P {H ∈ aB : h$, Hi > 0 for all $ ∈ ∆b Q}.

For each parabolic subgroup P there is the Harish-Chandra map

HP : G(A) → aP characterized by:

hχ,HP (m)i ∗ 1. |χ|(m) = e for all m ∈ M(A) and χ ∈ X (MP ),

2. HP (nmk) = HP (m), n ∈ N(A), m ∈ M(A), k ∈ K.

1 For a group G, denote by G(A) the kernel of the Harish-Chandra map HP : G(A) → aP .

For a parabolic subgroup with Levi decomposition P = MN, denote by ρP the unique

∗ 2hρP ,H(m)i element in aP satisfying e = | AdN (m)|. Here AdN (m) is the adjoint action of m on the Lie algebra of N.

13 × r Let F∞ denote F ⊗Q R. There is an isomorphism (F∞) ' TP (F∞) (see [MW], I.1.11 and I.1.13 also [JLR], III.4). Let AP for P 6= G and AG be the intersections of the

× r 1 image of (R+) in TP (F∞) with G(A) and Z(A), respectively. The Harish-Chandra map HP induces an isomorphism of AP onto aP and we have the decomposition

1 X M(A) = AG × AP × M(A) . For X ∈ aP , write e for the unique element in AP such

X ∗ λ hλ,HP (p)i that HP (e ) = X, and for λ ∈ aP , write e for the character p 7→ e of P (A). Then for a suitable normalization of the Haar measure, we have

Z Z Z Z Z f(x)dx = f(namk)e−2hρP ,HP (a)idndadmdk. 1 1 G(A) N(A) AP M(A) K

Let Ω = NG(MB)/MB be the Weyl group of MB in G, where NG(MB) denotes the normalizer of MB in G. For a Levi subgroup M, let ΩM = NM (MB)/MB be the Weyl group of M. For two parabolic subgroups P and Q with Levi factors MP and MQ respectively, let Ω(P,Q) be the set of elements ω ∈ Ω of minimal length in their class

−1 ωΩMP , such that ωMP ω = MQ. The minimal length condition is equivalent to the

P Q condition ω∆G = ∆G.

Let K be a maximal compact subgroup of G(A). Let z be the center of the universal Q enveloping algebra of the complexified Lie algebra of G∞ = v G(Fv). Here the product is over the archimedean places v of F . Then a function φ(g) on G(A) is called K-finite if the span of the functions g 7→ φ(gk)(∀k ∈ K) is finite dimensional.

It is called z-ﬁnite if there is an ideal I in z of ﬁnite codimension such that I · φ = 0.

The function φ(g) is called smooth if the following condition is satisﬁed. For g = g∞gf , with g∞ ∈ G∞ and gf ∈ G(Af ), there exist a neighborhood V∞ of g∞ in G∞ and

14 ∞ a neighborhood Vf of gf in G(Af ), and a C -function φ∞ : V∞ → C, such that

φ(g∞gf ) = φ∞(g∞) for all g∞ ∈ V∞ and gf ∈ Vf . In the function ﬁeld case φ is called smooth if it is locally constant.

Fix an embedding i : G,→ SL2n. For g ∈ G(A), write i(g) = (gkl)k,l=1,...,2n. Set Y ||g|| = sup{|gkl|v; k, l = 1,..., 2n} v where the product is over all places v of F . Then a function φ is called of moderate growth if there are C,N ∈ R such that for all g ∈ G(A) we have |φ(g)| ≤ C||g||N . The deﬁnition of moderate growth does not depend on the choice of the embedding i. As in [MW], I.2.17 we make:

Deﬁnition 1. A function

φ : G(F )\G(A) → C is called an automorphic form if

1. φ is smooth and of moderate growth,

2. φ is right K-ﬁnite,

3. φ is Z-ﬁnite.

The space of automorphic forms is denoted by A(G). For a parabolic subgroup

P = NM the space of automorphic forms of level P , denoted by AP (G), is the space of smooth right K-ﬁnite functions

(P ) φ : N(A)M(F )\G(A) → C

15 such that for every k ∈ K the function m 7→ φ(P )(mk) is an automorphic form of

(P ) M(A). For φ ∈ AP (G) and a parabolic subgroup Q ⊂ P with unipotent radical

NQ, the constant term is deﬁned by: Z (P ) (P ) φQ (g) = φ (ng)dn. NQ(F )\NQ(A)

For a cuspidal representation σ of M(A) denote by AP (G)σ the subspace of AP (G) consisting of functions φ which are AP -invariant and for every k ∈ K the function m 7→ φ(P )(mk) belongs to the space of σ.

We deﬁne the truncation operator by

Deﬁnition 2. Let φ : G(F )\G(A) → C be a smooth function. For h ∈ H(A) deﬁne

T X d(P )−d(G) X (Λ φ)(h) = (−1) φP (δh)χbP (HP (δh) − T ). P ⊂G δ∈P (F )∩H(F )\H(F ) Here the ﬁrst sum ranges over all standard parabolic subgroups P of G.

Similarly for a smooth function φ : P (F )\G(A) → C, for h ∈ H(A) deﬁne

T,P X d(R)−d(P ) X P (Λ φ)(h) = (−1) φR(δh)χbR(HR(δh) − T ). R⊂P δ∈R(F )∩H(F )\P (F )∩H(F ) Here the ﬁrst sum ranges over all standard parabolic subgroups R of G contained in

P .

Proposition 1. For all h ∈ H(A) we have

X X T,P φ(h) = (Λ φ)(δh)χP (HP (δh) − T ). P ⊂G δ∈P (F )∩H(F )\H(F ) Proof. This proof is exactly the same as that of Lemma 1.5 in [A3]; see also [JLR],

III.5.

16 Proposition 2. For T suﬃciently regular and ﬁxed h ∈ H(A) the function m 7→

(ΛT,P φ)(mh) is rapidly decreasing on (M(F ) ∩ H(F ))\(M(A) ∩ H(A)1).

Proof. Same as Proposition 8 in [JLR]. The proof of [A3], page 92, needs a small modiﬁcation for the “relative case” as in [JLR].

Deﬁnition of R #(following [JLR], pp. 178-185). Let V be a real vector space of ﬁnite dimension n. Let V ∗ be the space of complex valued linear forms on V and

V ∗ the space of real valued linear forms. An exponential polynomial function on V , R expoly for short, is a function f = f({λi,Pi}) of the form

X hλi,xi f(x) = e Pi(x), 1≤i≤r ∗ where λi are diﬀerent elements of V and Pi(x) are nonzero polynomials. By a cone in V we mean a closed subset of the form

C = {x ∈ V : hµi, xi ≥ 0}, where {µ } is a basis of V ∗. Let {e } be a basis of V dual to {µ }. i R j i Consider the integral Z −hλ,xi IC (λ; f) := f(x)e dx. C

For the expoly f = f({λi,Pi}) the integral converges absolutely for λ in the open set

∗ U = {λ ∈ V : Rehλi − λ, eji < 0 for all 1 ≤ i ≤ r, 1 ≤ j ≤ n}.

Fix an index k. Let Ck be the intersection of C and the hyperplane Vk = {x ∈ V : hµk, xi = 0}. In the integral we can write x = hµk, xiek + y with y ∈ Vk and then m X X 1 Z ∂ I (λ; f) = ehλ−λi,yi P (y)dy. C hλ − λ , e im+1 ∂y i 1≤i≤r m≥0 i k Ck k

17 This formula gives the analytic continuation of IC (λ; f) to the tube domain

∗ Uk = {λ ∈ V : Rehλi − λ, eki < 0 for all 1 ≤ i ≤ r}.

The analytic continuation is a meromorphic function with hyperplane singularities, the singular hyperplanes being:

∗ Hk,i = {λ ∈ V : hλ, eki = hλi, eki for all 1 ≤ i ≤ r}.

∗ Therefore the function IC (λ; f) has analytic continuation to V by Hartogs’ lemma (see [JLR], pp. 180). Lemma 3 of [JLR] asserts:

Lemma 3. The function IC (λ; f) is holomorphic at λ = 0 if and only if hλi, eki 6= 0 for every pair (i, k).

∗ Denote the characteristic function of C by χC . Let T ∈ V . For λ ∈ V such that

Rehλi − λ, eji < 0 for all i and j set Z −hλ,xi fb(λ; C,T ) = f(x)χC (x − T )e dx V

Z X −hλ−λi,xi = Pi(x)χC (x − T )e dx 1≤i≤r V

X hλ−λi,T i hλi,xi = e IC (λ; Pi(x + T )e ). 1≤i≤r

The integrals are absolutely convergent and fb(λ; C,T ) extends to a meromorphic function on V ∗.

18 Deﬁnition 3. The function f(x)χC (x−T ) is called #integrable if fb(λ; C,T ) is regular at λ = 0. In this case, we deﬁne the #integral

Z # f(x)χC (x − T )dx (2.1.1) V to be the value fb(0; C,T ).

From [MW], I.3.2, any automorphic form φ ∈ AP (G) has the decomposition

l (P ) X hλi+ρP ,HP (a)i φ (namk) = αi(HP (a))ψi(mk)e . (2.1.2) i=1

1 Here n ∈ N(A), a ∈ AP , m ∈ M(F )\M(A) and k ∈ K. The αi(X) are polynomials,

∗ ψi ∈ AP (G) and λi ∈ aP . The decomposition is not unique. The set {λi} is uniquely determined by φ.

Deﬁnition 4. For an automorphic form φ ∈ A(G) deﬁne

Z # T,P Λ (φP ) (h)χP (HP (h) − T )dh 1 b P (F )∩H(F )\H(A) to be

Z # Z Z T,P X −hρP ,2Xi Λ (φP ) (e mk)dmdk e · χP (2X − T )dX, 1 b aP K M(F )\M(A) where the integral R # is as deﬁned in (2.1.1). Write ψK(m) for R ψ(mk)dk. By aP K (2.1.2) this integral is equal to

l Z Z # X T,P K hλi,2Xi Λ (ψi ) (m)dm αi(X) · e · χP (2X − T )dX. 1 b i=1 M(F )\M(A) aP

19 R ∗ Deﬁnition 5. For an automorphic form φ ∈ A(G) deﬁne 1 φ(h)dh, also H(F )\H(A) denoted by ΠH (φ) and called the period of φ over H, to be

Z # X T,P (Λ (φP ))(h)χP (HP (h) − T )dh. 1 b P ⊂G P (F )∩H(F )\H(A)

(P ) For a level P automorphic form φ ∈ AP (G) with decomposition (2.1.2) deﬁne

Z ∗ (P ) φ (h)χP (HP (h) − T )dh 1 b P (F )∩H(F )\H(A) to be

l Z # X K hλi,2Xi ΠM∩H (ψi (m)) αi(X)e χbP (2X − T )dX. i=1 aP

R T Proposition 4. For φ ∈ A(G) we have that 1 (Λ φ)(h)dh is equal to H(F )\H(A) Z ∗ X d(P )−d(G) (−1) φP (h)χP (HP (h) − T )dh. 1 b P ⊂G P (F )∩H(F )\H(A) Proof. Same as Theorem 10 in [JLR].

2.2 The group

As in section 2.1, let G be the group GSp(2), the algebraic group deﬁned over F by G = GSp(2) = g ∈ GL(4) | tgJg = λ(g)J, λ(g) ∈ GL(1) ,

0 w 0 1 where J = ( −w 0 ) and w = ( 1 0 ). It is the group of symplectic similitudes of a 4-dimensional space over F , a local or global ﬁeld of characteristic other than two. Here tg denotes the transpose of g. The form J has the advantage that the upper triangular subgroup B of G is a minimal parabolic. The maximal parabolics

20 IX x y which contain B are the Siegel parabolic P = MN,N = {( 0 I ); X = ( z x )}, M =

A 0 A 0 1 0 0 λwtA−1w = ( 0 λεAε ) ), where ε = ( 0 −1 ); and the Heisenberg parabolic Q = a 0 0 1 x y z n o 0 1 0 y MQNQ,MQ = 0 A 0 ; det A = λ , with unipotent radical NQ = 0 0 λ/a 0 0 1 −x 0 0 0 1 1 0 0 z 0 1 0 0 which is an Heisenberg group with center ZQ = 0 0 1 0 . The centralizer H = 0 0 0 1 ε 0 ZG(Θ) of Θ = {( 0 −ε )} in G is

a 0 0 b 0 α β 0 a b α β 0 γ δ 0 ; det ( c d ) = det γ δ 6= 0 . c 0 0 d

Given θ ∈ F × put θ = ( 0 1 ) and Θ = θ1 0 . The centralizer H = Z (Θ) of 1 θ 0 0 θ1 G Θ in G consists of the matrices h = ( a b ); a = ( a1 a2 ) , ··· , d = d1 d2 , such c d θa2 a1 θd2 d1 √ √ a b that ( c d ) , a = a1 + a2 θ, ··· , d = d1 + d2 θ, has determinant λ(h) in GL(1). If θ ∈ F − F 2 is not a square then H(F ) is isomorphic via the map just deﬁned to √ {g ∈ GL(2,E); det g ∈ GL(1,F )}, where E = F ( θ) is the quadratic extension of F √ generated by the square root θ of θ. There is a natural injection G/H → X(θ), g 7→

1 t −1 x = λ gΘJ g = gΘg J, where

X(θ) = {x ∈ G; λ(x) = θ, (xJ)2 = θ} = {x ∈ G; λ(x) = θ, tx = −x}.

1 t In particular, G/H → X(1), g 7→ x = λ gΘJ g, is an injection. Denote by A the diagonal subgroup, and by NB the unipotent upper triangular subgroup (thus B =

ANB). Let W = Norm(A)/A be the Weyl group of A in G, where Norm(A) is the normalizer of A in G. Identify W with a set of representatives s in G with ts = s−1 if s is of order two in W . Double cosets decompositions as below are well known (see, e.g., [Sp]).

21 t Proposition 5. (a) Each x in X(θ) has the form x = nsa n with n ∈ NB, a ∈ A, s ∈ W with (s2 = 1 in W and) sa = −as−1. (b) The group G is the disjoint union

PH ∪ P γ1H = BH ∪ Bγ1H ∪ Bγ2H ∪ Bγ3H, where

1 0 2 II 1 0 0 1 −1 w 0 γ 0 γ1 = 1 ,I = ( 0 1 ) , γ = ( 1 1 ) , γ2 = ( 0 w ) , γ3 = 0 1 εγ0ε . − 2 II 2

0 1 G is the disjoint union PH ∪ P γ0H, γ0 = diag(1, ( −1 0 ) , 1). (d) G acts on X(θ) by √ 1 t −1 −1 g : x 7→ λ(g) gx g = gxJ g J. The variety X(θ) consists of the points ± θJ, which are ﬁxed by G, and the orbit

0 a b1 b −a 0 −b b2 ; b b + b2 − ad = θ of ΘJ. −b1 b 0 d 1 2 −b −b2 −d 0

The stabilizer of ΘJ is ZG(Θ) = H.

t Proof. (a) Each x in G can be expressed in the form x = n1sa n2 (a ∈ A, s ∈

t 2 W, ni ∈ NB). For x in X(θ) we have sa = −a · s (hence s = 1 in W ). We may

+ − + + assume that n2 = 1, and write n1 = n1 n1 . Here n1 lies in the group Ns which is generated by the root subgroups Nα of NB associated to the positive roots α such

− − that sα is positive; n1 lies in the group Ns which is generated by the Nα with

+ − t t −t + + α > 0 and sα < 0. Then n1 n1 sa = x = − x = sa n1 n1 , and so n1 = 1 and

− t t −1 −1 n1 ∈ Ns . The relation n1sa = sa n1 can be written as n1 = a s n1sa, or on

−1 −1t −1 −1 −1t −1 −1 applying transpose and inverse, as n1 = as n1 sa . Put σ(n) = as n sa .

− − This is an automorphism of Ns of order two. Since Ns is a unipotent group, the ﬁrst

22 1 − − cohomology set H (hσi,Ns ) of the group hσi generated by σ, with coeﬃcients in Ns ,

1 − is trivial. This H is the quotient of the set {n ∈ Ns ; σ(n)n = 1} by the equivalence

−1 − relation n ≡ n0nσ(n0) . In particular, our n1 ∈ Ns satisﬁes n1σ(n1) = 1, hence it

− −1 is in the equivalence class of 1, namely there is n ∈ Ns with n1 = nσ(n) . Hence

−1 −1t −1 t x = n1sa = nσ(n) sa = nas nsa sa = nsa n, as required. (b) The Weyl group W of G consists of 8 elements, represented by the reﬂections 1, (14)(23), (12)(34), (13)(24), (14), (23), (2431), (3421). The last two are not of order 2, while for s = 1 there are no a ∈ A with sas = −a. The transposition

0 1 (23) is represented in G by s = diag(1, ( −1 0 ) , 1), but there is no a ∈ A with sas = −a. An analogous statement holds for (14).

Concerning the remaining 3 Weyl group elements, we have the following. Choose

0 I t the representative s4 = −I 0 for (13)(24). If x = s4a, a ∈ A, then x = − x

w 0 implies that a = diag(b, b). Since −Jx = ( 0 w ) a has square I we have that b =

t diag(c, 1/c), c ∈ GL(1). If d = diag(c, 1, 1, 1/c) then ds4 d = s4a. Hence the part

t −1 t {ns4a n; n ∈ NB, a ∈ A} of X is equal to the B-orbit λ bs4 b, which is the image of

−1 t Bγ3H under G/H → X, g 7→ λ gΘJ g.

2 As s2 = J represents (14)(23), if x = Ja and (Jx) = I, then the diagonal entries of a ∈ A are ±1. Since x = −tx implies that Ja = aJ, we have that a = ±I or ±Θ.

Clearly there exists no g ∈ G such that λ−1gΘJ tg = gΘg−1J is equal to x = Ja if a = ±I. But when g = I (resp. g = γ2), then x = JΘ (resp. x = −JΘ) is obtained.

t We conclude that the part {nJa n; n ∈ NB, a ∈ A} of X is the union of the B-orbits

23 −1 t ±λ bJΘ b, namely the image of the union of the cosets BH and Bγ2H under the map G/H → X.

t w 0 Note that γ1JΘ γ1 = s3Θ represents (12)(34), where s3 = ( 0 w ). If x = s3Θa then x = −tx and (Jx)2 = I imply that a = diag(b, b, 1/b, 1/b), and d = diag(b, 1, 1, 1/b)

t t (b ∈ GL(1)) satisﬁes dJΘ d = x. Hence the part {ns3Θa n; n ∈ NB, a ∈ A} of X is

−1 t the B-orbit λ bs3Θ b, which is the image of Bγ1H under G/H → X.

The decomposition G = PH ∪P γ1H follows at once from the decomposition B\G/H. Note that we have proved also (d) in the case of θ = 1. The isolated points were obtained when s2 = J was discussed; they are x = ±J, not in the orbit of ΘJ. (c) It suﬃces to consider θ ∈ F − F 2. The proof follows closely that of (b). The relation sa = −as−1 implies that s 6= 1, (14), (23), and the relation (xJ)2 = θ implies that s 6= (14)(23). Two elements of W of order 2 are left.

0 0 1 0 −1 Choose the representative w0 = diag(( −1 0 ) , ( 1 0 )) for (12)(34) ∈ W . It satisﬁes t 0 0 0 1 0 −θ w0w0 = 1. It is more convenient to work with w0 the element diag(( −1 0 ) , θ 0 ). 0 −1 0 Since w0 = −w0, we have that x = x0a = aw0. Then a ∈ A has the form

2 a = diag(b, b, c, c). From (Jw0a) = θ it follows that bc = 1. But dw0d = w0a = x

t t if d = diag(b, 1, 1, 1/b). Since γ0ΘJ γ0 = w0, the part {naw0a n; n ∈ NB, a ∈ A} of

−1 t X(θ) is the B-orbit λ bw0 b, which is the image of Bγ0H under G/H → X(θ). Similarly we choose ΘJ to represent (13)(24) ∈ W (it is the product of the element diag(1, θ, 1, θ) and a representative s ∈ W with ts = s−1 = −s). If x = ΘJa then the relation “sa = −as−1” implies that a = diag(b, c, b, c). Further, bc = 1 from (xJ)2 = θ. But then dΘJd = ΘJa if d = diag(b, 1, 1, 1/b). Hence the part

24 t {naΘJa n; a ∈ A, n ∈ NB} of X(θ) is the image of BH under G/H → X(θ). Then (c) follows, and so does (d).

2.3 Geometric side

√ 2 Let F be a global ﬁeld (of characteristic 6= 2) , θ ∈ F − F ,E = F ( θ), A = AF

2 and AE the rings of ad´elesof F and E. Denote by L (Z(A)G(F )\G(A)) the space of complex valued functions on G(A) which are left invariant under G(F ), where Z is the center of G, and are absolutely square integrable on Z(A)G(F )\G(A). Let f = ⊗fv

∞ be a test function on G(A). Thus fv ∈ Cc (Gv/Zv),Gv = G(Fv) = GSp(2,Fv), for all

∞ v, where Cc means smooth (locally constant if v is ﬁnite) and compactly supported.

0 Moreover, for almost all v the component fv is the unit element fv in the convolution

∞ algebra of Kv = G(Rv)-biinvariant functions in Cc (Gv/Zv)(Rv is the ring of integers

0 in the nonarchimedean ﬁeld Fv). Thus fv is the quotient of the characteristic function of ZvKv in Gv by the volume |KvZv/Zv| with respect to the implicitly chosen Haar R measure on Gv/Zv. The convolution operator (r(f)φ)(h) = f(g)φ(hg)dg on Z(A)\G(A) 2 R L (Z( )G(F )\G( )) is an integral operator (= Kf (h, g)φ(g)dg) with A A Z(A)G(F )\G(A) P −1 kernel Kf (g, h) = γ∈Z(F )\G(F ) f(g γh). Our Fourier-Period summation formula is based on integrating this kernel on h in

Z(A)H(F )\H(A) and g in N(F )\N(A), against a character of N(F )\N(A), con- structed as follows. Let ψ be a ﬁxed nontrivial character of A/F . The unipotent

IX x y group N(A) consists of n = ( 0 I ) ,X = ( z x ). Any character of N(A)/N(F ) has the

t form ψT (n) = ψ(tr TX), where w T w = T is a 2 × 2 matrix with entries in F . The

25 t −1 Levi subgroup M(A), consisting of m = diag(U, λw U w), acts on ψT by

−1 −1 t −1 −1 ψT (mnm ) = ψ(λ tr T UXw Uw) = ψ(λ det U · tr εU εT UX).

Hence multiplying U, and consequently m, on the left by a suitable matrix, we may replace εT by a conjugate. Note that

wt(εg−1εT g)w = εg−1εT g (g ∈ GL(2)).

0 Moreover, the connected component StabM (ψT ) of the identity in the stabilizer

StabM (ψT ) of ψT in M is isomorphic to the centralizer Z(εT ) of εT in GL(2) via m ↔ U, λ = det U. The centralizer Z(εT )(A) is a torus in GL(2, A) when εT is

0 1 × nonsingular. Put ψθ for ψT when εT = ( θ 0 ) , θ ∈ F , and ψ0 = ψT when T = I. The absolute convergence of this integral is immediate: Let ||g|| denote the usual

P −1 N norm function on the group G(A) ([HCM], p. 6). Then γ∈G(F ) |f(g γh)| ≤ c||g|| (for some c = c(f) > 0,N = N(f) > 0) for all g, h. Integrating the last sum over h in the space Z(A)H(F )\H(A), which has ﬁnite volume, and over g in the compact

U(F )\U(A), where ||g|| is bounded, we obtain a ﬁnite number. We even have the following result.

Lemma 6. If f is compactly supported on Z(A)\G(A), then the function Kf is compactly supported on N(F )\N(A) × H(F )Z(A)\H(A).

Proof. If G is a connected linear algebraic group over F , and H is a reductive closed subgroup over F , then G/H is an aﬃne variety V over F ([Bo], Proposition 7.7).

Then V (F ) is discrete and closed in V (A). The natural map G(A)/H(A) → V (A) is continuous, and it maps G(F )/H(F ) ⊂ G(A)/H(A) to V (F ) ⊂ V (A). Hence

26 G(F )/H(F ) is closed in G(A)/H(A), namely G(F )H(A) is closed in G(A) and so

H(A)/H(F ) is closed in G(A)/G(F ). Moreover, for G over F as above, for any closed F -subgroup H of G, H(A)1/H(F ) is closed in G(A)/G(F ) ([Go], (2.1)). Now P −1 for our function f, since N(F )\N(A) is compact, Kf (u, h) = Z(F )\G(F ) f(u γh) has compact support on N(F )\N(A) × G(F )Z(A)\G(A), hence also on its closed subset N(F )\N(A) × H(F )Z(A)\H(A), by either of these results. A computational proof is as follows. The kernel

−1 X Kf (u , h) = f(uγh) γ∈G(F )/Z(F ) is equal to

X X X f(uνµηh) µ η ν

(µ ∈ N(F )\G(F )/H(F ), η ∈ H(F )/Z(F ), ν ∈ N(F )/N(F ) ∩ µH(F )µ−1).

By Proposition 1(c), a set of representatives for the µ is given by the elements (1)

α β α −β × × diag 0 1 , 0 1 , α ∈ F , β ∈ F ; and (2) diag(1, 1, λ, λ)γ0, λ ∈ F . If u lies in a ﬁxed compact subset of N(A), and f(uνµηh) 6= 0, then νµηh lies in a compact subset of G(A)/Z(A). Hence Ad(νµ)Θ = Ad(νµηh)Θ stays in a compact of G(A),

IX and consequently in a ﬁnite set. Put ν = ( 0 I ). Then for µ of the form (1), we have

−1 −1 AθA−1 −AθA−1X−XAθA−1 α β 0 1 νµΘµ ν = 0 −AθA−1 ,A = 0 1 , θ = ( θ 0 ) .

Since AθA−1 lies in a ﬁnite set, so do α and β. For µ of the form (2) we have

−1 −1 −θλX θλXX−λ−1 νµΘµ ν = −θλ θλX .

27 Hence λ lies in a ﬁnite set. Consequently only ﬁnitely many µ occur, and for each

µ we have a summation over ν in a ﬁnite subset of N(F )/N(F ) ∩ µH(F )µ−1 =

Ad(N)µΘµ−1. Finally, since νµηh lies in a compact of G(A)/Z(A), we conclude that h stays in a compact set modulo H(F )Z(A).

The geometric side of the Fourier summation formula is described as follows.

θ θ Proposition 7. For any f = ⊗fv on G(A) deﬁne the function f = ⊗fv on X(θ)(A) by f θ(gΘ g−1J) = R f (gh)dh. Then v θ Hv/Zv v Z Z X −1 f(n γh)ψθ(n)dndh (2.3.1) N(F )\N(A) Z(A)H(F )\H(A) γ∈Z(F )\G(F ) is absolutely convergent and equal to the (ﬁnite) sum

X X Ψ(λ, f θ) + Ψi(f θ), λ∈F × i=± where

θ Y θ i θ Y i θ Ψ(λ, f ) = Ψ(λ, fv ), Ψ (f ) = Ψ (fv ). v v Here, if u = λ−1(1 − yz − θ−1x2), we put

Z 0 u y x θ −2 −3 θ −u 0 −x θz −1 Ψ(λ, fv ) = |θ|v |λ|v fv −y x 0 −θλ ψv(−λ (y + z))dxdydz (2.3.2) 3 Fv −x −θz θλ 0 and Z 0 u 1 0 i θ θ −u 0 0 θ Ψ (fv ) = fv i −1 0 0 0 ψv(u)du. (2.3.3) Fv 0 −θ 0 0 Similarly, introduce f˜ = ⊗f˜ on X(1)( ) by f˜ (gΘg−1J) = R f (gh)dh. Then v A v Hv/Zv v Z Z X −1 f(n γh)ψ0(n)dndh (2.3.4) N(F )\N(A) Z(A)H(F )\H(A) γ∈Z(F )\G(F )

28 P ˜ P i ˜ is absolutely convergent and equal to the (ﬁnite) sum λ∈F × Ψ(λ, f) + i=± Ψ (f), where ˜ Y ˜ i ˜ Y i ˜ Ψ(λ, f) = Ψ(λ, fv), Ψ (f) = Ψ (fv). v v Here, with u = −λ−1(1 − yz − x2), we put

Z 0 u y x ˜ −3 ˜ −u 0 −x z Ψ(λ, fv) = |λ|v fv −y x 0 λ ψv(2x/λ)dxdydz (2.3.5) 3 Fv −x −z −λ 0 and Z 0 u 0 1 i ˜ ˜ −u 0 −1 0 Ψ (fv) = f i 0 1 0 0 ψv(u)du. (2.3.6) Fv −1 0 0 0 Proof. The integral (2.3.1) is a sum of two parts, according to Proposition 1(c). The main part is

Z Z X f(nγh)ψθ(n)dndh N(F )\N( ) Z( )H(F )\H( ) A A A γ∈B(F )γ0H(F ) which is equal to

Z X I 0 f (n ( 0 λ ) γ0h) ψ(z − θy)dndh, λ∈F × H(A)/Z(A)

I 0 since B(F )γ0H(F ) = N(F )Λγ0H(F ), Λ = {( 0 λ ); λ ∈ GL(1)}. By matrix multipli-

θ cation and the deﬁnition of fv we have that the local factor Z Z I 0 fv (n ( 0 λ ) γ0h) ψv(z − θy)dndh Nv Hv/Zv

θ is equal to Ψ(λ, fv ), as deﬁned in (2.3.2). The ﬁniteness of the sum over λ is easily

−1 proven on considering the map g 7→ gΘθg J, and using the fact that f is compactly supported.

29 The second part is

Z Z X f(nνmηh)ψθ(n)dhdn N(F )\N(A) Z(A)H(F )\H(A)

(here the summation is over ν ∈ N(F ), m ∈ M(F ) ∩ B(F )/Λ, η ∈ N(F ) ∩ H(F )\H(F )), which is equal to

Z Z X A 0 a b f (n ( 0 εAε ) h) ψθ(n)dndh (A = ( 0 1 )) . N(A) N(F )∩H(F )\H(A) a∈F ×;b∈F

I t x y Write h in the form ( 0 I ) h, where now t = θy x (x, y range over A/F ) and h ∈

N(A) ∩ H(A)\H(A). Since

A 0 I t A−1 0 I AtεA−1ε ( 0 εAε )( 0 I ) 0 εA−1ε = 0 I , and

tr [εA−1εT At] = a−1θy − θay − a−1b2θ2y − 2θbx, integrating over x in A/F we obtain 0 unless b = 0, in which case the volume |A/F | =

1 is obtained. Integrating over y in A/F again we get 0, unless a = ±1, in which case the volume |A/F | = 1 is obtained. Our integral is then the sum over i = ± of Z θ −1 f (inΘθn J)ψθ(n)dn. N(A)/N(A)∩H(A) The local factors of this integral are equal to those of (2.3.3).

The integral (2.3.4) is similarly handled. By Proposition 1(b) it is expressed as a sum of two parts. Since P (F )γ1H(F ) = N(F )Λγ1H(F ), the main part takes the form Z Z X I 0 f (nν ( 0 λ ) γ1ηh) ψ(tr X)dndh N(A)/N(F ) Z(A)H(F )\H(A) ν∈N(F ),λ∈F ×,η∈H(F )

30 which is equal to Z Z X I 0 f (n ( 0 λ ) γ1h) ψ(tr X)dndh. λ∈F × N(A) H(A)/Z(A) Note that

−1 A 0 γ1H(F )γ1 ∩ P (F ) = {( 0 εAε )} , since a 0 b 0 −1 A 0 a b 0 γ1 0 A 0 γ1 = ( 0 εAε ) ,A = ( ) ,A = wAw. c 0 d c d ˜ × ˜ Using the deﬁnition of fv we then obtain the sum over λ ∈ F of Ψ(λ, f) = Q ˜ ˜ v Ψ(λ, fv), where Ψ(λ, fv) is deﬁned by (2.3.5). The second part of (2.3.4) is

Z Z X f(nνd(ξ)ηh)ψ(tr X)dndh N(A)/N(F ) Z(A)H(F )\H(A) ν∈N(F )/N(F )∩H(F ) ξ∈H1(F )/A(F ),η∈H(F )/Z(F ) which is equal to

Z Z X f(nd(ξ)h)ψ(tr X)dndh, N( )/N(F )∩H(F ) H( )/Z( ) A A A ξ∈H1(F )/A(F ) t −1 where H1 = GL(2,F ),A = diagonal subgroup in H1, and d(ξ) = diag(ξ, w ξ w). Since

IY −1 I ξY wtξw 0 y d(ξ)( 0 I ) d(ξ) = 0 I ,Y = z 0 , and

t α β tr ξY w ξw = 2αγy + 2βδz if ξ = γ δ ,

IY changing h 7→ ( 0 I ) h, and integrating over y, z in A/F , we obtain 0 unless ξ is I or w, modulo the diagonal subgroup A. Using now the deﬁnition of f˜, we obtain the

i ˜ Q i ˜ i ˜ sum over i = ± of Ψ (f) = v Ψ (fv), where Ψ (fv) is deﬁned by (2.3.6).

31 2.4 Asymptotes

To compare the geometric side (2.3.1) (or (2.3.2)) of the Fourier summation formula with an analogous expression for a different group, we need to characterize the Fourier orbital integrals (2.3.2) and (2.3.3) (or (2.3.5) and (2.3.6)) of which the formula consists. To express this characterization, let F be a local field, ψ : F → C× a R nontrivial character, dx the autodual Haar measure on F, ϕˆ(x) = F ϕ(y)ψ(xy)dy ∞ × the Fourier transform of ϕ ∈ Cc (F ), and γψ(a)(a ∈ F ) the Weil function, defined by Z Z 1 2 −1/2 1 −1 2 ϕ(x)ψ ax dx = γψ(a)|a| ϕˆ(x)ψ − a x dx. F 2 F 2 × ×2 Then γ(a) = γψ(1)γψ(−a) is a function from F /F to the group of the complex fourth roots of unity, satisfying γ(a)γ(b) = γ(ab)(a, b) (see [We1]), where (a, b) is the Hilbert symbol (= 1 if a = x2 − by2, = −1 if not). Put G = GSp(2); | · | is the normalized absolute value on F .

Deﬁnition 6. The complex valued functions Ψ1, Ψ2 on F are called equivalent, and we write Ψ1 ≡ Ψ2, if Ψ1(λ) = Ψ2(λ) for λ in some neighborhood of 0 (F nonarchimedean), and if Ψ1, Ψ2 have equal derivatives of all orders at λ = 0 (F archimedean).

∞ × Proposition 8. (a) For every function f ∈ Cc (G(F )/Z(F )) and θ ∈ F , the function Ψ(λ, f θ) of (2.3.2) is compactly supported in λ on F , smooth (locally constant

32 if F is nonarchimedean) on F ×, and

θ −1 −3/2 Ψ(λ, f ) ≡ γψ(−1)γψ(θ)|2λ| |θ|

·{(θ, 2λ)ψ(−2/λ)Ψ+(λ, f θ) + (θ, −2λ)ψ(2/λ)Ψ−(λ, f θ)}, where Ψi(λ, f θ) are smooth functions in a neighborhood of λ = 0 whose values at

λ = 0 are the Ψi(f θ) which are deﬁned by (2.3.3) , i = ±.

Conversely, if Ψ(λ) is a complex valued compactly supported function in λ on F , smooth on F ×, and

−1 −3/2 + − Ψ(λ) ≡ γψ(−1)γψ(θ)|2λ| |θ| {(θ, 2λ)ψ(−2/λ)Ψ (λ) + (θ, −2λ)ψ(2/λ)Ψ (λ)}, for some smooth complex valued functions Ψ+(λ), Ψ−(λ), then there exists a func-

∞ θ × i θ tion f ∈ Cc (G(F )/Z(F )) with Ψ(λ, f ) = Ψ(λ) for all λ ∈ F , and Ψ (λ, f ) ≡ Ψi(λ)(i = ±).

∞ ˜ (b) For every function f ∈ Cc (G(F )/Z(F )) the function Ψ(λ, f) of (2.3.5) is compactly supported in λ on F , smooth on F ×, and

Ψ(λ, f˜) ≡ |2λ|−1ψ(2/λ)Ψ+(λ, f˜) + |2λ|−1ψ(−2/λ)Ψ−(λ, f˜), where Ψi(λ, f˜) are smooth near λ = 0 and whose values at λ = 0 are the Ψi(f˜) which are deﬁned by (2.3.6), i = ±.

Conversely, if Ψ(λ) is a complex valued compactly supported function in λ on F , smooth on F ×, and

Ψ(λ) ≡ |2λ|−1ψ(2/λ)Ψ+(λ) + |2λ|−1ψ(−2/λ)Ψ−(λ),

33 for some smooth complex valued functions Ψ+(λ), Ψ−(λ), then there exists a function

∞ ˜ × i ˜ i f ∈ Cc (G(F )/Z(F )) with Ψ(λ, f) = Ψ(λ) for all λ ∈ F , and Ψ (λ, f) ≡ Ψ (λ) (i = ±).

Proof. (a) Consider the case of θ ∈ F − F 2. The function f θ is smooth with compact

2 5 support on the quadric b1b2 + b − ad = θ (in F ) part of X(θ) (see Proposition 1(d)).

This quadric is the union of the open subsets {b1 6= 0} and {d 6= 0}, since θ is not a square in F . If f θ is supported on {d 6= 0}, the integral Ψ(λ, f θ) of (2.2) is zero

θ for λ in some neighborhood of zero. Assume then that f is supported on {b1 6= 0}.

−1 −1 2 This set is parametrized by b1, b, a, d. The substitution z 7→ u = λ (1 − yz − θ x ), dz = |λ/y|du, gives

ZZZ Ψ(λ, f θ) = |θλ|−2 ϕ(u, x, y, λ)ψ(−λ−1(y + y−1) + u/y + (θλy)−1x2)dudxd×y, where 0 u y x !! ϕ(u, x, y, λ) = f θ −u 0 −x (θ−x2−θλu)/y ∗ ∗ 0 −θλ ∗ ∗ θλ 0 is a smooth compactly supported function on the subset {y 6= 0} of F 4. To simplify the notations, assume that ϕ(u, x, y, λ) is a product ϕ1(u)ϕ2(x)ϕ3(y)ϕ4(λ), where

∞ ∞ × θ ϕi ∈ Cc (F )(i = 1, 2, 4) and ϕ3 ∈ Cc (F ). We then get that Ψ(λ, f ) equals Z Z −2 −1 −1 2 −1 −1 × |θλ| ϕ4(λ) ϕˆ1(y )ϕ3(y) ϕ2(x)ψ((θλy) x )dx ψ(−λ (y + y ))d y F × F Z −2 −1 1/2 = |θλ| ϕ4(λ) ϕˆ1(y )ϕ3(y)γψ(2θλy)|θλy/2| F × Z 1 2 −1 −1 × · ϕˆ2(x)ψ(− θλyx )dx ψ(−λ (y + y ))d y. F 4

34 Assume now that F is nonarchimedean. For λ suﬃciently close to 0, we have that

Ψ(λ, f θ) equals

Z −1/2 −3/2 −1 1/2 −1 −1 × |2| |θλ| ϕ2(0)ϕ4(0) ϕˆ1(y )ϕ3(y)|y| γψ(2θλy)ψ(−λ (y + y ))d y. F ×

We shall prove below the following

∞ × R −1 −1 × Lemma 9. For any ϕ in Cc (F ), the value of F × ϕ(y)ψ(λ (y + y ))d y at λ near zero is

1/2 1/2 γψ(2λ)|λ/2| ψ(2/λ)ϕ(1) + γψ(−2λ)|λ/2| ψ(−2/λ)ϕ(−1).

Hence for λ near zero we obtain

θ −1 −3/2 −1 Ψ(λ, f ) = |2| |θ| |λ| ϕ2(0)ϕ4(0)

·{γψ(−2λ)ψ(−2/λ)ϕ ˆ1(1)ϕ3(1)γψ(2θλ) + γψ(2λ)ψ(2/λ)ϕ ˆ1(−1)ϕ3(−1)γψ(−2θλ)}.

Since γψ(−2λ)γψ(2θλ) = γψ(−1)γψ(θ)(2λ, θ), we obtain

−1 −3/2 −1 = γψ(−1)γψ(θ)|2| |θ| |λ| ϕ2(0)ϕ4(0){ϕˆ1(1)ϕ3(1)(θ, 2λ)ψ(−2/λ)

+ϕ ˆ1(−1)ϕ3(−1)(θ, −2λ)ψ(2/λ)} Z −1 −3/2 −1 = γψ(−1)γψ(θ)|2| |θ| |λ| {(θ, 2λ)ψ(−2/λ) ϕ(u, 0, 1, 0)ψ(u)du F Z +(θ, −2λ)ψ(2/λ) ϕ(−u, 0, −1, 0)ψ(u)du}. F

This is the asserted asymptotic behavior.

35 −1 1 Assume next that F = R, the ﬁeld of real numbers. Put ϕ5(y) =ϕ ˆ1(y )ϕ3(y)|y| 2 , ∗ R 1 2 θ and ϕ (t) = ϕˆ2(x)ψ( tx )dx. Then Ψ(λ, f ) is equal to 2 R 2 Z −1/2 −3/2 ∗ 1 −1 −1 × 2 |θλ| ϕ4(λ){γψ(2θλ) ϕ5(y)ϕ2 − θλy ψ(−λ (y + y ))d y × 2 R+ Z ∗ 1 −1 −1 × +γψ(−2θλ) ϕ5(−y)ϕ2 θλy ψ(−λ (y + y ))d y}. × 2 R+ ∞ × For h ∈ Cc (R+) we have Z y + y−1 Z (x − x−1)2 h(y)ψ d×y = 2ψ(λ−1) h(x2)ψ d×x × 2λ × 2λ R+ R+ √ (y = x2). On changing x − x−1 = 2u, x = u + u2 + 1, this becomes

Z √ du = 2ψ(λ−1) h((u + u2 + 1)2)ψ(2u2/λ)√ 2 R u + 1 Z λ = γ (λ)|λ|1/2ψ(λ−1) gˆ(u)ψ − u2 du, ψ 8 R √ √ where g(u) = h((u + u2 + 1)2)/ u2 + 1. To use this formula in our context, put gi(λ) to be Z Z λ 2 2 − 1 p 2 ∗ i p 2 ψ(i v )dv (u + 1) 2 ϕ i(u + u2 + 1) ϕ − θλ(u + u2 + 1) ψ(uv)du 16 5 2 2 R R ∗ for i = ±. Note that gi are smooth functions on R (although ϕ2 is not a Schwartz

θ function), and that gi(0) = ϕ2(0)ϕ5(i1). Then Ψ(λ, f ) is equal to

−1 −3/2 −1 γψ(−1)γψ(θ)2 |θ| |λ| ϕ4(λ){(2λ, θ)ψ(−2/λ)g+(λ) + (−2λ, θ)ψ(2/λ)g−(λ)}, and the asymptotic behavior asserted in (a) of the proposition follows.

The case of θ ∈ F ×2 is similar to that of (b), which is done next.

(b) Consider the integral Ψ(λ, f˜) of (2.3.5). The function f˜ is smooth and compactly

2 5 supported on the quadric X = {b1b2 + b − ad = 1 in F } described by Proposition

36 i 1(d). This X is the union of four open subsets: Vi = {b+ 2 (b1−b2) 6= 0} (i = ±),Vb1 = ˜ ˜ {b1 6= 0},Vd = {d 6= 0}. If f is supported on {d 6= 0} then Ψ(λ, f) is zero for λ in ˜ 2 some neighborhood of zero. If f is supported on {b1 6= 0}, writing z = (1+λu−x )/y we see that Ψ(λ, f˜) is rapidly decreasing as λ → 0. Assume then that f˜ is supported

1 1 1 on V+. Change variables: x 7→ 2 (y + z), y 7→ x + 2 (y − z), z 7→ x − 2 (y − z), and note that x2 + yz, hence u, are not changed. Then we get that Ψ(λ, f˜) equals to

0 u x+ 1 (y−z) 1 (y+z) !! ZZZ 2 2 −3 −u 0 − 1 (y+z) x− 1 (y−z) −1 |λ| f˜ 2 2 ψ(λ (y + z))dxdydz, F 3 ∗ ∗ 0 λ ∗ ∗ −λ 0 and Ψ(−λ, f˜) equals to ZZ Z |λ|−2 ϕ(u, x, y, λ)ψ(−λ−1(y + y−1))ψ(u/y)ψ(λ−1y−1x2)dudxd×y, F 2 F × where 0 u ∗ ∗ ˜ −u 0 ∗ ∗ ϕ(u, x, y, λ) = f ∗ ∗ 0 −λ ∗ ∗ λ 0 is smooth with compact support on the subset {y 6= 0} of F 4. We obtained precisely the same integral as in the nonsplit case (a), where θ ∈ F − F 2, but with θ = 1.

The computation there applies with θ = 1 too. The leading term in the asymptotic expansion of Ψ(λ, f˜) is then Z Z |2λ|−1 ψ(2/λ) ϕ(u, 0, 1, 0)ψ(u)du + ψ(−2/λ) ϕ(−u, 0, −1, 0)ψ(u)du F F = |2λ|−1ψ(2/λ)Ψ+(λ, f˜) + |2λ|−1ψ(−2/λ)Ψ−(λ, f˜).

˜ ˜ The treatment of f which is supported on V− is similarly carried out. A general f

can be expressed as f+ + f− + fb1 + fd, with f∗ supported on the open set V∗. The case where F is the ﬁeld of complex numbers is similarly handled.

37 i θ For the opposite direction(s), given Ψ(λ) choose f1 with Ψ (λ, f1 ) of (2.3.3) (or i ˜ i θ ˜ Ψ (λ, f1) of (2.3.6)) equivalent to Ψ (λ). Then Ψ(λ) − Ψ(λ, f1 ) (or Ψ(λ) − Ψ(λ, f1))

× θ is smooth and compactly supported on F , and it is easy to ﬁnd f2 with Ψ(λ, f2 ) (or ˜ × Ψ(λ, f2)) equal to this function on F . Then f = f1 +f2 is the required function.

It remains to prove Lemma 9. It follows on taking a = b = ±2/λ in the following

Lemma. Thus let F be a nonarchimedean local ﬁeld, R its ring of integers, π a generator of its maximal ideal (π), and v the valuation on F ×, normalized by v(π) = 1.

Lemma 10. Given a nontrivial character ψ of F , and N ≥ 2v(2) + 1, there is a

× R 1 constant A such that for any a, b ∈ F with v(b) ≤ A we have that 1+(πN ) ψ( 2 (at + −1 −1/2 2 N bt ))dt is equal to ψ(ac)|a| γψ(ac) if b/a = c , c ≡ 1 mod (π ), and to zero otherwise.

Proof. It suﬃces to prove this for ψ such that R⊥ = {x ∈ F ; ψ(xy) = 1 for all y ∈ R} is R. In this case one may take A = −2N − v(2). The integral of our lemma is equal to

X Z 1 1 ψ (atu + bt−1u−1) du, n = (1 + v(2) − v(b)) . 1+(πn) 2 2 t∈1+(πN )/1+(πn)

If t ∈ 1 + (πN ) is such that the integral over 1 + (πn) is nonzero, then on writing u = 1 + πnx, |x| ≤ 1, we see that v(at − bt−1) + n − v(2) ≥ 0. Hence v(b−1at2 − 1) ≥

N + v(2), namely b−1a ≡ 1 mod (2πN ), and so a−1b = c2 with c ≡ 1 mod (πN ). If a−1b = c2 and c ≡ 1 mod (πN ), then

Z 1 Z 1 ψ( (at + bt−1))dt = ψ( ac(t + t−1))dt. 1+(πn) 2 1+(πN ) 2

38 If now λ ∈ F × has v(λ) ≥ 2N + v(2), then

Z t + t−1 Z (x − x−1)2 ψ( )dt = |2|ψ(λ−1) ψ( )dx (t = x2) N 2λ 1 N 2λ 1+(π ) 1+( 2 π ) Z −1 2 −1 1/2 = |2|ψ(λ ) ψ(2y /λ)dy = ψ(λ )|λ| γψ(λ), 1 N ( 2 π ) where x − x−1 = 2y, namely x = y + p1 + y2. The lemma follows, as does Lemma

2.5 Matching

The geometric side of the Fourier summation formula on G = GSp(2) is analogous

− and will be compared to − a Fourier summation formula on G0 = GL(2). Let F

0 0 0 0 ∞ 0 be a global ﬁeld, and f = ⊗fv a test function on G (A). Thus fv ∈ Cc (Gv/Zv)

0 0 0 for all v, and for almost all v this fv is the unit element fv in the convolution

0 0 0 algebra of K = Gv(Rv)-biinvariant function on Gv/Zv. Here Z denotes (also) the

0 0 R 0 center of G . The convolution operator (r(f )φ)(x) = 0 f (y)φ(xy)dy on G (A)/Z(A) 2 0 0 R L (Z( )G (F )\G ( )) is an integral operator (= 0 0 Kf 0 (x, y)φ(y)dy) with A A Z(A)G (F )\G (A) P 0 −1 0 the kernel Kf 0 (x, y) = f (x γy)(γ ∈ Z(F )\G (F )). The geometric side of the

Fourier summation formula for G0(A) is obtained on integrating this kernel against

0 0 0 1 ∗ ψ(x) on x over N (F )\N (A),N = {( 0 1 )} , ψ(x) = ψ(∗) if ∗ ∈ A/F , and ψ is our

a 0 × × ﬁxed nontrivial character of A/F , and against χ(a) over y = ( 0 1 ), a ∈ A /F ,

× × 0 −1 where χ is a character of A /F whose square is 1. Put ψ = ψ, and w = ( 1 0 ).

39 0 0 0 × × Proposition 11. For any f = ⊗fv on G (A) and a character χ of A /F with χ2 = 1, the integral Z Z X 0 1 x a 0 × f (( 0 1 ) γ ( 0 1 )) ψ(x)χ(a)dxd a × × A/F A /F γ∈Z(F )\G0(F )

P 0 + 0 − 0 0 Q 0 is equal to the sum of λ∈F × Ψχ(λ, f ), Ψχ (f ) and Ψχ (f ), here Ψχ(f ) = v Ψχ(fv) and Z Z 0 0 1 x 1 λ a 0 × Ψχv (λ, fv) = fv (( 0 1 ) w ( 0 1 )( 0 1 )) ψv(x)χv(a)dxd a, × Fv Fv Z Z − 0 0 1 x a 0 × + 0 0 Ψχv (fv) = fv (( 0 1 )( 0 1 )) ψv(x)χv(a)dxd a, Ψχv (fv) = Ψχv (0, fv). × Fv Fv This follows at once from the Bruhat decomposition G0 = B0 ∪ N 0wB0. From

1 x 0 1 1 λ 1 0 1 −λ 0 1 1 −x 0 −1 2x(1+λ) 1+2xλ ( 0 1 )( 1 0 )( 0 1 )( 0 −1 )( 0 1 )( 1 0 )( 0 1 )( 1 0 ) = 1+2xλ 2λ it transpires that the sum over λ ranges over a ﬁnite set, depending on the support

0 0 of f , that x ranges (in Ψχv (fv)) over a compact set in Fv, and that a ranges there

× 0 over a compact set in Fv , again depending on the support of f .

0 As in the case of G, we shall characterize next the Fourier orbital integrals Ψχv (λ, fv), as in Jacquet [J2], Proposition 4.2, p. 129. Let F be a local ﬁeld, and χ : F × → {±1} a character with χ2 = 1.

0 ∞ 0 Proposition 12. (a) For every function f ∈ Cc (G (F )/Z(F )) there are smooth

+ 0 − 0 i 0 i 0 compactly supported functions Ψχ (λ, f ) and Ψχ (λ, f ) on F with Ψχ(0, f ) = Ψχ(f )

0 + 0 −1 − 0 × (i = ±) such that Ψχ(λ, f ) = Ψχ (λ, f ) + ψ(λ )Ψχ (λ, f ) for all λ in F . (b) For any smooth compactly supported functions Ψ+(λ) and Ψ−(λ) on F there is a function

0 0 ∞ 0 0 + −1 − f = fχ ∈ Cc (G (F )/Z(F )) with Ψχ(λ, f ) = Ψ (λ) + ψ(λ )Ψ (λ) on F .

40 0 Proof. Write Ψ for Ψχ in the course of the proof. Expressing G (F ) as the union of the two open sets N(F )wN(F )A(F ) and N(F )N(F )A(F )(N(F ) = wN(F )w−1), N =

1 ∗ ∗ 0 0 {( 0 1 )},A = {( 0 ∗ )}, we may assume that f is supported on one of these sets. If f is supported on N(F )wN(F )A(F ), then Ψ(λ, f 0) is smooth and compactly supported on F , and Ψ(0, f 0) = Ψ+(f 0). Any smooth compactly supported function on F is of

0 0 ∞ 0 the form Ψ(λ, f ) for some f ∈ Cc (G (F )/Z(F )) supported on N(F )wN(F )A(F ). Suppose that the function f 0 is supported on N(F )N(F )A(F ). Denote by T (u, v)

R 0 1 0 1 u a 0 × the integral f (( v 1 )( 0 1 )( 0 1 )) χ(a)d a. It is smooth and compactly supported on F ×F , and any smooth compactly supported function on F ×F is of the form T (u, v) for some f 0 supported on N(F )N(F )A(F ). Since f 0 is trivial on the center and χ2 = 1, we have that Ψ(λ, f 0) = R T (x(xλ − 1), x−1)ψ(x)dx, Ψ−(f 0) = R T (x, 0)ψ(x)dx.

−1 −1 If T (x(xλ − 1), x ) 6= 0 then |x(xλ − 1)| ≤ c1, |x| ≤ c2 for some positive constants

0 −1 −1 depending on f (or T ). Then c2 ≤ |x| ≤ (1 + c1c2)|λ| and |λ| ≤ (1 + c1c2)c2. Hence Ψ(λ, f 0) is smooth on F × and compactly supported on F . If T vanishes unless v lies in a compact of F ×, then Ψ(λ, f 0) is smooth at 0, and Ψ−(f 0) = 0. Hence we may assume that T is supported on {(u, v); |u| ≤ c1, |v| ≤ c2}, for some ﬁxed c1, and

+ 0 0 a suﬃciently small c2 > 0, say with c1c2 < 1. In particular Ψ (f ) (= Ψ(0, f )) is 0. Changing x to z = x − λ−1 we obtain

Z Ψ(λ, f 0) = ψ(λ−1) T (z(λz + 1), λ/(1 + λz))ψ(z)dz.

−1 −1 If the integrand is not zero then c2 |λ| ≤ |1 + λz| ≤ c1|z| , hence |λz| ≤ c1c2 < 1,

−1 −1 and |1 + λz| ≥ 1 − c1c2 > 0, and ﬁnally |z| ≤ c1|1 + λz| ≤ c1(1 − c1c2) . The integrand is then a compactly supported function of z in F . If λ is near zero, so is

41 λz, and 1 + λz lies in a neighborhood of 1. The value of the integral over z at λ = 0 is clearly Ψ−(f 0), hence (a) follows.

i 0 For (b), given smooth compactly supported functions Ψ (λ) on F (i = ±), there is f2

− 0 − 0 supported on N(F )N(F )A(F ) with Ψ (f2) = Ψ (0), and there is f1 supported on

0 N(F )wN(F )A(F ) with Ψ(λ, f1) equals to

+ −1 − 0 Ψ (λ) + ψ(λ )Ψ (λ) − Ψ(λ, f2), (∗) since the last expression is smooth and compactly supported on F . The required

0 0 0 function is f = f1 + f2.

0 In fact, when F is archimedean we need to show that f2 can be chosen such that all

− 0 − derivatives of Ψ (λ, f2) and Ψ (λ) are equal at λ = 0, since then (∗) is smooth on F

0 and f1 can be found. For this purpose let T1,T2 be smooth functions on F supported R on |u| ≤ c1, |v| ≤ c2, with c1c2 < 1, and with T1(z)ψ(z)dz = 1. Then the integrand in Z T1(z(1 + λz))T2(λ/(1 + λz))ψ(z)dz is supported on |z| ≤ c1/(1−c1c2), and the value of the integral at λ = 0 is T2(0). The value of the derivatives of the integral can be computed in terms of the derivatives of

0 − 0 T2 at λ = 0, hence f2 can be chosen so that Ψ (λ, f2) and all of its derivatives would take at λ = 0 any prescribed values, as asserted.

0 Note that Ψχ(λ, f ) depends on a choice of a quadratic character χ. In matching general test functions, any χ can be taken in the following deﬁnition. But the matching of spherical functions (cf. [J2], Proposition 5.1, p. 132) forces the choice of χ = χθ.

42 × Deﬁnition 7. Fix θ in F , choose χ = χθ, and ﬁx ψ 6= 1. The functions f ∈

∞ 0 ∞ 0 × Cc (G(F )/Z(F )) and f ∈ Cc (G (F )/Z(F )) are called matching if for all λ ∈ F we have

θ −1 −3/2 0 Ψ(λ, f ) = γψ(−1)γψ(θ)|2λ| |θ| (θ, 2λ)ψ(−2/λ)Ψχ(λ/4, f ).

∞ 0 ∞ 0 Corrolary 13. For every f ∈ Cc (G(F )/Z(F )) there exists f ∈ Cc (G (F )/Z(F )), and for every f 0 there exists an f, such that f and f 0 are matching. If f and f 0 are

i θ i 0 matching, then Ψ (f ) = (θ, i)Ψχ(f )(i = ±).

Proof. This follows at once from Propositions 8 and 12.

Proposition 14. Suppose that F is a p-adic ﬁeld, p > 2, θ is a unit, χ is unramiﬁed and χ2 = 1, and ψ is 1 on R but not on π−1R. Then the unit elements f 0 and f 00 are matching.

Proof. If f = f 0 then f θ (see Proposition 7) is the characteristic function of the set of points in X(θ) (see Proposition 1(d)) with integral coordinates on the quartic

2 ± θ b1b2 + b − ad = θ. Using (2.3.3) it is clear that Ψ (f ) = 1. From (2.3.2) it follows that Ψ(λ, f θ) is 0 if λ 6∈ R, it is 1 if λ ∈ R×. Suppose that |λ| < 1. As in the beginning of the proof of Proposition 3(a), we have ZZZ Ψ(λ, f θ) = |λ|−2 ψ(−λ−1(y + y−1) + uy−1 + x2(θλy)−1)dxd×ydu.

The integration ranges over (x, y, u) in R3, with θ − x2 − θλu in yR. But the integral of ψ(uy−1)du over R ∩ [λ−1(1 − θ−1x2) + λ−1yR] is 1 if y ∈ R× and 0 if y ∈ R − R×.

Hence Z Z Ψ(λ, f θ) = |λ|−2 ψ(−λ−1(y + y−1) + x2(θλy)−1)dxd×y. R R×

43 Using the deﬁnition of γψ, recorded before Proposition 3, this is Z −3/2 −1 −1 × = |λ| ψ(−λ (y + y ))γψ(2θλy)d y. R× If |λ| < q−1 = |π|, this can be written as (y = tu) Z −3/2 X −1 −1 −1 |λ| γψ(2θλt) ψ(−λ (tu + t u ))du. π t∈R×/(1+ππR) 1+ππR

R −1 −1 −1 Lemma 10, with N = 1 and A = −2, asserts that 1+ππR ψ(−λ (tu + t u ))du is 0 1/2 −1 unless t ≡ ±1 mod ππR, and it is |λ| ψ(−2tλ )γψ(−2λt) if t = ±1. Hence

θ −1 X −1 Ψ(λ, f ) = |λ| γψ(2θλt)γψ(−2λt)ψ(−2t/λ) = |λ| (θ, λ)(ψ(2/λ) + ψ(−2/λ)), t=±1 since γψ(2θλt)γψ(−2λt) = γψ(−1)γψ(θ)(θ, 2λt) = (θ, λ): γψ is 1 on units, and (u, v) = 1 for units u, v.

θ If |λ| = |π|, since γψ(2θλy) = γψ(2λ)(θ, λ)(λ, y), we have that Ψ(λ, f ) is equal to Z −3/2 −1 × |λ| (θ, λ)γψ(2λ) (λ, y)ψ(−(y + y )/λ)d y R× −1/2 X −1 = |λ| (θ, λ)γψ(2λ) (λ, y)ψ(−(y + y )/λ). y∈R×/(1+ππR) Lemma 15. Let k be a ﬁnite ﬁeld of odd characteristic. Let τ be a nontrivial character of k. Put ε(x) = 1 if x is a square in k×, ε(x) = −1 if x is not a square, ε(0) = 0.

Then X X ε(y)τ(y + y−1) = (τ(2) + τ(−2)) τ(x2). y∈k× x∈k Apply this to k = R/πππR, τ(x) = ψ(−λ−1x). Then

X (λ, y)ψ(−(y + y−1)/λ) y∈R×/(1+ππR)

44 is equal to

X −1 2 −1/2 (ψ(2/λ) + ψ(−2/λ)) ψ(−λ x ) = (ψ(2/λ) + ψ(−2/λ))γψ(−2λ)|λ| . x∈R/πππR

Hence

Ψ(λ, f θ) = |λ|−1(θ, λ)(ψ(2/λ) + ψ(−2/λ)).

Proof. For z ∈ k, the quadratic equation y + y−1 = z has (1) two solutions, both squares in k×, if both z + 2 and z − 2 are nonzero squares; (2) two solutions, none of which is a square, if neither z + 2 nor z − 2 is a square; (3) the solution y = 1 if z = 2; (4) the solution y = −1 if z = −2; and (5) no solutions if precisely one of z + 2, z − 2, is a square. Then

X X ε(y)τ(y + y−1) = (ε(z − 2) + ε(z + 2))τ(z) y∈k× z∈k X X = (ε(z − 2) + 1)τ(z) + (ε(z + 2) + 1)τ(z) z∈k z∈k X X X = τ(x2 + 2) + τ(x2 − 2) = (τ(2) + τ(−2)) τ(x2). z∈k z∈k x∈k The lemma follows.

0 00 0 ± 00 00 If f is f on G , and χ is unramiﬁed, we have Ψχ (f ) = 1, and Ψχ(f ) is 0 if λ 6∈ R, 1 if λ ∈ R×, 1 + ψ(λ−1) if 0 < |λ| < 1. The proposition follows.

2.6 Corresponding

The Fourier summation formula for G0(A) = GL(2, A) is an identity of sums of distributions. The geometric side is described by Proposition 11. The spectral side

45 will be recorded next, following e.g. [F5], Section D. The notations are those of [F5], to be recalled below.

0 0 0 × × Proposition 16. For any f = ⊗fv on G (A) and a character χ of A /F with χ2 = 1, the integral of Proposition 11 is equal to the sum of

X X 0 0 0 1 0 W (π (f )Φ )L 0 , π ⊗ χ (2.6.1) ψ Φ 2 π0 Φ0∈π0 and

X 1 1 4π E I(f 0, χ, )Φ0, χ, Φ0(1) (2.6.2) ψ 2 2 Φ0 and

Z 1 X X 0 0 −1 1 E (I(f , ω, ζ)Φ , ω, ζ) ·L 0 ω , − ζ dζ, (2.6.3) 2 ψ Φ ,χ 2 ω Φ0 iR where

2 2 −1 LΦ,χ(ω, ζ) = LΦ(ζ, ωχ) LΦ(2ζ, ω ) .

Here π0 ranges over all cuspidal irreducible representations of PGL(2, A); Φ0 ranges over an orthonormal basis of smooth vectors for the space of the representation π0 in

2 0 0 0 the space L0(Z(A)G (F )\G (A)). The Whittaker functional Wψ(Φ ) is deﬁned by the integral R Φ0(u)ψ(u)du, and N(F )\N(A) Z 0 0 a 0 ζ−1/2 × LΦ0 (ζ, π ⊗ ω) = Φ (( 0 1 )) |a| ω(a)d a × × F \A Z 1 0 a 0 0 ζ− 2 × = Wψ(π (( 0 1 )) Φ )|a| ω(a)d a × A

46 is the L-function of π0⊗ω at ζ associated with Φ. The ω range over a set of representatives for the set of connected components of the complex manifold of unitary characters x 7→ ω(x) of A×/F ×, a connected component consisting of ωνiζ , ν(x) = |x|, ζ ∈ R

(= ﬁeld of real numbers). The normalizedly induced G0(A)-module I(ω, ζ) consists of smooth Φ0 with

0 a ∗ ζ+1/2 0 Φ (( 0 b ) g, ζ) = ω(a/b)|a/b| Φ (g, ζ), and

Z 0 0 Eψ(Φ , ω, ζ) = E(u, Φ , ω, ζ)ψ(u)du, N(F )\N(A) X E(h, Φ0, ω, ζ) = Φ0(δh, ζ). δ∈B(F )\G(F )

The sums in (2.6.2), (2.6.3), range over an orthonormal basis for I(ω, ζ), indepen- dently of ζ. All sums and integrals are absolutely convergent.

Our next aim is then to develop an analogous Fourier summation formula for the group G = GSp(2). We shall do that on multiplying the spectral expression Kf (u, h) for the kernel by ψθ(u) and integrating over u in N(F )\N(A) and h in the set

H(F )Z(A)\H(A). It will be useful to ﬁx conventions for induction. These should be the same as those used in the theory of Eisenstein series, which are used to describe the contribution to the kernel from its spectral decomposition, as recalled in the next few sections. Thus let F be a local ﬁeld, π0 an admissible representation of GL(2,F ) with a trivial central

× character, ω a character of F , and ζ2 a complex number. Then the G(F )-module

47 0 0 normalizedly induced from the data π , ζ2, ω on P (F ) will be denoted by IP (π , ζ2, ω). It consists of all smooth functions φ : G(F ) → π0 which satisfy

3 A ∗ ζ2+ 2 0 φ 0 λwtA−1w g = ||A|/λ| ω(|A|/λ)π (A)(φ(g))

(A ∈ GL(2,F ), |A| = det A, g ∈ G(F ), λ ∈ F ×).

0 × 2 If π = I(ζ1, −ζ1) ⊗ χ,(χ a character of F with χ = 1, ζ1 a complex number), is

a ∗ the GL(2,F )-module consisting of the smooth φ : PGL(2,F ) → C with φ (( 0 b ) g) =

ζ1+1/2 0 χ(ab)|a/b| φ(g), then IP (π , ζ2, ω) is IB(ζ1, ζ2 − ζ1, χ, ω/χ). Here IB(ζ1, ζ2, χ, ω) is the G(F )-module induced from the Borel subgroup B(F ), consisting of the smooth functions φ : G(F ) → C which satisfy

a ∗ 4 2 1/2 2 ζ1 ζ2 2 b a b a ab a ab λ/b φ g = 3 χ ω φ(g). 0 λ/a λ λ λ λ λ

× 0 Here a, b and λ are in F . If π is a GL(2,F )-module with central character ωπ0 , and

0 0 ζ3 a complex number, the induced IQ(π , ζ3) consists of φ : G → π ,

a ∗ ∗ 0 A ∗ 2 1+ζ3 −1 0 φ g = |a /λ| ω 0 (a)(π (A)φ)(g) 0 0 λ/a π

(A ∈ GL(2,F ), λ = |A|, a ∈ F ×).

0 2 0 −1 If π = I(ζ4, −ζ4) ⊗ χ, then ωπ0 = χ , and IQ(π , ζ3) = IB(ζ3 − ζ4, 2ζ4, χ , 1). Analo- gous notations will be used also in the global case .

It will be useful to compute the trace tr π(f) of π = I(ζ1, ζ2). For this purpose take

48 ∞ φ ∈ I(ζ1, ζ2), and f ∈ Cc (Z(F )\G(F )), and note that Z (π(f)φ)(h) = f(g)φ(hg)dg Z(F )\G(F ) Z Z Z 2 ζ1 ζ2 −1 −1/2 a ab = f(h nak˜ )δB(˜a) φ(k)dndadk˜ N(F ) Z(F )\A(F ) K λ λ ZZZ 2 ζ1 ζ2 −1 −1 a ab = ∆(˜a)f(h n ank˜ ) φ(k)dndadk,˜ λ λ where

4 2 3 a˜ = diag(a, b, λ/b, λ/a), δB(˜a) = |a b /λ |, and

|(a − b)(ab − λ)(a2 − λ)(b2 − λ)| ∆(ã) = δ (ã)−1/2| det(ad(ã) − 1) | = . B LieN |a4b4λ3|1/2

Put Z Z −1 −1 Ff (ã) = ∆(ã)Φf (ã), Φf (ã) = f(k n ank˜ )dkdn. N(F ) K

Then

Z 2 ζ1 ζ2 a ab tr I(ζ1, ζ2; f) = Ff (˜a) da.˜ Z(F )\A(F ) λ λ

When f is K-biinvariant, Ff (˜a) depends only on the valuations of a, b, λ in the nonarchimedean case. As usual π denotes a generator of the maximal ideal in the ring of integers R of F, q the cardinality of the residue ﬁeld R/(π), and the absolute value is normalized by |π| = q−1. Put

n m m+n Ff (n, m) = Ff (diag(1,ππ ,ππ ,ππ )).

49 Then

X X (m+n)ζ1 mζ2 tr I(ζ1, ζ2; f) = q q Ff (n, m). n m

0 ∞ 0 a 0 If f ∈ Cc (Z(F )\G (F )), and Ff 0 (ã) = ∆H (ã)Φf 0 (ã), herea ˜ = ( 0 b ), ∆H (ã) =

2 1/2 R R 0 −1 −1 0 0 |(a − b) /ab| and Φf (˜a) = K0 N 0(F ) f (k n ank˜ )dndk, we put Ff (n) to be −n 0 0 Ff 0 (diag(π , 1)) if f is K -biinvariant. A similar computation shows that

Z a ζ1 0 X nζ1 tr IH (ζ1, −ζ1; f ) = Ff 0 (˜a) da˜ = Ff 0 (n)q , 0 0 b Z (F )\A (F ) n the last equality holds for a K0-biinvariant function f 0.

Deﬁnition 8. The K-biinvariant function f on Z(F )\G(F ), and the K0-biinvariant

0 0 0 1 function f on Z(F )\G (F ), are corresponding, if tr IH (ζ1, −ζ1; f ) = tr I(ζ1, 2 − ζ1; f) 0 00 for every complex number ζ1. In particular, the unit elements f and f are corresponding.

If f 0 and f are corresponding, then

X nζ1 X X (m+n)ζ1 m(1/2−ζ1) q Ff 0 (n) = q q Ff (n, m) n n m for all ζ1, hence

X m/2 Ff 0 (n) = q Ff (n, m). m Proposition 17. Corresponding f 0 and f are matching.

This statement is proven by Zinoviev in [Z], using Proposition 14, which establishes this statement in the case of f 00 and f 0. It will be interesting to ﬁnd a simple and

50 conceptual proof of our assertion, as one has in the case of base-change and ordinary orbital integrals. A suitable “symmetric space” analogue of the regular functions technique introduced in [F3] in the group case may yield a reduction of the spherical case (Proposition 17) to the germ expansion for a general test function, which is analyzed in Propositions 8 and 12, or to the case of the unit element (Proposition

14).

0 0 0 Remark 1. Since IP (π , ζ2) = IB(ζ1, ζ2 −ζ1) for π = IH (ζ1, −ζ1), and tr π is invariant

0 0 under the replacement ζ1 7→ −ζ1, and since IQ(π , ζ3) = IB(ζ3 − ζ4, 2ζ4) for π =

0 I(ζ4, −ζ4), and tr π is invariant under ζ4 7→ −ζ4, we conclude that tr IB(ζ1, ζ2) is invariant under (ζ1, ζ2) 7→ (−ζ1, ζ2 + 2ζ1), as well as under (ζ1, ζ2) 7→ (ζ1 + ζ2, −ζ2). Of course these relations are induced by the action of the two reﬂections in the Weyl group, α1 which is associated with the parabolic P , and α2 which is associated with Q.

2.7 Spectral analysis

The spectral expansion of the kernel Kf (x, y)(x, y ∈ Z(A)\G(A)) of the convolution operator r(f) on L2(Z(A)G(F )\G(A)) is a sum of contributions associated to G, P, Q, and B. The contribution associated to G is analogous to that described in the case of G0 = GL(2) above. Its cuspidal part is the bounded function

X X KG(x, y) = m(π) (π(f)φ)(x)φ(y). π φ The ﬁrst sum ranges over a set of representatives for the set of equivalence classes of the cuspidal representations in L2(Z(A)G(F )\G(A)). The multiplicity of π in the

51 cuspidal spectrum of L2(Z(A)G(F )\G(A)) is a ﬁnite integer, denoted by m(π). The inner sum ranges over an orthonormal basis for the space of π, consisting of smooth bounded functions φ. In the case of G0, the multiplicity one theorem asserts that m(ρ) are all equal to 1. There we let ρ range over the discrete series representations,

0 and Wψ(φ ) 6= 0 implied that ρ is generic, hence cuspidal (not one dimensional). In the case of G, it has recently been shown ([F9]) that the m(π) are 1 for G = PGSp(2).

Since ψθ is not a generic character of NB, Wψθ does not kill the noncuspidal discrete spectrum representations. Multiplying by ψθ(x) and integrating over x in N(F )\N(A) and y in Z(A)H(F )\H(A), we obtain X X m(π) Wψθ (π(f)φ)ΠH (φ), (2.7.1) π φ where Z Z

Wψθ (φ) = φ(n)ψθ(n)dn, ΠH (φ) = φ(h)dh. N(F )\N(A) Z(A)H(F )\H(A) The contribution of the continuous and the residual spectrum to the trace formula is computed in Sections 2.8, 2.8, 2.9 and 2.10.

The following proposition gives the relative trace formula for G = GSp(2). The proof will be given in the rest of this section and the following sections.

∗ ∗ −1 Proposition 18. Suppose that f = f1 ∗ f2 , where f2 (g) = f 2(g ), and f1, f2 are

K-ﬁnite elements of X which are spherical (Kv-biinvariant) outside V . Then the integral of (2.3.1) is equal to the sum (2.7.1) + (2.7.6).

∞ For f ∈ Cc (G(A)/Z(A)), let

X −1 Kf (x, y) = f(x γy) γ∈G(F )/Z(F )

52 be the kernel of the convolution operator r(f), where r is the right regular representation of G(A) on L2(Z(A)G(F )\G(A)). A choice of a Tamagawa measure is implicit. We want to ﬁnd the spectral expansion of the integral

Z Z J(f) = Kf (x, y)ψ(x)dxdy. (2.7.2) N(F )\N(A) Z(A)H(F )\H(A)

Following [A2] we recall the spectral formula for the kernel Kf (x, y). Consider pairs of the form χ = (M1, σ), where M1 is a Levi subgroup of G and σ a discrete spectrum

1 representation of Z(A)\M1(A) , such a pair will be called a discrete pair. Two such

0 pairs (M1, σ) and (M2, σ ) are called equivalent if there is an element ω in the Weyl

−1 ω 0 ω group Ω such that M1 = ω M2ω and σ is unitary equivalent to σ . Here σ is deﬁned by σω(m) = σ(ω−1mω).

Then the spectral formula for the automorphic kernel is

X Kf (x, y) = Kf,χ(x, y). χ

Here the sum over χ = (M1, σ) ranges over the set of equivalent classes of discrete pairs and Kf,χ(x, y) is given by the formula Z −1 X X n(P1) E(x, I(f, σ, λ)Φ, σ, λ)E(y, Φ, σ, λ)dλ. ∗ iaP (M1,σ) 1 Φ∈LM1 (σ)

Here LP1 (σ) is an orthonormal basis of the (normalizedly) induced representation I(σ).

Set Z Z Jχ(f) = Kf,χ(x, y)ψ(x)dxdy, N(F )\N(A) Z(A)H(F )\H(A)

53 then (2.7.2) becomes X J(f) = Jχ(f). (2.7.3) χ ∗ The test function f will be assumed to be the convolution of two functions f = f1 ∗f2 ,

∞ ∗ for fi = ⊗vfiv ∈ Cc (G(A)/Z(A)). Here we denote by f (g) the function given by

−1 g 7→ f(g ). Then the formula for the kernel Kf (x, y) becomes Z −1 X X n(P1) E(x, I(f1, σ, λ)Φ, σ, λ)E(y, I(f2, σ, λ)Φ, σ, λ)dλ. ∗ iaP (M1,σ) 1 Φ∈LP1 (σ) Recall the following majorization ([A2], Lemma 4.4).

Proposition 19. For any X,Y in the universal enveloping algebra of G∞, and for any left and right K-ﬁnite function f, there is a continuous seminorm ν on the space of smooth compactly supported functions on G∞, and an integer N, such that: Z −1 X X ∗ n(P1) R(X)E(x, I(f, σ, λ)Φ, σ, λ)R(Y )E(y, Φ, σ, λ) dλ ∗ iaP (M1,σ) 1 Φ∈LP1 (σ) is bounded by

N N ν(X ∗ f∞ ∗ Y )||x|| ||y|| .

It follows that if x is in a compact set the function y 7→ Kf (x, y) is of moderate

T growth and therefore the truncated function Λ2 Kf (x, y) is rapidly decreasing.

Proposition 20. Let C be a compact subset of G(A). Suppose T is a suﬃciently regular element in aB. Then for all x ∈ C, y ∈ H(A) and any χ we have

T T Kf (x, y) = Λ2 Kf (x, y),Kf,χ(x, y) = Λ2 Kf,χ(x, y).

Proof. Same as Lemma 10 in [GJR] (which is the same as Lemma 2.1 in [J3]).

54 From this Proposition we have the second equality in

Z Z J(f) = Kf (x, y)ψ(x)dxdy N(F )\N(A) Z(A)H(F )\H(A) Z Z T = Λ2,mKf (x, y)ψ(x)dxdy N(F )\N(A) Z(A)H(F )\H(A) Z −1 X X = n(P1) Wψ(λ, I(f2, σ, λ)Φ) ·I· dλ, ia∗ (M1,σ) P1 Φ where I is Z T Λ E(y, I(f1, σ, λ)Φ, σ, λ)dy (2.7.4) Z(A)H(F )\H(A) and Z Wψ(λ, Φ) = E(x, Φ, σ, λ)ψ(x)dx. N(F )\N(A) Consequently, we obtain the following expression J(f).

Proposition 21. For T suﬃciently regular and functions fi, i = 1, 2, of the form

∞ fi = fi,∞fi with fi,∞ compactly supported, we have: Z −1 X X J(f) = n(P1) Wψ(λ, I(f2, σ, λ)Φ) ·I· dλ. ia∗ (M1,σ) P1 Φ

The integral being absolutely convergent in the sense that

Z −1 X X J(f) = n(P1) Wψ(λ, I(f2, σ, λ)Φ) ·I dλ ia∗ (M1,σ) P1 Φ is ﬁnite. The same is true for Jχ(f).

55 The Bessel distribution.

Let P1 ⊂ P2 be two parabolic subgroups deﬁne:

P X P B 2 (f, λ) = Π (EP2 (h, M(ω, λ)I(f, λ)φ, (ω) 2 )K) ·W (λ, φ). (2.7.5) P1 MP2∩H P1 ψ

φ∈LP1 (σ) In the next four sections we will prove the following proposition. It follows from

Propositions 26, 28 and 29 bellow.

Proposition 22. The spectral side of the Fourier-Period trace formula for G =

GSp(2), namely 2.7.2, is

X Z J(f) = ν BP1 (f, λ)dλ. (2.7.6) P1 P0 i(aP1 )∗ (P0⊂P1,σ) P0

Here the sum is over the pairs of standard parabolic subgroups P0 ⊆ P1 of G, and over the discrete spectrum representations σ of the standard Levi of P0. The Bessel distribution BP1 (f, λ) is given by equation 2.7.5. The expression is absolutely covergent; P0

X Z ν |BP1 (f, λ)|dλ < ∞. P1 P0 i(aP1 )∗ (P0⊂P1,σ) P0

For a parabolic subgroup P of G with standard Levi M, denote by HM the group

M ∩ H. A discrete spectrum representation σ of M(A) will be called distinguished by H if there is a function in the representation space of σ such that ΠHM (ψ) 6= 0.

The summands corresponding to the case P0 = P1 give the discrete part of the trace formula which comes from the noncuspidal spectrum.

56 Proposition 23. The discrete part of the spectral side of the Fourier-Period formula is the sum of the following:

X X m(π) Wψ(π(f)φ)ΠH (φ), π φ∈πKS X X νP JP (P,I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ), σ {φ} X X νQ JQ(Q, I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ), σ {φ} X X νB JB(B,I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). σ {φ} The sum over σ, in any of the three terms, is over the discrete spectrum representations of the standard Levi component of the indicated parabolic subgroup, which are distinguished by H.

2.8 Computing the integrals of truncated Eisenstein series

The proof of Proposition 22 occupies this and the next three sections. In this section we compute, following [JLR], the periods of the truncated Eisenstein series, namely the integrals (2.7.4). Here and in the following sections P1 will denote a general standard (containing the upper triangular Borel subgroup) parabolic subgroup of G.

The Siegel parabolic subgroup will be denoted by P and the Heisenberg parabolic subgroup by Q.

For a parabolic subgroup P1 = N1M1 of G denote by νP1 the constant     d(G)−d(P1) X ∨ 2 · vol aαα : 0 ≤ aα ≤ 1 .   α∈∆P1

57 Given a discrete spectrum representation σ of M1(A), a function φ ∈ AP1 (G)σ and λ ∈ a∗ P1 , deﬁne the Eisenstein series E(g, φ, σ, λ) (in its domain of absolute convergence) by

X hλ+ρ ,H (δg)i E(g, φ, σ, λ) = φ(δg)e P1 P1 . (2.8.1)

δ∈P1(F )\G(F ) −1 For an element ω ∈ Ω(P1,P1), set N1ω to be N1 ∩ ωN1ω . Deﬁne the intertwining operator M(ω, λφ)(g) as usual by Z −hωλ+ρ ,H (g)i −1 hλ+ρ ,H (ω−1ng)i e P1 P1 φ(ω ng)e P1 P1 dn. (2.8.2) N1ω(A)\N1(A)

Eisenstein series induced from B.

Consider E(g, φ, σ, λ), given by equation (2.8.1), when P1 is the standard (upper triangular supgroup) minimal parabolic subgroup B. Then by Proposition 4 Z ΛT E(h, φ, σ, λ)dh 1 H(F )\H(A) is the sum of:

Z ∗ IG(E, φ, σ, λ) = E(h, φ, σ, λ)dh; 1 H(F )\H(A)

Z ∗ −IP (E, φ, σ, λ) = − EP (h, φ, σ, λ)χP (HP (h) − T )dh; 1 b P (F )∩H(F )\H(A)

Z ∗ −IQ(E, φ, σ, λ) = − EQ(h, φ, σ, λ)χQ(HQ(h) − T )dh; 1 b Q(F )∩H(F )\H(A)

Z ∗ IB(E, φ, σ, λ) = EB(h, φ, σ, λ)χB(HB(h) − T )dh. 1 b B(F )∩H(F )\H(A)

58 The integral IB(E, φ, σ, λ). Set

EB(g, ψ, λ) = ψ(g)ehλ+ρB ,HB (g)i.

Then (from the Proposition in [MW], II.1.7) the constant term of E(g, φ, σ, λ) is

X B EB(g, φ, σ, λ) = E (g, M(ω, λ)φ, ωλ), ω∈Ω and IB(E, φ, σ, λ) is given by

X eh(ωλ)B ,T i ν Π (EB(h, M(ω, λ)φ, 0)K) . B MB ∩H Q ∨ h(ωλ)B, α i ω∈Ω α∈∆B

The integral IP (E, φ, σ, λ). Set

X EP (g, ψ, λ) = ψ(δg)ehλ+ρB ,HB (δg)i. δ∈B(F )\P (F )

Then (from the Proposition in [MW], II.1.7) the constant term of E(g, φ, σ, λ) is

X P EP (g, φ, σ, λ) = E (g, M(ω, λ)φ, ωλ). ω∈Ω(B,B) −1 ω α2>0

B ∗ ∗ ∗P Write ωλ = (ωλ)P + (ωλ)P , according to the decomposition aB = aP ⊕ aB . Then

IP (E, φ, σ, λ) is given by

X eh(ωλ)P ,T i ν Π (EP (h, M(ω, λ)φ, (ωλ)P )K) . P MP ∩H B Q ∨ h(ωλ)P , α i ω∈Ω(B,B) α∈∆P −1 ω α2>0

The integral IQ(E, φ, σ, λ). Set

X EQ(g, ψ, λ) = ψ(δg)ehλ+ρB ,HB (δg)i. δ∈B(F )\Q(F )

59 Then (from the Proposition in [MW], II.1.7) the constant term of E(g, φ, σ, λ) is

X Q EQ(g, φ, σ, λ) = E (g, M(ω, λ)φ, ωλ). ω∈Ω(B,B) −1 ω α1>0

B ∗ ∗ ∗Q Write ωλ = (ωλ)Q + (ωλ)Q, according to the decomposition aB = aQ ⊕ aB . Then

IQ(E, φ, σ, λ) is given by

X eh(ωλ)Q,T i ν Π (EQ(h, M(ω, λ)φ, (ωλ)Q)K) . Q MQ∩H B Q ∨ h(ωλ)Q, α i ω∈Ω(B,B) α∈∆Q −1 ω α1>0

Eisenstein series induced from P .

Consider E(g, φ, σ, λ), given by equation (2.8.1), for P1 = P the Siegel parabolic subgroup. Then by Proposition 4, Z ΛT E(h, φ, σ, λ)dh 1 H(F )\H(A) is equal to Z ∗ IG(E, φ, σ, λ) = E(h, φ, σ, λ)dh (2.8.3) 1 H(F )\H(A) minus

Z ∗ IP (E, φ, σ, λ) = EP (h, φ, σ, λ)χP (HP (h) − T )dh. (2.8.4) 1 b P (F )∩H(F )\H(A) Set

EP (g, ψ, λ) = ψ(g)ehλ+ρP ,HP (g)i.

Then (from the Proposition in [MW], II.1.7) the constant term of E(g, φ, σ, λ) is

X P EP (g, φ, σ, λ) = E (g, M(ω, λ)φ, ωλ), ω∈Ω(P,P )

60 and IP (E, φ, σ, λ) is given by

X eh(ωλ)P ,T i ν Π (EP (h, M(ω, λ)φ, 0)K) . P MP ∩H Q ∨ h(ωλ)P , α i ω∈Ω(P,P ) α∈∆P

Eisenstein series induced from Q.

Consider E(g, φ, σ, λ), given by equation (2.8.1), when P1 is the Heisenberg parabolic subgroup Q. Then by Proposition 4,

Z ΛT E(h, φ, σ, λ)dh 1 H(F )\H(A) is equal to Z ∗ IG(E, φ, σ, λ) = E(h, φ, σ, λ)dh 1 H(F )\H(A) minus

Z ∗ IQ(E, φ, σ, λ) = EQ(h, φ, σ, λ)χQ(HQ(h) − T )dh. 1 b Q(F )∩H(F )\H(A) Set

EQ(g, ψ, λ) = ψ(g)ehλ+ρQ,HQ(g)i.

Then (from the Proposition in [MW], II.1.7) the constant term of E(g, φ, σ, λ) is

X Q EQ(g, φ, σ, λ) = E (g, M(ω, λ)φ, ωλ), ω∈Ω(Q,Q) and IQ(E, φ, σ, λ) is given by

X eh(ωλ)Q,T i ν Π (EQ(h, M(ω, λ)φ, 0)K)φ, 0) . Q MQ∩H Q ∨ h(ωλ)Q, α i ω∈Ω(Q,Q) α∈∆Q

61 (G, M)-families

In order to compute the contribution to the spectral side of the trace formula from the proper parabolic subgroups we record, in this section, the basic deﬁnitions and theorems about (G, M)-families as introduced by Arthur in [A4] (see also [G] and

[Lap2]). The techniques from this section will be used in the next two sections where we compute the contribution from the Borel subgroup and the maximal parabolic subgroups. Let M be a ﬁxed Levi subgroup of G. Denote by P(M) the set of parabolic subgroups of G for which M is a Levi component.

∗ ∗ ∗ In this section denote by aM the vector space X (M)⊗Z R, where X (M) is the group of rational characters of M.

∗ Deﬁnition 9. A collection {cP (Λ) : P ∈ P(M)} of smooth functions cP (Λ) on iaM is called a (G, M)-family if the following condition is satisﬁed: for any two elements

P1 and P2 of P(M) contained in a given parabolic subgroup Q, and Λ belonging to

∗ iaQ, we have

cP1 (Λ) = cP2 (Λ).

Let νM be the covolume of the lattice spanned by the coroots in aM . Deﬁne

−1 Y ∨ θP (Λ) = νM Λ(α ). α∈∆P The following is Lemma 6.2 of [A4].

Lemma 24. For a (G, M)-family {cP (Λ) : P ∈ P(M)}, the function

X cP (Λ) cM (Λ) = , θP (Λ) P ∈P(M)

62 ∗ ∨ initially deﬁned away from the hyperplanes {Λ ∈ iaM : hΛ, α i = 0}, extends to a

∗ smooth function on iaM .

Denote by F(M) the set of parabolic subgroups of G whose Levi component contains

M. Fix Q in F(M). Let L be its Levi component. Denote by PL(M) the set of parabolic subgroups of L whose Levi component is M. For each R ∈ PL(M), set Q(R) to be the unique group in P(M) which is contained in Q and such that

Q(R) ∩ L = R. Then the family of functions

Q L {cR(Λ) = cQ(R)(Λ) : R ∈ P (M)}

Q is an (L, M)-family. For this (L, M)-family denote by cM (Λ) the function

X Q Q −1 cR(Λ)θR (Λ) . R∈PL(M)

Suppose {cP (Λ)} and {dP (Λ)} are two (G, M)-families. Put eP (Λ) = cP (Λ)dP (Λ).

Then {eP (Λ)} is also a (G, M)-family. For a pair of parabolic subgroups Q ⊂ R, deﬁne

R R −1 Y ∨ θbQ(Λ) = (ˆνM ) Λ($ ). R $∈∆Q

R R ∨ R Hereν ˆM is the co-volume of the lattice in aP spaned by the elements α for α ∈ ∆P .

∨ R The vector $ in aQ is deﬁned by

∨ ∨ ∨ R α($ ) = $ (α ), α ∈ ∆Q.

Deﬁne the functions

0 dQ(λ),Q ⊃ P

63 by

0 X d(Q)−d(P ) R −1 −1 dQ(λ) = (−1) θbQ(Λ) dR(Λ)θR(Λ) . {R:R⊃P } The following is Lemma 6.3 of [A4].

Proposition 25. With the notations as above we have

X Q 0 eM (Λ) = cM (Λ)dQ(Λ). Q∈F(M)

2.9 The contribution from the maximal parabolics

Following [GJR] in this and the next section we compute the contribution to the spectral side of the (Fourier-Period) trace formula from spectral data χ = (M1, σ), with M1 a proper Levi subgroup of G and σ a discrete spectrum representation of M1(A). To simplify notations we will denote, for a pair of parabolic subgroups

P1 ⊂ P2, the period integral

Π (EP2 (h, M(ω, λ)I(f, λ)φ, (ωλ)P2 )K) MP2∩H P1 by

J (P ,M(ω, λ)I(f, λ)φ, (ωλ)P2 ). P2 1 P1

The contribution from the maximal parabolic P (the Siegel parabolic).

In this section we compute the contribution to (2.7.2) from the discrete pairs (MP , σ).

Here MP is the standard Levi subgroup of P and σ a discrete spectrum representation of MP (A).

64 G Recall from (2.7.5) that when P1 = P and P2 = G the Bessel distribution BP (f, λ) is X Z ∗ E(h, I(λ, σ, f1)φ, σ, λ)dh ·Wψ(λ, I(λ, σ, f2)φ). 1 φ H(F )\H(A) Proposition 26. The contribution from the Siegel parabolic P to the trace formula is Z X X P1 νP1 BP (f, λ)dλ. (2.9.1) P1 ∗ i(aP ) {P1:P ⊆P1⊆G} (MP1 ,σ)

P1 ∗ Remark 2. When P1 = P the space i(aP ) is a pont. The summand in this case occurs discretely in the trace formula. It is equal to

X X νP JP (P,I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). σ {φ} Proof. Let Z Z X T JP (f) = νP Λ E(h, I(λ, σ, f1)φ, σ, λ)dh ·Wψ(λ, I(λ, σ, f2)φ)dλ. ∗ iaP φ Recall from Section 2.8 that the integral of the truncated Eisenstein series, induced from the parabolic subgroup P , is equal to the diﬀerence of (2.8.3) and (2.8.4). There- fore the distribution JP (f) can be written as

1 2 JP (f) = JP (f) + JP (f).

The ﬁrst term her is Z 1 JP (f) = νP BP (f, λ)dλ. ∗ iaP The second term is Z 2 X P JP (f) = νP IP (E ,I(λ, σ, f1)φ, σ, λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ. ∗ iaP φ

65 2 To compute JP (f), recall that Ω(P,P ) = {1, ωP = sα1 sα2 sα1 }. Hence the integral

P IP (E ,I(λ, σ, f1)φ, σ, λ) equals

ehλ,T i ehωP λ,T i JP (P,M(1, λ)φ, 0) ∨ + JP (P,M(ωP , λ)φ, 0) ∨ . hλ, α1 i hωP λ, α1 i

∨ ∨ Since hωP λ, α1 i = −hλ, α1 i, hωP λ, T i = −hλ, T i, we have:

P IP (E ,I(λ, σ, f1)φ, σ, λ) (2.9.2)

ehλ,T i e−hλ,T i = JP (P,M(1, λ)φ, 0) ∨ − JP (P,M(ωP , λ)φ, 0) ∨ . hλ, α1 i hλ, α1 i Set

P + 0 dφ,φ0 (λ) = JP (P,M(1, λ)φ , 0) ·Wψ(λ, I(λ, σ, f2)φ),

P − 0 dφ,φ0 (λ) = JP (P,M(ωP , λ)φ , 0) ·Wψ(λ, I(λ, σ, f2)φ).

In formula (2.9.2), the left hand side is holomorphic at λ = 0, hence the right hand side is holomorphic at λ = 0, which implies that J(P,M(1, 0)φ, 0) is equal

P + P − to J(P,M(ωP , 0)φ, 0). Therefore we have dφ,φ0 (0) = dφ,φ0 (0). From Proposition 21 the integral is absolutely convergent therefore we can change

2 the order of integration and summation. Thus the expression JP (f) is equal to

Z hλ,TP i −hλ,TP i X e P + e P − 0 νP ∨ dφ,φ0 (λ) − ∨ dφ,φ0 (λ) hI(λ, σ, f1)φ , φidλ. ∗ hλ, α i hλ, α i φ,φ0 iaP 1 1

Here TP is the projection of T on the space aP .

66 Now we shall use the theory of (G, M)-families as developed in [A4]. In this case it amounts to writing the last integrand as the sum of the following three terms

P + P + d 0 (λ) − d 0 (0) hλ,TP i φ,φ φ,φ 0 e ∨ hI(λ, σ, f1)φ , φi, hλ, α1 i P − P − d 0 (λ) − d 0 (0) −hλ,TP i φ,φ φ,φ 0 −e ∨ hI(λ, σ, f1)φ , φi, hλ, α1 i hλ,TP i −hλ,TP i e e P + 0 ∨ − ∨ dφ,φ0 (0)hI(λ, σ, f1)φ , φi. hλ, α1 i hλ, α1 i

Following the argument of [GJR], page 47, we claim that the ﬁrst two terms are square integrable. Indeed, in those terms the quotients are smooth even at λ = 0. The reciprocals of the denominators are bounded in the complement of a neighborhood of

0 0, and hI(λ, σ, f1)φ , φi is integrable and square integrable. If x and y are in compact sets we have that

Z X E(x, I(λ, f2)φ, λ)E(y, I(λ, f2)φ, λ) dλ φ is ﬁnite. Thus integrating over N(F )\N(A) × N(F )\N(A) we get that Z X 2 |Wψ(λ, I(λ, σ, f2)φ)| dλ φ

P + P − is ﬁnite as well. Therefore the functions dφ,φ0 and dφ,φ0 are square integrable. Hence the integral over λ of those two terms is the Fourier transform evaluated at the projection TP of T on aP . Hence this tends to zero as T goes to inﬁnity by the following lemma.

Lemma 27. Let f be in L1(Rn) and f ∧(T ) its Fourier transform. Then f ∧(T ) → 0 as T tends to inﬁnity.

67 Proof. This is the Riemann-Lebesgue lemma. See Theorem 8.22 in [Fo].

The contribution of the last term is

Z hλ,TP i −hλ,TP i e e P + 0 lim ∨ − ∨ dφ,φ0 (0)hI(λ, σ, f1)φ , φidλ T →∞ hλ, α1 i hλ, α1 i

Z Z TP −iλx P + 0 = lim e dx dφ,φ0 (0)hI(λ, σ, f1)φ , φidλ T →∞ −TP Z Z ∞ −iλ1x P + 0 = lim χTP (x)e dx dφ,φ0 (0)hI(λ, σ, f1)φ , φidλ T →∞ −∞ Z ∧ P + 0 = lim χT (λ) dφ,φ0 (0)hI(λ, σ, f1)φ , φi dλ, T →∞ P by Parseval’s identity it is equal to

Z P + 0 ∧ lim χTP (λ) dφ,φ0 (0)hI(λ, σ, f1)φ , φi (λ)dλ T →∞ Z P + 0 ∧ = dφ,φ0 (0)hI(λ, σ, f1)φ , φi (λ)dλ

P + 0 = dφ,φ0 (0)hI(0, σ, f1)φ , φi, by Fourier inversion formula. Therefore the contribution is

X P + 0 νP dφ,φ0 (0)hI(0, σ, f1)φ , φi φ,φ0 X = νP JP (P,M(1, 0)I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). {φ}

68 The contribution from the maximal parabolic Q.

In this section we compute the contribution to (2.7.2) from the discrete pairs (MQ, σ).

Here MQ is the standard Levi subgroup of Q and σ a discrete spectrum representation of MQ(A).

G Recall from (2.7.5) that when P1 = Q and P2 = G the Bessel distribution BQ(f, λ) is X Z ∗ E(h, I(λ, σ, f1)φ, σ, λ)dh ·Wψ(λ, I(λ, σ, f2)φ). 1 φ H(F )\H(A) Proposition 28. The contribution from the maximal parabolic Q to the trace formula is Z X X P1 νP1 BQ (f, λ)dλ. (2.9.3) P1 ∗ i(aQ ) {P1:Q⊆P1⊆G} (MP1 ,σ)

P1 ∗ Remark 3. When P1 = Q the space i(aQ ) is a point. The summand in this case occurs discretely in the trace formula and is equal to

Z X Q X X νQ BQ(f, λ)dλ = νQ JQ(Q, I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). i(aQ)∗ σ Q σ {φ}

Proof. Let

Z Z X T JQ(f) = νQ Λ E(h, I(λ, σ, f1)φ, σ, λ)dh ·Wψ(λ, I(λ, σ, f2)φ)dλ. ∗ iaQ φ

Then

1 2 JQ(f) = JQ(f) + JQ(f).

Here Z 1 JQ(f) = νQ BQ(f, λ)dλ. ∗ iaQ

69 The second term is Z 2 X Q JQ(f) = νQ IQ(E ,I(λ, σ, f1)φ, λ) ·Wψ(λ, I(λ, σ, f2, φ))dλ. ∗ iaQ φ

Recall that Ω(Q, Q) = {1, ωQ = sα2 sα1 sα2 }, therefore

IQ(E,I(λ, σ, f1)φ, λ)

ehλ,T i ehωQλ,T i = JQ(Q, M(1, λ)φ, 0) ∨ + JQ(Q, M(ωQ, λ)φ, 0) ∨ . hλ, α2 i hωQλ, α2 i ∨ ∨ We have hωQλ, α2 i = −hλ, α2 i, hωQλ, T i = −hλ, T i. Set

Q+ 0 dφ,φ0 (λ) = JQ(Q, M(1, λ)φ , 0) ·Wψ(λ, I(λ, σ, f2)φ),

Q− 0 dφ,φ0 (λ) = JQ(Q, M(ωQ, λ)φ , 0) ·Wψ(λ, I(λ, σ, f2)φ).

Then the contribution from IQ(E, φ, σ, λ) is equal to

Z hλ,TQi −hλ,TQi X e Q+ e Q− 0 νQ ∨ dφ,φ0 (λ) − ∨ dφ,φ0 (λ) hI(λ, σ, f1)φ , φidλ. ∗ hλ, α i hλ, α i φ,φ0 iaQ 2 2 The same computations as in Section 2.9 complete the proof.

2.10 The contribution from the minimal parabolic

In this section we compute the contribution to (2.7.2) from the discrete pairs (MB, σ).

Here MB is the Levi (i.e. the diagonal subgroup) of B and σ a discrete spectrum representation of MB(A), namely a character.

Proposition 29. The contribution from the minimal parabolic subgroup B to the trace formula is Z X X P1 νP1 BB (f, λ)dλ. (2.10.1) P1 ∗ i(aB ) {P1:B⊆P1⊆G} (MP1 ,σ)

70 The term corresponding to P1 = B occurs discretely in the trace formula and is equal to

X X B νB JB(B,E ,M(ω, 0)I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). σ {φ} The proof of this proposition occupies the rest of this section. Put

Z Z X T JB(f) = νB Λ E(h, I(λ, σ, f1)φ, λ)dh ·Wψ(λ, I(λ, σ, f2)φ)dλ. ∗ iaB φ

Since Z T Λ E(h, I(λ, σ, f1)φ, λ)dh is the sum

IH (E,I(λ, σ, f1)φ, σ, λ) + IB(E,I(λ, σ, f1)φ, σ, λ)

+IP (E,I(λ, σ, f1)φ, σ, λ) + IQ(E,I(λ, σ, f1)φ, σ, λ) we have

1 2 3 4 JB(f) = JB(f) + JB(f) + JB(f) + JB(f).

Here Z 1 G JB(f) = νB BB(f, λ)dλ ∗ iaB and Z i X JB(f) = νB IBi (E,I(λ, σ, f1)φ, λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ, ∗ φ iaB where B2 = B, B3 = P and B4 = Q.

71 2 The term JB(f).

1 2 Assume that the function f1 is itself the convolution f1 ∗ f1 of two functions. Then

2 X 2 0 0 I(λ, σ, f1 ) = hI(λ, σ, f1 )φ, φ iφ , φ0

Z hωλ,T i 2 X X e J (f) = b · A 0 (λ) · B 0 (λ)dλ, B Q hωλ, α∨i φ,φ φ,φ φ,φ0 ω∈Ω α∈∆B where

1 0 Aφ,φ0 (λ) = JB(B,M(ω, λ)I(λ, σ, f1 )φ , 0) · Wψ(λ, I(λ, σ, f2)φ) and

2 0 Bφ,φ0 (λ) = hI(λ, σ, f1 )φ, φ i.

Let M = MB be the Levi of the standard Borel subgroup. Denote by P(M) the set of F -parabolics with M as a Levi factor. Set

hωλ,T i cω(λ) = e , dω(λ) = Aφ,φ0 (λ).

If P ∈ P(M) then there exists an ω0 ∈ Ω such that ω0P = B. Deﬁne

cP (λ) = cω0 (λ), dP (λ) = dω0 (λ).

Lemma 30. The families {cP } and {dP } are (G, M)-families.

Proof. One has to show that if s is a simple reﬂection and ss1 = s2, then for λ such that s1λ = s2λ the families above satisfy

cs1 (λ) = cs2 (λ), ds1 (λ) = ds2 (λ).

72 This is obvious for the ﬁrst family. For the second, it is enough to show that if s1λ = s2λ, then

2 0 JB(B,M(s1, λ)I(λ, σ, f1 )φ , 0) is equal to

2 0 JB(B,M(s2, λ)I(λ, σ, f1 )φ , 0).

It is enough to show

2 0 2 0 JB(B,M(s1, λ)I(λ, σ, f1 )φ , 0) = JB(B,M(s2, λ)I(λ, σ, f1 )φ , 0).

By linearity this reduces to

0 0 JB(B,M(s1, λ)φ , 0) = JB(B,M(s2, λ)φ , 0).

Since ss1 = s2, the functional equation for the intertwining operators gives that

M(s2, λ) is equal to M(s, s1λ)M(s1, λ). Therefore the equality to be established reduces to

0 0 JB(B,M(s, λ)φ , 0) = JB(B, φ , 0).

Recall that by deﬁnition the intertwining operator M(s, λ)φ(g) is equal to

Z −1 e−hsλ+ρB ,HB (g)i φ(s−1ng)ehλ+ρB ,HB (s ng)idn. Ns(A)\NB (A) Then Z hρB ,HB (g)i JB(B,M(s, λ)φ, 0) = M(s, λ)φ (g)e dg

73 Z Z −1 = e−hsλ,HB (g)i φ(s−1ng)ehλ+ρB ,HB (s ng)idndg. (2.10.2) Ns(A)\NB (A)

Write s−1ng as s−1gg−1ng. Since g and s−1g are in M(A) we have that

−1 −1 −1 −1 −1 HB(s gg ng) = HB(s g) + HB(g ng) and HB(s g) = HB(sg) = HB(g).

We also have

e−hλ,HB (g)iehλ+ρB ,HB (g)i = ehρB ,HB (g)i.

Then the left side of (2.10.2) is equal to

Z Z −1 ehρB ,HB (g)i σ(s−1g)φ(g−1ng)ehλ+ρB ,HB (g ng)idndg Ns(A)\NB (A)

Z Z = ehρB ,HB (g)iσ(s−1g) φ(n)ehλ+ρB ,HB (n)idndg Ns(A)\NB (A)

Z Z = ehρB ,HB (g)iσ(s−1g)φ(1)dg = ehρB ,HB (g)iφ(g)dg.

The last integral is equal to JB(B, φ, 0), which proves the lemma.

We have   Z 2 X X 1 JB(f) = b  cP (λ)dP (λ)  B(λ)dλ θP (λ) φ P ∈P(M)

2 where B(λ) = hI(λ, σ, f1 )φ, φi. Using the splitting formula for the product of (G, M)-families ([A4], Lemma 6.3), we get

X 1 X R 0 cP (λ)dP (λ) = cM (λ)dR(λ), θP (λ) P ∈P(M) R∈F(M)

74 where F(M) is the set of F -parabolics whose Levi contains M.

R Deﬁne cM (λ) to be

X −1 cP (λ)θP (λ) . R∈P(M),P ⊂R

R hωλ,T i If R ∈ P(M) then there is a ω ∈ Ω such that R = ωB. Then cM (λ) = e . If R = G then X ehωλ,T i cG (λ) = . M hωλ, α∨ihωλ, α∨i ω 1 2 If R is a maximal parabolic, then R = ωP or R = ωQ. In the ﬁrst case

hωλ,T i hsα1 ωλ,T i R e e cM (λ) = ∨ + ∨ . hωλ, α1 i hsα1 ωλ, α1 i

In the second case

hωλ,T i hsα2 ωλ,T i R e e cM (λ) = ∨ + ∨ . hωλ, α2 i hsα2 ωλ, α2 i Proposition 31. The limit

Z R 0 lim cM (λ)dR(λ)B(λ)dλ T →∞ exists.

Proof. Following [A4], let χT be the characteristic function of the convex hull of the points {ω−1T } for ω ∈ Ω.

The case R = G.

G When R = G the function cM (λ) is the Fourier transform of χT . Thus Z Z G 0 0 ∧ lim cM (λ)dG(λ)B(λ)dλ = lim χT (x) dGB (λ)dx, T →∞ T →∞

75 by Parseval’s equality it is equal to Z 0 ∧ dGB (λ)dx

0 which is equal to dG(0)B(0) = d1(0)B(0) by Fourier inversion formula. The case R = ωB.

The Riemann-Lebesgue lemma (Theorem 8.22 in [Fo]) implies that Z Z R 0 hωλ,T i 0 lim cM (λ)dR(λ)B(λ)dλ = lim e dR(λ)B(λ)dλ T →∞ T →∞ is zero.

The case R = ωP or R = ωQ.

0 0 If R = ωP , write T = T1 + T2 and λ = λ1 + λ2 according to the decomposition

∗ P ∗ ∗ −1 −1 0 −1 0 aB = aB ⊕ aP . Set α = ω α1, T1 = ω T1 and T2 = ω T2 . When R is P the

R deﬁnition of cM (λ) reduces to

ehλ,T2i e−hλ,T2i cP (λ) = ehλ,T1i − . M hλ, α∨i hλ, α∨i The integral Z P 0 cM (λ)dP (λ)B(λ)dλ is equal to ! Z Z ehλ,T2i e−hλ,T2i hλP ,T1i 0 e dR(λ)B(λ) ∨ − ∨ B(λ)dλ1 dλ2. ∗ P ∗ hλ, α i hλ, α i aP aB The inner integral is 0 by a computation similar to the one in Section 2.9.

2 Therefore the term JB(f) is equal to

X 1 0 X B νB d1(0)hI(0, σ, f1 )φ , φi = νB JB(B,E ,I(0, σ, f1)φ, 0) ·Wψ(0,I(0, σ, f2)φ). φ,φ0 {φ}

76 3 The term JB(f).

1 2 M As in Section 2.10 let f1 = f1 ∗ f1 . Then JB (f) is equal to Z X 2 0 νB IP (E,I(λ, σ, f1)φ, λ) · Wψ(λ, I(λ, σ, f2)φ) · hI(λ, σ, f1 )φ, φ idλ. ∗ iaB φ,φ0

1 Here, putting φ1 = I(λ, σ, f1 )φ,

X eh(ωλ)P ,T i I (E,I(λ, σ, f )φ, σ, λ) = ν J (B,M(ω, λ)φ , ωλ) , P 1 P P 1 h(ωλ) , α∨i ω P 1

−1 where the sum ranges over ω ∈ Ω(B,B) such that ω α2 > 0. The set {ω ∈

−1 Ω|ω α2 > 0} consists of the elements {1, ω1 = sα1 , ω2 = sα1 sα2 sα1 , ω3 = sα1 sα2 }. We have that

∨ ∨ ∨ ∨ h(ω2λ)P , α1 i = −hλP , α1 i, h(ω3λ)P , α1 i = −h(ω1λ)P , α1 i,

h(ω2λ)P ,T i = −hλP ,T i, h(ω3λ)P ,T i = −h(ω1λ)P ,T i.

Therefore IP (E,I(λ, σ, f1)φ, σ, λ) is equal to the sum of

ehλ,TP i e−hλ,TP i JP (B,M(ω, λ)φ1, λ) ∨ − JP (B,M(ω, λ)φ1, ω2λ) ∨ hλ, α1 i hλ, α1 i and

ehω1λ,TP i e−hω1λ,TP i JP (B,M(ω, λ)φ1, ω1λ) ∨ − JP (B,M(ω, λ)φ1, ω3λ) ∨ . hω1λ, α1 i hω1λ, α1 i

77 Similar analysis as in Section 2.9 shows that Z 3 X JB(f) = c1 JP (B,M(1, λ)φ1, λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ P ∗ φ i(aB )

X Z +c2 JP (B,M(ω1, λ)φ1, ω1λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ. P ∗ φ i(aB )

4 The term JB(f).

1 2 M As before let f1 = f1 ∗ f1 . Then JB (f) is equal to Z X 2 0 νB IQ(E,I(λ, σ, f1)φ, σ, λ) · Wψ(λ, I(λ, σ, f2)φ) · hI(λ, σ, f1 )φ, φ idλ. ∗ iaB φ,φ0 Here

X eh(ωλ)Q,T i IQ(E,I(λ, σ, f1)φ, σ, λ) = JP (B,M(ω, λ)φ1, ωλ)νQ ∨ , h(ωλ)Q, α i ω∈Ω(B,B) 2 −1 ω α1>0

−1 where the sum ranges over ω ∈ Ω(B,B) such that ω α2 > 0. The set {ω ∈

−1 Ω|ω α1 > 0} consists of the elements {1, ω1 = sα2 , ω2 = sα2 sα1 sα2 , ω3 = sα2 sα1 }. We have that

∨ ∨ ∨ ∨ h(ω2λ)Q, α1 i = −hλQ, α1 i, h(ω3λ)Q, α1 i = −h(ω1λ)Q, α1 i,

h(ω2λ)Q,T i = −hλQ,T i, h(ω3λ)Q,T i = −h(ω1λ)Q,T i.

Therefore IQ(E,I(λ, f1)φ, λ) is equal to the sum of

ehλ,TQi e−hλ,TQi JQ(B,M(ω, λ)φ1, λ) ∨ − JQ(B,M(ω, λ)φ1, ω2λ) ∨ hλ, α2 i hλ, α2 i

78 and

ehω1λ,TQi e−hω1λ,TQi JQ(B,M(ω, λ)φ1, ω1λ) ∨ − JQ(B,M(ω, λ)φ1, ω3λ) ∨ . hω1λ, α2 i hω1λ, α2 i As in Section 12.2 Z 4 X JB(f) = d1 JQ(B,M(1, λ)φ1, λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ Q ∗ φ i(aB )

X Z +d2 JQ(B,M(ω1, λ)φ1, ω1λ) ·Wψ(λ, I(λ, σ, f2)φ)dλ. Q ∗ φ i(aB )

2.11 Comparison

In this section we reproduce the argument of Section 11 of [L2] (see also Theorem

2 in [FK]) to show that the equality of the spectral sides of the two trace formulas implies an equality of the discretely occurring terms.

Let G0 be the group GL(2), B0 the Borel subgroup and K0 a maximal compact sub-

0 0 0S 0 0S Q 0 group of G (A). For a ﬁnite set of places S write K = K KS, here K = v6∈S Kv 0 Q 0 S and KS = v∈S Kv. Similarly ﬁx a maximal compact subgroup K = K KS, with S Q Q K = v6∈S Kv and KS = v∈S Kv, of G(A).

0 0 Let Gc = GL2(C) be the dual group of G and Gb = GSp(2, C) the dual group of G. Deﬁne the map η : Gc0 × Gc0 → Gb by

a2 b2 a1 b1 × a2 b2 7→ a1 b1 . c1 d1 c2 d2 c1 d1 c2 d2 The local unramified representations are uniquely determined by the semisimple conjugacy classes that they define in the dual groups. The map above defines a transfer

79 of such representations. Here we consider the case of an unramiﬁed local repre-

0 sentation πv, which is the unramiﬁed constituent of of the induced representation

0 ξv −ξv IG0 (ξv, −ξv). Its parameter is t(πv) = diag(|qv| , |qv| ). The trivial representation

1 − 1 has parameter diag(|qv| 2 , |qv| 2 ). The map η will map these conjugacy classes to the

1 1 0 ξv −ξv − class η(t(πv)) = diag(|qv| 2 , |qv| , |qv| , |qv| 2 ) in Gb, which is the parameter of the

1 unramiﬁed constituent of IG(ξv, 2 − ξv). 0 ∞ 0 ∞ Test functions f in Cc (G (A)) and f in Cc (G(A)) are called corresponding if their

0 ∨ 0 ∨ ∨ Satake transforms (fv) (η(t(πv))) and fv (t(πv)) are equal. Here fv (t(πv)) is tr πv(fv)

0 0 0 0 ∗ and fv(t(πv)) is tr πv(fv). The map η gives a dual map of the Hecke algebras η : Hv →

0 ∗ ∨ 0 ∨ 0 Hv which is deﬁned on the Satake transforms by (η (fv )) (t(πv)) := fv (η(t(πv))). Let S be a ﬁnite set of places of F , which contains all the archimedean places and

0 0 S 0 0 those that ramify in E. Choose product test functions f = (f ) fS on G (A), where

0 0 0 S 0 0 0 0 0 fS = ⊗v∈Sfv and (f ) = ⊗v6∈Sfv with fv the unit element in Hv, the Hecke algebra

0 ∞ 0 S of spherical (Kv-biinvariant) functions in Cc (Gv/Zv). Similarly for f = f fS on

G(A). With the above notations deﬁne the following complex numbers:

0 0 0 Y 0 0 0 α0(π , φ ) = tr πv(fv) ·Wψ0 (φ ), v∈S

0 0 0 X 0 0 0 1 0 A (π , f ) = α (π , φ ) · L 0 ( , π ⊗ χ), 0 S 0 φ 2 {φ0}

0 0 0 X 1 0 1 0 0 A (π , f ) = 4π W 0 ( ,I(f , χ, )φ ) · φ (1) r S ψ 2 S 2 {φ0} and

0 0 0 1 X 0 0 0 −1 1 A (π , f , λ) = W 0 (λ, I(f , ω , λ)φ ) ·L 0 (ω , − λ). c S 2 ψ S φ ,χ 2 {φ0}

80 Then the spectral side of the relative trace formula for G0 = GL(2) given by Propo- sition 16 can be written as the sum of the following three terms:

The contribution from the cuspidal spectrum ! X Y ∨ 0 0 0 0 fv (η(t(πv))) · A0(π , fS), 0 {π0;π0Kv 6=0} v6∈S the contribution from the residual spectrum

Y 1 f ∨ η(χ), · A0 (χ0, f 0 ), v 2 r S v6∈S and the contribution from the continuous spectrum

Z X ∨ 0 0 0 0 fS (η(ω , λ)) · Ac(ω , fS, λ)dλ. ω0 iR Similarly, for the group G, deﬁne the following functions:

Y α0(π, φ) = tr πv(fv) ·Wψ(φ), v∈S X A0(π, fS) = m(π) α0(π, φ) · ΠH (φ), {φ} Y αc,P (σ, φ, λ) = tr I(λ, σv, fv) ·Wψ(λ, φ), v∈S X X Ac,P (σ, fS, λ) = csαc,P (σ, φ, λ) · JP (B,M(1, λ)I(λ, σ, fS)φ, sλ),

s∈{1,ω1} {φ} Y αc,Q(σ, φ, λ) = tr I(λ, σv, fv) ·Wψ(λ, φ), v∈S X X Ac,Q(σ, fS, λ) = dsαc,Q(σ, φ, λ) · JQ(B,M(1, λ)I(λ, σ, fS)φ, sλ),

s∈{1,ω1} {φ}

81 and for any proper standard parabolic subgroup P1 of G deﬁne

Y αd,P1 (σ, φ) = tr I(0, σv, fv) ·Wψ(0, φ), v∈S X Ad,P1 (σ, fS) = αd,P1 (σ, φ) · JP1 (P1,I(0, σ, fS)φ, 0). {φ} Then the spectral side of the relative trace formula for G = GSp(2) can be written as the sum of the following terms: ! X Y ∨ fv (t(πv)) · A0(π, fS), {π;πKv 6=0} v6∈S Z ! X X Y ∨ fv (σ, λ) · Ac,P1 (σ, fS, λ)dλ, i P1∈{P,Q} σ R v6∈S and ! X X Y ∨ fv (t(I(0, σv))) · Ad,P1 (σ, fS). P ∈{B,P,Q} σ v6∈S 1 distinguished The equality of the geometric sides for corresponding test functions implies that the spectral sides of the two trace formulas are equal. The equality of the spectral sides can be written as ! X Y Y 1 A0 (π0, f 0 ) · f ∨ (η(t(π0 ))) + A0 (χ0, f 0 ) · f ∨ η(χ), (2.11.1) 0 S v v r S v 2 π0 v6∈S v6∈S ! ! X Y ∨ X X Y ∨ − A0(π, fS) · fv (t(πv)) − Ad,P1 (σ, fS) · fv (t(I(0, σv))) π v6∈S P1 σ v6∈S (2.11.2) Z X 0 0 0 Y ∨ 0 = Ac(ω , fS, λ) · fv (η(ωv, λ))dλ (2.11.3) ω0 iR v6∈S Z X X Y ∨ − Ac,P1 (σ, fS, λ) · fv (σv, λ)dλ. (2.11.4) i P1 σ R v6∈S

82 ×3 Q Fix a place v 6∈ S. Following [FK], let T be the set {(zi) ∈ C : i zi = 1}. The trace tr πv(fv) is a linear combination of characters of T of the form t 7→ t(ωx).

3 Here ω ∈ Ω is an element in the Weyl group Ω, x = (xi) is in Z /Z and (zi)(ωx) =

Q xi 3 i(ωzi) . We identify the group A of diagonal elements of GSp(2) with (Gm) via λ(g) λ(g) the isomorphism sending the diagonal element g = diag(a, b, b , a ) to the element 3 0 ¯ −1 (a, b, λ) of (Gm) . Denote by T the set of t in T such that t = ωt for some ω ∈ Ω 1 1 − 2 ∨ 2 0 and qv ≤ |α (t)| ≤ qv for all roots α. Let T˜ be the quotient of T by Ω and C(T˜) the ˜ algebra of continuous complex-valued functions on T . Deﬁne Px(t) to be the function P ω t(ωx) and C the C-linear span of all such functions.

Lemma 32. The space C is dense in C(T˜).

Proof. The space T˜ is compact and Hausdorﬀ, the subalgebra C of C(T˜) separates points, contains the constants and the complex conjugate of each of its elements.

Therefore, by the Stone-Weierstrass theorem, C is dense in C(T˜). (This is the lemma in [FK] on page 198).

∨ Fix v. Each term in the sums appearing in (2.11.1) and (2.11.2) has a factor fv (t(πv)).

∨ Let tv,i, i = 0, 1, 2,... be the distinct elements t(πv) in Gb which occur in fv (t(πv)) of (2.11.1) and (2.11.2). Set

X Y ∨ ci,v = αk fw (tw,k).

t(πv)k=tv,i w6∈S∪{v}

Then the sum of (2.11.1) and (2.11.2) can be written as

X ∨ ci,vfv (tv,i). i

83 For a standard parabolic subgroup P of G, with Levi decomposition P = MN and

hλ,HP (m)i λ ∈ aP denote by χλ the character of P deﬁned by χλ(mn) = e , here m is in M and n in N.

1 1 For λ ∈ iaP denote by χλ the restriction of χλ to the torus A. The map λ 7→ χλ deﬁnes a homomorphism from iaP to the group of unramiﬁed characters of A.

∨ 1 Its kernel is a lattice ΛP and tr I(λ, πv, fv) = fv (χλ). Set

0 X Y ∨ cP,v(λ) = βk fv (τv, λ). τ w6∈S∪{v} Here the sum over τ ranges over ω, ω0 or σ in (2.11.3) and (2.11.4). Then each of the integrals in (2.11.3) and (2.11.4) can be written as

Z 0 ∨ cP,v(λ)fv (λ)dλ. iR Set

X 0 cP,v(λ) = cP,v(λ + l).

l∈ΛP Then each of the integrals in (2.11.3) and (2.11.4) can be written as

Z ∨ cP,v(λ)fv (λ)dλ. iR/ΛP With these notations the equality of the spectral sides can be written as:

Z X ∨ X ∨ ci,vfv (tv,i) = cj,v(t)fv (t)dt. (2.11.5) i j Tj

˜ Here tv,i belong to the compact manifold T , ci,v are complex numbers, Tj are compact ˜ submanifolds of T and cj,v(t) are measurable complex valued functions.

84 Proposition 33. With the above notations, we have that ci,v = 0 far all i.

Proof. We follow the argument of the Proposition on page 198 in [FK]. Suppose not.

We may assume that c0,v 6= 0 and after multiplying by a scalar we may assume that ˜ c0,v = 1. Let P (t) be a function in C such that P (t0) = 1, |P (t)| ≤ 2 for t ∈ T

2 and |P (t)| < ε for all t such that |t − t0| > ε . From Lemma 32 it follows that any

∨ continuous function in C can be approximated by an element fv in the Hecke algebra

Hv. Since the trace formulas give absolutely convergent sums and integrals we have that

X X Z M = |ci,v| + |cj,v(t)||dt| i j Tj is ﬁnite. Hence for every ε > 0 there is N > 0 such that

X X Z |ci,v| + |cj,v(t)||dt| < ε. i>N j>N

∨ With the choice of fv as above equation (2.11.5) gives the following

N N X X Z X X Z |c0,v| ≤ |ci,v|ε + ε |cj,v(t)||dt| + |ci,v| + |cj,v(t)||dt| i=1 j=0 i>N j>N N N ! X X Z ≤ ε |ci,v| + |cj,v(t)||dt| + 1 ≤ ε(M + 1). i=1 j=0

This is true for every ε. This contradicts the assumption that c0,v = 1.

Hence we have

X Y ∨ αk fw (tw,k) = 0.

t(πv)k=tv,i w6∈S∪{v}

85 Repeating the argument a ﬁnite number of times, we get that for a ﬁnite set of places

V disjoint from S we have

X Y ∨ αk fw (tw,k) = 0. (2.11.6) w6∈S∪V

Corrolary 34. All the αk are equal to zero.

Proof. Suppose that α0 6= 0. Choose an N such that

X |α0| |α | ≤ . (2.11.7) k 2 k≥N

Then choose a set V disjoint from S so that if 1 ≤ k < N then for some v ∈ V we have

∨ that tv,0 6= tv,k. Then applying (2.11.6) with all fv equal to 1 we get a contradiction to (2.11.7).

This gives us the following equality. ! X Y 1 A0 (π0, f 0 ) · f ∨ (η(t(π0 ))) + A0 (χ0, f 0 ) · f ∨ η(χ), 0 v v v r v v 2 π0 v6∈S ! X Y ∨ − A0(π, fv) · fv (t(πv)) π v6∈S ! X X Y ∨ = Ad,P1 (σ, fv) · fv (t(I(0, σv))) . P1 σ v6∈S

Summarizing the above we have the following proposition.

Proposition 35. Fix θ ∈ F − F 2. Fix a finite set S of F -places, including the archimedean places and those that ramify in E/F . At each v 6∈ S fix an unramified

0 0 1 0 Gv-module πv = IG (ξv, −ξv). Put πv = I(ξv, 2 −ξv); this is an unramiﬁed Gv-module.

86 0 0 Then, with the notations as above, for any matching test functions fv on Gv/Zv and fv on Gv/Zv we have

X X 0 0S 0 1 0 0 X 0S 0 1 0 1 0 W 0 (π (f )φ )L 0 ( , π ⊗ χ ) + 4π E 0 (I(f , χ , ), χ , )φ (1) ψ φ 2 ψ 2 2 0 0S 0 π φ0∈π0K φ

X X S = m(π) Wψ(π(f )φ)ΠH (φ) π φ∈πKS

X X X S S + νP1 JP1 (P1,I(0, σ, f )φ, 0) ·Wψ(0,I(0, σ, f )φ).

P1 π {φ} The sum over π0 is over the cuspidal representations of PGL(2, A) which have a

0S 0 nonzero vector ﬁxed by K , such that πv is the unramiﬁed constituent of IG0 (ξv, −ξv) for all v 6∈ S. The sum over φ0 ranges over an orthonormal basis of smooth vectors

0S in the space π0K of K0S-ﬁxed vectors in π0. The second sum over φ0 is empty unless

1 0 0 0 IG (χθv, 2 ) ' IG (ξv, −ξv) for all v 6∈ S, in which case the ﬁrst sum, over π , is empty. 0 1 K0S 0 Then φ ranges over IG (χθ, 2 ) .

The ﬁrst sum over π is over all cuspidal representations of G such that πv is the

1 unramified constituent of I(ξv, 2 −ξv) for all v 6∈ S. The φ range over an orthonormal KS S basis of smooth vectors in the space π of K -fixed vectors in π. The sum over P1 is over the standard proper parabolic subgroups of G. The second sum over π is over discrete series automorphic representations of G such that πv is the unramified

1 constituent of I(ξv, 2 − ξv) for all v 6∈ S.

The main representation theoretic application of this trace identity is the following proposition. Let χθ is the quadratic character corresponding to the quadratic exten- √ sion E = F ( θ).

87 Proposition 36. Let π0 be a cuspidal automorphic representation of PGL(2, A) with

1 0 2 L( 2 , π ⊗ χθ) 6= 0 for some θ in F − F . Then there exists a discrete spectrum automorphic representation π of PGSp(2, A) which is distinguished by H, has a nonzero

Fourier coeﬃcient and πv is the unramiﬁed constituent of the induced representation

0 1 1 0 1 0 I(πv, 2 ) for almost all v. If L( 2 , π ) = 0 then π is cuspidal. If L( 2 , π ) 6= 0 then π is residual.

Proof. Fix a set S of F -places containing the archimedean ones and those that ramify √ 0 in E = F ( θ), such that πv is unramiﬁed for v 6∈ S. For any matching test functions

0 0 fv on Gv and fv on Gv the identity from Proposition 35 holds. Namely it holds where

0 0 0 the ﬁxed unramiﬁed Gv-modules at all v 6∈ S are the components πv of π . In this case π0 parameterizes the only term in the left side of the equality of Proposition 35.

0 1 1 0 0 Note that tr IG (ξ, −ξ; fv) is equal to tr IG(ξ, 2 − ξ; fv). Since L( 2 , π ⊗ χθ) 6= 0 and 0 0 0 0 L(ζ, π ⊗ χθ) lies in the span of the L-functions Lφ0 (ζ, π ⊗ χθ) for φ ∈ π , there is

0 0 0 some φ ∈ π with L 0 (ζ, π ⊗ χ ) 6= 0. 1 φ1 θ 0 0 0K0S Since π is cuspidal, it is generic. Hence there is a vector φ2 ∈ π such that the

0 0 value Wψ0 (φ2) of the Whittaker function of φ2 at the identity is not zero. Note that the matching from Corollary 13 permits us to use an arbitrary test function f 0S. Since the operators π0(f 0S) span the endomorphism algebra of π0K0S , we may choose f 0S to

0 0S 0 0 have the property that π (f ) acts as 0 on each vector in π which is orthogonal to φ1,

0 0 0S but maps φ1 to φ2. With this choice of f the left side of the identity of Proposition 35 reduces to a single term, which is

0 1 0 Wψ0 (φ2)Lφ0 , π ⊗ χθ . 1 2

88 This is nonzero, hence so is the left side, and also the right side of the identity. Hence there exists a discrete spectrum automorphic representation π of PGSp(2, A), with

ΠH (φ1) 6= 0 and Wψ(φ2) 6= 0 for some φ1, φ2 in π.

0 1 This π has the property that tr πv(fv) = tr I(πv, 2 ) for all v 6∈ S, namely the Gv- 0 1 module πv is the unramiﬁed irreducible constituent of the representation I(πv, 2 ) 0 1 0 1 of Gv. This constituent is the Langlands quotient J(πv, 2 ) of I(πv, 2 ). Langlands 0 1 0 1 0 1 theory of Eisenstein series implies that the quotient J(π , 2 ) = ⊗J(πv, 2 ) of I(π , 2 ) 1 is automorphic, and deﬁnes a cuspidal representation precisely when L(π, 2 ) = 0. Hence the representation π which we obtained is as required.

2.12 Converse

We start with a description of some GSp(2)-packets, establishted in [F9] using a comparison of the trace formula of PGL(4) twisted by the outer automorphism g 7→ tg−1, with the stable trace formula of GSp(2) − in analogy with the theory of base change from U(3, E/F ) to GL(3,E) of [F2]. These packets are those obtained by the

Saito-Kurokawa lifting, see [PS1].

If ρ is a cuspidal representation of PGL(2, A) there is a global packet of 2r global automorphic representations of PGSp(2, A) whose components are the Langlands 1 1 1 quotient J(ρv, 2 ) of the induced I(ρv, 2 ) = IP (ρv, 2 ) at almost all places v of F . Here r is the number of discrete spectrum components of ρ. Exactly half of these are discrete spectrum automorphic representations, when r ≥ 1. When r = 0 the single

1 representation is discrete spectrum, necessarily residual, precisely when L(ρ, 2 ) 6= 0.

89 One of these 2r−1 representations (r ≥ 1) is residual, namely noncuspidal, precisely

1 1 1 when L(ρ, 2 ) 6= 0. It is the quotient J(ρ, 2 ) = ⊗vJ(ρv, 2 ). Its space is generated by 1 the residues of the Eisenstein series E(h, Φ, ρ, ζ) at ζ = 2 . 1 When ρv is square integrable, the induced I(ρv, 2 ) is reducible, of length two (see [Sh],

Proposition 6.1, p. 287, for a proof in the case where ρv is cuspidal). The constituent

1 + other than J(ρv, 2 ) is a noncuspidal square integrable, denoted here by π(ρv) . The + − packet of π(ρv) will contain a second member, a cuspidal π(ρv) .

1 The cuspidal automorphic representations in the packet of J(ρ, 2 ) are of the form − π = ⊗πv, where πv = π(ρv) for an even number of places v where ρv is square-

1 1 1 integrable and π 6= J(ρ, 2 ) if L(ρ, 2 ) 6= 0, in which case π = J(ρ, 2 ) is discrete spectrum, residual but not cuspidal. Namely there are 2r−1 cuspidal representations

1 r−1 1 in the packet when L(ρ, 2 ) = 0, and 2 − 1 when L(ρ, 2 ) 6= 0, provided r ≥ 1, and 0 when r = 0.

When ρ is a (nontrivial) quadratic character of F ×\A×, the automorphic induced rep- 1 1 1 resentation IP (ρ ◦ det, 2 ) = IB( 2 , 0; ρ, ρ) of PGSp(2, A) has a quotient J( 2 , 0; ρ, ρ) = 1 1 ⊗vJ( 2 , 0; ρv, ρv). Here J( 2 , 0; ρv, ρv) is the Langlands quotient of the induced IP (ρv ◦ 1 1 det, 2 ) = IB( 2 , 0; ρv, ρv). This global quotient is irreducible and residual discrete spectrum.

1 The induced I( 2 , 0; ρv, ρv), ρv 6= 1, is reducible of length two (by [KR]), and in 1 addition to the nontempered constituent J( 2 , 0; ρv, ρv) there is a square-integrable + + but not cuspidal constituent π(ρv) . The packet of π(ρv) contains also a cuspidal

− member π(ρv) .

90 1 The automorphic members in the global packet of J( 2 , 0; ρ, ρ) are obtained by replac- 1 − ing the component J( 2 , 0; ρv, ρv), ρv 6= 1, by the cuspidal π(ρv) , at an even number 1 of places. They are all cuspidal, except the residual J( 2 , 0; ρ, ρ). These representations were studied by Howe and Piatetski-Shapiro [HPS] by means of the Theta lifting. These examples are similar to those found in [F2] in the case of U(3): these are cuspidal representations with a ﬁnite number of cuspidal components and all other components are nontempered.

1 The examples related to J(ρ, 2 ), where ρ is cuspidal, and the packet contains only a ﬁnite number of elements, were studied in [PS1] and [F9]. They are diﬀerent from the examples of [F2] on U(3), but similar to packets of the two-fold covering group of SL(2, A), described by Waldspurger [Wa1]. Our Proposition 36 establishes the existence of a cuspidal element in the packet of

1 J(ρ, 2 ), ρ cuspidal, using its properties of being cyclic (Pθ 6= 0) and having a nonzero

Fourier coeﬃcient Wψθ . To prove that this packet contains a cuspidal element we used

1 the fact that there is no residual representation when L(ρ, 2 ) = 0 and ρ is cuspidal. But we have not proven the existence of a cuspidal member in the packet when ρ

1 is cuspidal and L(ρ, 2 ) 6= 0, nor when ρ 6= 1 is a quadratic character (When ρ = 1 1 there are no discrete spectrum representation in the packet of J(ρ, 2 )). This however is proven in [F9].

We shall also consider a converse to Proposition 36. For this purpose note that an unpublished theorem of Bernstein shows that on any irreducible admissible representation πv of Gv = PGSp(2,Fv) = SO(5,Fv) there exists at most one − up to a scalar

multiple − Hv = SO(4,Fv)-invariant nonzero linear form Pπv . Fix such a form. Let

91 us also ﬁx a linear form Wπv,ψθ on such a πv which transforms under the action of

IX Nv = {u = ( 0 I )} according to multiplication by the character ψθ(u).

If Pπ˜v denotes the invariant form on the contragredientπ ˜v of πv, then it lies in the

∗ ∞ dualπ ˜v ofπ ˜v, and for each fv ∈ Cc (Gv/Zv) the vector πv(fv)Pπ˜v lies in the smooth ˜ ∗ part π˜v ' πv ofπ ˜v . Associated to πv and ψv we then obtain a linear form

X ∨ WPπv,ψθ : fv 7→ hWπv,ψθ , πv(fv)Pπ˜v i = Wπv,ψv (πv(fv)ξ)Pπ˜v (ξ ) ξ

∞ n h n h on Cc (Gv/Zv) with the property WPπv,ψθ ( fv ) = ψθ(n)WPπv,ψθ (fv) for fv (g) =

−1 fv(n gh), n ∈ Nv, h ∈ Hv, g ∈ Gv. The sum over ξ on the right ranges over an

∨ orthonormal basis for πv, and ξ is the dual basis forπ ˜v. Also, Wπv,ψθ lies in the dual

∗ πv of πv, hence its value at πv(fv)Pπ˜v is well deﬁned.

The Whittaker-Period distributions WPπv,ψθ are analogous to Harish-Chandra’s characters. They have interesting properties, but we shall use only the simple fact

− see Proposition 0.1 of [F7] (the distribution (WψP )π(f) is obtained on taking

C1 = N, ζ1 = ψ, C2 = C, ζ2 = 1,P1 = Wψ and P2 = P there) − that WPπv is independent of the choice of the basis ξ, and that if π1v, ··· , πkv are inequivalent irreducible

representations, then WPπ1v,ψθ , ··· ,WPπkv,ψθ are linearly independent distributions

∞ on Cc (Gv/Zv).

Proposition 37. Let F be a function ﬁeld, and ﬁx θ ∈ F − F 2. Let π be a discrete

spectrum representation of PGSp(2, A) with Pθ(Φ1) 6= 0 and Wψθ (Φ2) 6= 0 for some

Φ1, Φ2 in π. Then either there exists a cuspidal representation ρ of GL(2, A) with 1 1 1 L( 2 , ρ ⊗ χθ) 6= 0 and πv ' J(ρv, 2 ) for almost all v, or πv ' J( 2 , 0; χθ,v , χθ,v ) for almost all v.

92 Proof. Choose a suﬃciently large ﬁnite set V containing all the F -places where π

0 or E ramify, and at each v ∈ V ﬁx a congruence subgroup Kv ⊂ Kv such that πv

0 ∞ 0 0 contains a nonzero Kv-ﬁxed vector. We shall work only with fv in Cc (Kv\Gv/ZvKv), v ∈ V . The right side of the identity of Proposition 35 can be written in the form

X Y m(π) WPπv,ψθ (fv). π v∈V The sum is finite, since we fixed the ramification at all places, and F is a func-

tion ﬁeld. The linear independence of the forms WPπv,ψθ implies that there are

∞ fv ∈ Cc (Gv//ZvKv)(v ∈ V ) for which the sum over π reduces to a single nonzero contribution, that which is parametrized by π. Here we used Corollary 5.1, which permits us to use any test functions fv. In particular, the right side, associated with G,

1 is nonzero. Hence so is the left side, establishing the existence of ρ, or J( 2 , 0; χθ, χθ), associated with π, as asserted.

Remark 4. An analogous argument applies in the number ﬁeld case too, on using the

K-finiteness of f, since by [F9] fixing π at almost at all places limits its infinitesimal character to a finite set.

2.13 Split cycles

We have studied above cusp forms on G(A) = PGSp(2, A) cyclic with respect to the subgroup H(A), θ ∈ F − F 2, deﬁned in Section 2.2. There is no analogous theory when θ ∈ F 2, since in this case there are no cusp forms on G(A) with nonzero H(A)- cycles. This is the content of [PS1], Corollary 6.2, proven by means of the Theta

93 correspondence. We shall outline next a spectral proof, using a Fourier summation formula, in the spirit of this paper, of that fact. As usual, ﬁx a character ψ 6=

1 of A/F into the multiplicative groups of the complex numbers, and introduce a 1 x ∗ ∗ 0 1 y ∗ character ψB of NB(A) = n = 0 0 1 −x by ψB(n) = ψ(x + y). The Whittaker 0 0 0 1 R function of a cusp form Φ on G( ) is deﬁned by WΦ(g) = Φ(ng)ψ(n)dn. A NB (F )\NB (A)

A cuspidal G(A)-module π is called generic if WΦ 6= 0 for some Φ in π. Put zΦ(g) = R Φ(zg)dz, where − as in Section 2.2 − ZQ is the center of the unipotent ZQ(F )\ZQ(A) radical NQ of the parabolic subgroup Q of G = GSp(2). Denote by R the standard maximal unipotent subgroup of the standard Levi subgroup MQ of Q. As in [PS1], Lemma 6.2, we have the following.

Lemma 38. For any cusp form Φ on G(A), we have that X zΦ(g) = WΦ(γg).

γ∈R(F )\MQ(F )

n ∗ ∗ ∗ o Proof. The quotient Z\Q/Z is naturally isomorphic to the subgroup D = ∗ ∗ ∗ Q 0 0 1 R of GL(3), and zΦ(δg) = zΦ(g) for all δ ∈ D. Since Φ is cuspidal, zΦ(yd)dy Y (F )\Y (A) is equal to zero for all d ∈ D(A), where Y is any of the unipotent subgroups Y1 = n 1 0 ∗ o n 1 ∗ ∗ o 0 1 ∗ or Y2 = 0 1 0 of D. The stabilizer of the character ψ1(y) = ψ(y2), y = 0 0 1 0 0 1 1 0 y1 n ∗ ∗ ∗ o 0 1 y2 , of Y1(A) is D1(A),D1 = 0 1 ∗ . Hence the Fourier expansion of zΦ along 0 0 1 0 0 1

Y1(A) is X Z zΦ(d) = zΦ,1(δd), zΦ,1(d) = zΦ(yd)ψ1(y)dy. Y1(F )\Y1( ) δ∈D1(F )\D(F ) A

n 1 ∗ 0 o Taking the Fourier expansion of zΦ,1 along the subgroup Y1 = 0 1 0 , and 0 0 1 noting that the constant term is zero (since Φ is cuspidal), we obtain zΦ(d) =

94 P δ∈N(F )\D(F ) WΦ(δd), where N is the unipotent upper triangular subgroup of D '

Q/ZQ, as required.

Lemma 38 implies the following. Suppose Φ is a cusp form on G(A) with the integral R Φ(h)dh not equal to zero , namely its restriction to H( ) is not orthog- Z(A)H(F )\H(A) A onal to the constant functions and hence it is not cuspidal (on H(A)). Then zΦ 6= 0

(otherwise the restriction of Φ to H(A) would be cuspidal), and WΦ 6= 0, namely Φ lies in a generic π. We are led then to consider the sum in the following.

Proposition 39. Let f be a cuspidal test function on G(A)(e.g., it has a cuspidal component). Let {Φ} denote an orthonormal basis of the space of cusp forms on

G(A). Then X Z Z (π(f)Φ)(n)Φ(h)ψB(n)dndh = 0. Φ∈{Φ} NB (F )\NB (A) Z(A)H(F )\H(A) Proof. The sum of the proposition is the spectral side of a Fourier summation formula.

It suﬃces to compute its geometric side,

Z Z X f(nγh)ψB(n)dndh. NB (A)/NB (F ) Z(A)H(F )\H(A) γ∈Z(F )\G(F )

Proposition 6(b) asserts that G = NH ∪ NAγ1H ∪ Nγ2H ∪ NAγ3H. To show that our sum vanishes it suﬃces to note that ψB is nontrivial on

−1 γiH(A)γi ∩ N(A)B, (i = 1, 2, 3, 4), γ4 = I.

This is clear for γ4 = I. When i = 1, 2, 3 take

1 4x 1 y 1 y 1 0 1 0 1 y x 1 , 0 1 and 0 1 in H, respectively. 0 1 0 1 0 1 The proposition follows.

95 To show that there are no cusp forms on G(A) with nonzero H(A)-cycles it remains to (extend Proposition 39 to a general test function f and) isolate any single cusp form Φ which occurs in the sum. At least when F is a function ﬁeld, this separation is performed in a similar context in the section below.

2.14 Invariance of Fourier coeﬃcients of cyclic automorphic

forms

Let G be a reductive group scheme over a global ﬁeld F , and P = MN a parabolic F - subgroup with Levi subgroup M and unipotent radical N. Let ψ : N(F )\N(A) → C× be a character, and let φ be an automorphic form on G(A) which transforms trivially under Z(A), where Z is the center of G (thus φ ∈ L2(G(F )Z(A)\G(A))). The

(ψ-) Fourier coeﬃcient of φ (along the compact homogeneous space N(F )\N(A)) R is deﬁned to be Wψ(φ) = φ(n)ψ(n)dn. The Levi subgroup M( ) acts on N(F )\N(A) A N(A) by conjugation. The stabilizer of ψ is

−1 StabM(A)(ψ) = {m ∈ M(A); ψ(mnm ) = ψ(n) all n ∈ N(A)}.

It is a subgroup of M(A), and its subgroup of rational points is denoted by StabM (ψ).

The “generalized Whittaker” functional Wψ(φ) is clearly invariant under the action of StabM (ψ), Wψ(r(m)φ) = Wψ(φ), where (r(g)φ)(h) = φ(hg), since φ is an automorphic form. In general Wψ(φ) is not StabM(A)(ψ)-invariant. The purpose of this Section is to show that under some natural geometric conditions (on the group), cyclic cusp forms do have the property that Wψ is invariant under the group of ad´elepoints

0 of the connected component of the identity StabM (ψ) in StabM (ψ).

96 Denote by L0(G(F )Z(A)\G(A)) the space of cusp forms, namely functions φ in 2 R 0 0 L (G(F )Z( )\G( )) such that 0 0 φ(n g)dn = 0 for all g ∈ G( ) and A A N (F )\N (A) A any proper F -parabolic subgroup P 0 = M 0N 0 of G.A cuspidal G(A)-module is an irreducible constituent of the representation of G(A) by right translation on

L0 = L0(G(F )Z(A)\G(A)). Cusp forms are rapidly decreasing on a Siegel domain.

Let H be an F -subgroup of G. Assuming that the cycle Z(A)H(F )\H(A) is of ﬁ- R nite volume, the period integral PH (φ) = φ(h)dh converges. A cuspidal Z(A)H(F )\H(A) representation π ⊂ L0 is called cyclic if it contains a form φ with a nonzero period

PH (φ) over the cycle Z(A)H(F )\H(A). Any form φ in such π has a nonzero period

0 PH0 (φ) 6= 0, where H (A) is conjugate to H(A) in G(A).

Theorem 40. Denote by {δ} a set of representatives in G(F ) for the double coset space N(F )\G(F )/H(F ), and by {δ}0 its subset of δ such that ψ is 1 on N(A) ∩ δH( )δ−1. Suppose that Stab0 (ψ) is contained in δH( )δ−1 for all δ ∈ {δ}0. A M(A) A

If π is a cyclic cuspidal G(A)-module with a cuspidal component, then Wψ(φ) is Stab0 (ψ)-invariant for all φ in π. M(A)

Remark 5. When G = GL(k), M = the diagonal subgroup, N = the unipotent ! P × upper triangular subgroup, and ψ(n) = ψ ni,k+1 , where ψ : A/F → C is 1≤i

Wψ(φ). The Theorem applies (nontrivially) when the character ψ is a degenerate character, when viewed as a character of the unipotent radical of the minimal parabolic

97 subgroup. We shall discuss below an example where the Theorem applies nontrivially. This example, concerning GSp(2), has also been treated by Piatetski-Shapiro

[PS2] by means of the theta lifting. Our approach relies on an application of a

Fourier summation formula. Although much more recent, this approach has the advantage of being conceptually simpler. A disadvantage of this technique is that it deals with the entire automorphic spectrum. The presence of contributions from the continuous spectrum, expressed in terms of Eisenstein series, leads to technical difficulties. To avoid encountering these in this Section, we work with cuspidal representations with a cuspidal component. These technical difficulties can be handled as in [F5], where G = PGL(k),H = GL(k − 1), and ψ is the degenerate character ψ(n) = ψ(n1,2 + nk−1,k) of the unipotent upper triangular subgroup. Yet in the case of [F5] there are no cuspidal G(A)-modules cyclic over H(F )\H(A), while in the PGSp(2) example discussed below there are such cuspidal representations, and we content ourselves here with the form the Theorem takes, without launching into deep analysis. Another simplifying assumption which we make is to take F to be a function field, to avoid dealing with the archimedean places.

Proof. This is based on an application of the Fourier summation formula for a test function f = ⊗fv, product over all places v of the global ﬁeld F , where fv ∈

∞ 0 Cc (Zv\Gv) for all v, and fv is the unit element fv of the convolution algebra of spherical, namely Kv-biinvariant, compactly supported functions on Zv\Gv, for almost all v. Our notations are standard: Fv is the completion of F at the place v, we put Gv = G(Fv), Zv = Z(Fv),... , and Kv is a hyperspecial maximal compact

98 open subgroup of Gv. Implicit is a choice of a Haar measure dgv on Gv/Zv such Q that the product |Kv|, |Kv| = vol(Kv), converges, thus of a global Haar measure v dg = ⊗dgv. The subscript “c” means compactly supported, and the superscript “∞” indicates smooth, namely locally constant in the nonarchimedean case.

The convolution operator (r(f)φ)(g) = R f(h)φ(gh)dh on the space of square Z(A)\G(A) integrable functions L2(G(F )Z(A)\G(A)) is easily seen to be an integral operator Z (r(f)φ)(g) = Kf (g, h)φ(h)dh Z(A)G(F )\G(A)

P −1 with kernel Kf (g, h) = γ∈Z(F )\G(F ) f(g γh). On the other hand, if some component – say fv1 – of f, is a cusp form (we shall assume this from now on), then the operator r(f) factorizes through the natural projection from L2(G(F )Z(A)\G(A)) onto L0(G(F )Z(A)\G(A)) (see [F1]). The restriction r0(f) of r(f) to L0 has the 0 P P kernel Kf (g, h) = π φ(π(f)φ)(g)φ(h). Here π ranges over a set of representatives for the equivalence classes of the irreducible constituents of L0, while φ ranges over an orthonormal basis {φ} of smooth vectors in the π-isotypic component of L0. It is well known that the multiplicity of π in L0 is ﬁnite.

0 Now for our f, which has a cuspidal component, we have Kf (n, h) = Kf (n, h). We multiply both sides by ψ(n) and integrate over n in N(F )\N(A) and h in H(F )\H(A), to obtain the Fourier summation formula. It is

X X Z Z X Wψ(π(f)φ)P (φ) = f(nγh)ψ(n)dndh π φ N(A)/N(F ) H(F )\H(A) γ∈Z(F )\G(F ) Z Z X X X = f(nνδζh)ψ(n)dndh, N(A)/N(F ) H(F )\H(A) δ ζ∈H(F ) ν∈N(F )/N(F )∩δH(F )δ−1

99 where the sum over δ ranges over a set of representatives in G(F ) for the double coset S space N(F )\G(F )/H(F ), thus G(F ) = δ N(F )δH(F ) (disjoint union). This is

X Z Z = f(nδh)ψ(n)dndh −1 δ H(A) N(A)/N(F )∩δH(F )δ X Z Z = 0vol(δ) f(nδh)ψ(n)dndh. −1 δ H(A) N(A)/N(A)∩δH(A)δ

−1 The last sum ranges over the subset of the δ for which ψ is 1 on N(A) ∩ δH(A)δ . Here −1 −1 vol(δ) = |N(A)∩δH(A)δ /N(F )∩δH(F )δ | The last expression is named the “geometric side” of the Fourier summation formula, while the initial expression is the “spectral side”, for our test function f.

We shall compare the summation formula for f with that for sf(g) = f(s−1g), for any s in

Stab0 (ψ). Note that M(A) Z Z (π(sf)φ)(u) = sf(g)φ(ug) = f(s−1g)φ(ug) Z = f(g)φ(usg) = (π(f)φ)(us) = (π(s)(π(f)φ)) (u).

If {φα} denotes an orthonormal basis of the π-isotypic component of L0, then φ =

P 0 R 0 (φ, φα)φα, where (φ, φ ) = φ(g)φ (g)dg is a nondegenerate sesqui-linear α Z(A)G(F )\G(A) form on L0. Then

s X X −1 π( f)φ = π(s)π(f)φ = (π(s)π(f)φ, φα)φα = (π(f)φ, π(s )φα)φα α α

X ∗ −1 = (φ, π(f )π(s )φα)φα, α

100 where f ∗(g) = f(g−1). Consequently

X s X X ∗ −1 Wψ(π( f)φ)P (φ) = Wψ(φα)(φ, π(f )π(s )φα)P (φ) φ φ α   X X ∗ −1 = Wψ(φα)P  (π(f )π(s )φα, φ)φ α φ

X ∗ −1 X ∗ = Wψ(φα)P (π(f )π(s )φα) = Wψ(π(s)φ)P (π(f )φ), α φ where to obtain the last expression we choose the orthonormal basis {φα} to be {π(s)φ}. In summary,

X X ∗ X X s Wψ(π(s)φ)P (π(f )φ) = Wψ(π( f)φ)P (φ) π φ π φ X Z Z = 0 vol(δ) f(s−1nδh)ψ(n)dndh, −1 δ H(A) N(A)/N(A)∩δH(A)δ by the Fourier summation formula for sf,

X Z Z = 0 vol(δ) · f(ns−1δh)ψ(n)dndh, −1 δ H(A) N(A)/N(A)∩δH(A)δ since s ∈ Stab (ψ), M(A)

X Z Z = 0 vol(δ) · f(nδh)ψ(n)dndh, −1 δ H(A) N(A)/N(A)∩δH(A)δ

−1 since s ∈ δH(A)δ by the assumption of the theorem,

X X ∗ = Wψ(φ)P (π(f )φ), π φ by the Fourier summation formula for f (and the line following the deﬁnition of f ∗).

The theorem concerns a form φ1 in a fixed cuspidal G(A)-module π1, whose component at the finite place v1 is cuspidal. There is a finite set V of F -places, containing the archimedean

101 places and v1, such that not only π1v is unramiﬁed for all v outside V , but also φ1 is Kv-

V N invariant for all such v. The part f = fv of f outside V can be chosen such that fv is v6∈V 0 the unit element fv for almost all v, and fv is any Kv-spherical function at the remaining

ﬁnite set of places. For any cusp form φ, say in the cuspidal G(A)-module π, the operator

πv(fv) acts on φ by multiplication by the character tr πv(fv) if φ is Kv-invariant and as 0 otherwise. A standard argument of “generalized linear independence of characters” (see, e.g., [FK], or Section 2.11), based on the absolute convergence of the spectral side of the

Fourier summation formula, standard unitarity estimates on the Hecke eigenvalues of the unramiﬁed πv, and the Stone-Weierstrass theorem, permits deducing the following identity for all fV = ⊗v∈V fv such that fv1 is a cuspidal function.

X X X X Wψ(π(s)φ)P (π(f)φ) = Wψ(φ)P (π(f)φ), π φ π φ where π ranges over the cuspidal G(A)-modules with πv ' π1v for all v 6∈ V , and φ ranges K(V ) Q over an orthonormal basis for the space π of K(V ) = Kv-invariant vectors in the v6∈V π-isotypic part of L0.

We shall now view the cuspidal representation π as an abstract representation of G(A), and fix an isomorphism of π with ⊗πv. Any K(V )-fixed smooth form φ in the space of the cuspidal π = ⊗πv corresponds to a linear combination of finitely many vectors

V N V ξ ⊗ ( v∈V ξiv), 1 ≤ i ≤ k, where ξ is the (unique up to scalar multiple) K(V )-ﬁxed V N nonzero vector in π = v6∈V πv, and ξiv ∈ πv.

We claim that each of these products corresponds to a cusp form. Since πv is admissible there is some suﬃciently small good ([B]) compact open subgroup (e.g. a congruence subgroup) K1v of Gv such that ξiv(1 ≤ i ≤ k) are πv(K1v)-invariant. The algebra

{πv(fv); fv ∈ Cc(K1v\Gv/K1v)} is the algebra of endomorphisms of the ﬁnite dimensional

Kv (since πv is admissible) module πv . In particular there exists a K1v-biinvariant fv such

102 that πv(fv) acts as an orthogonal projection on the space generated by ξ1v. For a suitable choice of such fv(v ∈ V ) we have that πV (fV )φ corresponds to a factorizable vector

V N ξ ⊗ ( ξ1v). Consequently, in order to prove that Wψ(π(s)φ1) = Wψ(φ1), we may as- v∈V sume that the cusp form φ1 corresponds to a factorizable vector. Moreover, multiplying by a scalar we may assume that the orthonormal basis {φ} of π1 is chosen to include our form

φ1, which corresponds to a factorizable vector.

We are assuming that π1 is cyclic, namely that there exists a form φ2, which can be taken to be in the orthonormal basis {φ} of π1, such that P (φ2) 6= 0. Thus φ2 = φ1 or (φ1, φ2) = 0.

The argument of the previous paragraph implies that φ2 can also be assumed to correspond

V N to a factorizable vector. Thus φ2 corresponds to ξ ⊗ ( ξ2v). At the place v1 we take v∈V fv1 (g) = d(π1v1 )(π1v1 (g)ξ1v, ξ2v). The complex number d(π1v1 ) is the formal degree of π1v1 .

∞ This is a matrix coeﬃcient of the cuspidal Gv-module π1v1 . It lies in Cc (Zv1 \Gv1 ). By the

Schur orthonormality relations it has the property that πv1 (fv1 ) acts as 0 unless πv1 = π1v1 , and then π1v1 (fv1 ) acts as 0 on any vector perpendicular to ξ1v1 , while π1v1 (fv1 )ξ1v1 = ξ2v1 .

We conclude that the identity (1) holds where π ranges over the cuspidal G(A)-modules with πv ' π1v for all v 6∈ V and for v = v1, and φ ranges over an orthonormal basis of the

K(V ) V N space π of the form ξ ⊗ ξ1v1 ⊗ ξV1 , where V1 = V − {v1} and ξV1 ∈ πV1 = πv. v∈V1 P Note that the distribution f 7→ φ Wψ(φ)P (π(f)φ), where φ ranges over an orthonormal basis consisting of smooth vectors in the space of the irreducible cuspidal π, is independent of the choice of the basis {φ}. Consequently, in (1), the double sums can be expressed as the product by the multiplicity of π1 in L0(Z(A)G(F )\G(A)) (same finite number on both sides), of a sum over an orthonormal basis {φ} of the K(V1)-fixed smooth vectors in π1. Moreover, we may assume that the basis {φ} consists of factorizable vectors. In fact, fix a compact open subgroup K1v(v ∈ V1) such that φ1, φ2 are π1v(K1v)-invariant for all v ∈ V1.

103 0 For any irreducible πv choose an orthonormal basis {ξv}, containing the Kv-ﬁxed vector ξv

K1v if v 6∈ V , the vector ξv1 at v = v1, and a basis of the space πv of K1v-ﬁxed vectors for N 0 v ∈ V1. Then {φ} = { ξv} (ξv = ξv for all v 6∈ V ) makes an orthonormal smooth basis of v πK(V ).

We shall assume that our global field is a function field, or that the fv can be chosen at the archimedean places so that only π with infinitesimal character in a fixed finite set at each archimedean place v may occur in (1). Further, at the finite places v ∈ V1 we let fv range only over the space of K1v-biinvariant functions. This we do in order to use Harish-Chandra’s result that there exist only finitely many cuspidal representations with

fixed infinitesimal characters at the archimedean places, and fixed ramification at all finite places. For such f the sum over π in (1) is then finite.

Applying Bernstein’s decomposition theorem ([B], see also [F4], p. 165), we may even choose the fv(v ∈ V1) so that the components πv of the π which occur in (1) have inﬁnitesimal character in the same connected component as that of π1v(v ∈ V1); the terminology is that of [B] (and [F4]).

For each ﬁnite v ∈ V1, the set of inﬁnitesimal characters of the πv which occur in (1) is

∞ ﬁnite. Moreover, {πv(fv); fv ∈ Cc (ZvK1v\Gv/K1v)} is the algebra of endomorphisms of

K1v the space πv of K1v-ﬁxed vectors in π1v. Consequently, if ξ1v is the component at v of ∞ φi(i = 1, 2), there is fv in Cc (ZvK1v\Gv/K1v) such that πv(fv) = 0 for each πv 6' π1v which occurs in our sum, and such that π1v(fv) acts as 0 on each vector orthogonal to

ξ1v, while π1v(fv)ξ1v = ξ2v. Both sums over π and φ reduce then to a single contribution, parametrized by π1 and φ1, namely to

Wψ(φ1)P (π1(f)φ1) = Wψ(π(s)φ1)P (π1(f)φ1).

Since P (π1(f)φ1) = P (φ2) 6= 0, as φ2 is a cyclic vector, we conclude that Wψ(π(s)φ1)

104 0 is equal to Wψ(φ1) for all s ∈ Stab (ψ), in fact for any smooth cusp form φ1, as M(A) required.

Geometric conditions of Theorem 40

Let us verify that 40 applies with the group G = GSp(2) of Section 2.2. We need to determine a set of representatives {δ} for N(F )\G(F )/H(F ) such that ψT is 1 on

−1 N(A) ∩ δH(A)δ . By Proposition 6(b) such δ can be of the form m or mγ1, m ∈ M.

Singular case Consider ﬁrst δ = m ∈ M. Then N(A) ∩ δH(A)δ−1 consists of

−1 IX 0 y δnδ , n = ( 0 I ) ∈ H(A), thus X = z 0 . Replacing εT by a conjugate if necessary

0 1 2 we may assume, as in Section 2.3, that εT = ( θ 0 ) , θ ∈ F − F , or that εT =

u 0 × ( 0 v ) , u, v ∈ F , u − v 6= 0. Since

2 2 t 0 1 d b c d 0 y cd−abθ d −θb 0 y a b w Uw ( −θ 0 ) UX = ( c a ) −θa −θb z 0 = c2−θa2 cd−θab z 0 (U = ( c d ))

2 2 2 2 −1 2 2 has trace (c −θa )y +(d −θb )z, we have ψT (N(A)∩δH(A)δ ) = 1 only if c = θa and d2 = θb2, contradicting the assumption that θ ∈ F is not a square.

u 0 When T = ( 0 −v ) we need the coeﬃcients of z and y in

h i d b ua ub 0 y uad−vbc (u−v)bd 0 y tr ( c a ) −vc −vd z 0 = tr (u−v)ac ubc−vad z 0 = z(u − v)bd + y(u − v)ac

a 0 0 b to vanish. Since u 6= v, we have bd = 0 = ac, and so U = ( 0 d ) or U = ( c 0 ). Since Stab0 (ψ ) consists of g = diag(α, β, α, β) (with λ = αβ), we have that M(A) T δ−1Stab0 (ψ )δ ⊂ H( ), which is the geometric requirement of the Theorem. M(A) T A

105 Regular case Next we consider the δ of the form mγ, m ∈ M. As noted in Propo-

t −ω 0 sition 6, the image γJΘ γ of γ under the map G/H → X is ( 0 ω ). The geometric requirement of the Theorem, that Stab0 (ψ) lies in δH( )δ−1, follows from M(A) A

1 1 u ω−1 0 t t U 0 −ω 0 tU 0 λ mγJΘ γ m = t −1 ( ) = , 0 λw U w 0 ω 0 λwU −1w 0 λ ω λ λ u where u = det U, since for any T, Stab0 (ψ) consists of m = diag(U, λwtU −1w) M(A) with λ = u.

We shall consider next a similar example, where again G = GSp(2), and H is the

τ 0 0 1 2 centralizer of Θ = ( 0 τ ) ,ττ = ( τ 0 ), in G, where τ ∈ F − F . To verify the geometric condition of the theorem, once again, we need to produce a set

−1 of representatives {δ} for N(F )\G(F )/H(F ) such that ψT is 1 on N(A)∩δH(A)τ δ .

By Proposition 6(c), such δ can be of the form m or mγ0, m = m(U) is the element diag(U, λwtU −1w) ∈ M(F ).

Singular case Suppose ﬁrst that δ = m ∈ M(F ). Then N(A) ∩ δH(A)δ−1 consists

−1 IX x y of δnδ , n = ( 0 I ) ∈ H(A), thus X = ( τy x ). As above we may assume that

0 1 2 u 0 × εT = ( θ 0 ) , θ ∈ F − F , or that εT = ( 0 v ) , u 6= v ∈ F ; moreover, we take θ = τ if

×2 a b θ ∈ τF . Since (for U = ( c d ))

t 0 1 2 2 2 2 tr[w Uw ( −θ 0 ) UX] = 2(cd − abθ)x + [(c − θa ) + τ(d − θb )]y,

−1 ×2 ψT (N(A) ∩ δH(A)δ ) can be 1 (for all x, y) only if θ/τ ∈ F , in which case we take θ = τ, and then a = ±d and c = ±τb, namely δ = m(U) or m(εU), where m(U) ∈ Stab0 (ψ ), and so δ−1Stab0 (ψ )δ ⊂ H( ), which is the geometric M(A) T M(A) T A requirement of the Theorem.

106 u 0 t u 0 When εT = ( 0 v ) we have tr(w Uw ( 0 v ) UX) = (u − v)[x(ad + bc) + y(ac + bdτ)],

−1 2 2 and it is clear that ψT (N(A) ∩ δH(A)δ ) = 1 implies that c = τd , a contradiction.

Regular case It remains to consider the δ of the form mγ0, m = m(U) ∈ M(F ).

t ω 0 As λ(γ0) = 1, the image γ0ΘJ γ0 of γ0 in G/H → Xτ is ( 0 −τω ), and once again, since for every T the group Stab0 (ψ ) consists of m = diag(U, λwtU −1w) with M(A) T −1 λ = det U, it is contained in δH(A)τ δ .

Remark 6. According to an unpublished theorem of J. Bernstein, any irreducible admissible SO(n, Fv)-module πv has at most one − up to a scalar − O(n − 1,Fv)- invariant linear form on its space. In our case n = 5 and Gv = PGSp(2,Fv) '

SO(5,Fv), and Hv = O(4,Fv). Denote by Pv such an invariant form on πv, if it exists. Moreover, if πv is unramiﬁed we normalize Pv to take the value 1 at the

0 chosen Kv-invariant vector ξv . According to a theorem of Novodvorski and Piatetski-Shapiro [NPS], an irreducible

Gv-module πv has at most one − up to a scalar − linear form Wψv which transforms

0 via Wψv (πv(sn)w) = ψv(n)Wψv (w) for all n ∈ Nv, s ∈ StabMv (ψv), and w ∈ πv.

If π = ⊗πv is a cyclic cuspidal G(A)-module, let {ξv} denote an orthonormal basis

0 of πv including the chosen Kv-ﬁxed vector ξv if πv is unramiﬁed, and consider the distribution

X ∗ fv 7→ (Wψv Pv)πv (fv) = Wψv (ξv)Pv(ψv(fv )ξv). ξv

It is independent of the choice of the basis {ξv}, and for a set of inequivalent πv,

P ∗ these distributions are linearly independent. If (WψP )π(f) = φ Wψ(φ)P (π(f )φ)

107 then there is a constant c(π) such that for all f = ⊗fv we have a factorization

Y (WψP )π(f) = c(π) (Wψv Pv)πv (fv). v

108 CHAPTER 3

A TRACE FORMULA FOR THE SYMMETRIC SPACE

PGL(N)/ GL(N − 1)

3.1 Notations

In this section we set the notations and recall the deﬁnitions we need (for a general connected reductive quasisplit group G). Let B be a minimal parabolic subgroup of G over F . Fix a Levi subgroup MB of B. Denote by P any standard parabolic subgroup of G over F . Let MP be its unique Levi subgroup containing MB and NP the unipotent radical.

Let TP be the maximal split torus in the center of MP . Denote by X∗(TP ) =

∗ Hom(Gm,TP ) the group of co-characters of TP , and by X (TP ) = Hom(TP , Gm) the group of characters. Set

eaP = X∗(TP ) ⊗ R and d(P ) = dim(eaP ).

For any two parabolic subgroups Q ⊂ P , there are canonical injection eaP ,→ eaQ and surjection eaQ eaP , hence a canonical decomposition

P eaQ = aQ ⊕ eaP .

109 Since eaG is a summand of eaP for any P , set aP = eaP /eaG . Then we have the decomposition

P aQ = aQ ⊕ aP .

(T ) endomorphisms g → g. Hence they are simultaneously diagonalizable. Set gα =

∗ (T ) {X ∈ g|Ad(t)(X) = α(t)X} for α ∈ X (TB). If the space gα is not zero, α is called a weight. Hence we have

(T ) (T ) g = g0 ⊕ ⊕α∈Φgα .

Only ﬁnitely many nonzero α appear. These α are called roots of G with respect to

TB. The ﬁnite set of roots is denoted by Φ.

∗ There is a natural pairing h, i : X (TB) × X∗(TB) → Z, deﬁned as follows: For

∨ For each root α ∈ Φ the coroot α ∈ X∗(TB) is the unique homomorphism GL(1) →

∨ TB satisfying hα, α i = 2. A choice of a minimal parabolic subgroup determines an ordering of the set of roots

Φ. A root α is called simple if α > 0 and it cannot be written as α = β1 + β2 with βi both positive.

∗ Let ∆ and ∆b be the subsets of simple roots and simple weights in aG, respectively.

110 ∨ ∨ We write ∆ for the basis of aG dual to ∆,b and ∆b for the basis of aG dual to ∆.

Thus, ∆∨ is the set of coroots and ∆b ∨ is the set of coweights.

For every P , let ∆P ⊂ aG be the set of nontrivial restrictions of elements of ∆ to

∨ ∨ aP . For each α ∈ ∆P , let α be the projection of β to aP , where β is the root in

∨ ∨ ∆ whose restriction to aP is α. Set ∆P = {α : α ∈ ∆P }. Then we may also deﬁne

∨ their dual bases. Namely, we denote the dual basis of ∆P by ∆b P and the dual basis

∨ of ∆P by ∆b P .

P If Q ⊂ P , we write ∆Q to denote the subset of α ∈ ∆Q appearing in the action of

TQ on the unipotent radical of Q ∩ MP . Then aP is the subspace of aQ annihilated

P P ∨ ∨ P ∨ P P by ∆Q. Let (∆Q) = {α : α ∈ ∆Q}. We then deﬁne (∆b )Q and ∆b Q to be the basis

P P ∨ dual to ∆Q and (∆Q) , respectively.

P P Given two parabolic subgroups Q ⊂ P denote by τQ and τbQ respectively the characteristic functions of the sets

P {H ∈ aB : hα, Hi > 0 for all α ∈ ∆Q} and

P {H ∈ aB : h$, Hi > 0 for all $ ∈ ∆b Q}.

There exists (see [MW], I.1.4) a maximal compact subgroup K of G(A), which will be ﬁxed throughout the paper, satisfying the following conditions.

1. G(A) = B(A)K;

2. K∩M(A) is a maximal compact group of M(A) for all Levi subgroups M ⊂ G;

111 3. P (A) ∩ K = (MP (A) ∩ K)(NP (A) ∩ K).

For each parabolic subgroup P there is a Harish-Chandra map

HP : G(A) → aP .

Deﬁned by:

hχ,HP (m)i ∗ 1. |χ|(m) = e for all m ∈ M(A) and χ ∈ X (MP );

2. HP (nmk) = HP (m), n ∈ N(A), m ∈ M(A), k ∈ K.

1 For a group G, denote by G(A) the kernel of the Harish-Chandra map HG.

For a parabolic subgroup with Levi decomposition P = MN, denote by ρP the unique

∗ 2hρP ,HP (m)i element in aP satisfying e = | AdN (m)|. Here AdN (m) is the adjoint action of m on the Lie algebra of N.

× r Let F∞ denote F ⊗Q R. There is an isomorphism (F∞) ' TP (F∞) (see [MW], I.1.11 and I.1.13, and also [JLR], III.4). Let AP for P 6= G and AG be the intersections of the

× r 1 image of (R+) in TP (F∞) with G(A) and Z(A), respectively. The Harish-Chandra map HP induces an isomorphism of AP onto aP and we have the decomposition

1 X M(A) = AG × AP × (M(A) ∩ G(A) ). For X ∈ aP , write e for the unique element

X ∗ λ hλ,HP (p)i in AP such that HP (e ) = X. For λ ∈ aP , write e for the character p 7→ e of P (A). For a suitable normalization of the Haar measures, we have

Z Z Z Z Z f(x)dx = f(namk)e−2hρP ,HP (a)idndadmdk. 1 1 G(A) N(A) AP M(A)∩G(A) K

112 Let Ω = NG(MB)/MB be the Weyl group of MB in G, where NG(MB) denotes the normalizer of MB in G. For a Levi subgroup M, let ΩM = NM (MB)/MB be the Weyl group of M. For two parabolic subgroups P and Q with Levi factors MP and MQ respectively, let Ω(P,Q) be the set of elements ω ∈ Ω of minimal length in their class

−1 ωΩMP , such that ωMP ω = MQ. The minimal length condition is equivalent to the

P Q condition ω∆G = ∆G. Let z be the center of the universal enveloping algebra of the complexified Lie algebra Q of G∞ = v G(Fv). Here the product is over the archimedean places v of F . Then a function φ(g) on G(A) is called z-finite if there is an ideal I in z of finite codimension such that I · φ = 0.

Let K be a maximal compact subgroup of G(A) as above. The function φ is called K-ﬁnite if the span of the functions g 7→ φ(gk)(∀k ∈ K) is ﬁnite dimensional.

The function φ(g) is called smooth if the following condition is satisﬁed. For g = g∞gf , with g∞ ∈ G∞ and gf ∈ G(Af ), there exist a neighborhood V∞ of g∞ in G∞ and

∞ a neighborhood Vf of gf in G(Af ), and a C -function φ∞ : V∞ → C, such that

φ(g∞gf ) = φ∞(g∞) for all g∞ ∈ V∞ and gf ∈ Vf . In the function ﬁeld case φ is called smooth if it is locally constant.

Fix an embedding i : G,→ SL2n. For g ∈ G(A), write i(g) = (gkl)k,l=1,...,2n. Set

Y ||g|| = sup{|gkl|v; k, l = 1,..., 2n} v where the product is over all places v of F . Then a function φ is called of moderate

N growth if there are C,N ∈ R>0 such that for all g ∈ G(A) we have |φ(g)| ≤ C||g|| .

113 The deﬁnition of moderate growth does not depend on the choice of the embedding i. As in [MW], I.2.17 we make:

Deﬁnition 10. A function

φ : G(F )\G(A) → C is called an automorphic form if

1. φ is smooth and of moderate growth,

2. φ is right K-ﬁnite,

3. φ is Z-ﬁnite.

The space of automorphic forms is denoted by A(G). For a parabolic subgroup

P = NM the space of automorphic forms of level P , denoted by AP (G), is the space of smooth right K-ﬁnite functions

(P ) φ : N(A)M(F )\G(A) → C such that for every k ∈ K the function m 7→ φ(P )(mk) is an automorphic form of

(P ) M(A). For φ ∈ AP (G) and a parabolic subgroup Q ⊂ P with unipotent radical

NQ, the constant term is deﬁned by: Z (P ) (P ) φQ (g) = φ (ng)dn. NQ(F )\NQ(A)

(G) When P = G we will omit the superscript and write φQ instead of φQ . Note that the deﬁnition of the constant term applies also for any locally integrable function on

N(A)M(F )\G(A), not only for automorphic forms. A function will be called cuspidal

114 if its constant term is zero for all proper parabolic subgroups. A representation of

G(A) will be called cuspidal if its representation space consists of cuspidal functions.

3.2 Truncation

From here until section 7, G will be the group GL(n) and S will be the subgroup

h 0 GL(n − 1) × GL(1) embedded in G via (h, a) 7→ ( 0 a ) for h ∈ GL(n − 1) and a ∈ GL(1). Let B be the Borel subgroup of G consisting of upper triangular matrices and B = M0N0 its usual Levi decomposition. A standard parabolic subgroup P of G is determined by a partition (n1, n2, . . . , nr) with n1 + n2 + ··· + nr = n. For a given parabolic subgroup P of G we will denote by PS the standard parabolic subgroup P ∩ S of S.

1 For T ∈ a0 and a smooth function φ : G(F )\G(A) → C, deﬁne

T X d(P )−d(G) X (Λmφ)(h) = (−1) φP (δh)τbP (HP (δh) − T ). P ⊂G δ∈PS (F )\S(F )

The subscript m stands for “mixed”, as we take parabolic subgroups P of G and matrices δ in S(F ); we do not work with only G or only S.

Similarly, for a smooth function φ : P (F )\G(A)1 → C deﬁne

T,P X d(R)−d(P ) X P (Λm φ)(h) = (−1) φR(δh)τbR (HP (δh) − T ). R⊂P δ∈RS (F )\PS (F )

115 The following inversion formula holds, see (19) in [JLR] page 189,

X X T,P φ(h) = (Λm φ)(δh)τP (HP (δh) − T ). (3.2.1)

P ⊂G δ∈PS (F )\S(F )

We will call T suﬃciently regular if for all α ∈ ∆0 we have α(T ) >> 0.

T,P Proposition 41. For T suﬃciently regular the function h 7→ (Λm φ)(h) is rapidly

1 decreasing on MS(F )\MS(A) ∩ G(A) , where MS is the standard Levi subgroup of

PS = P ∩ S.

Proof. It is enough to consider the case of MS = S. The proof for the a Levi subgroups

1 + is the same. Suppose ω is a compact subset of N(A)(M(A) ∩ G(A) ) and T0 ∈ −a .

P Then for any parabolic subgroup P of S denote by S (T0, ω) the set of pak, with

0 P p ∈ ω, a ∈ A(R) , k ∈ K, such that α(HB(a) − T0) is positive for every α ∈ ∆B.

P P Let S (T0, T, ω) be the set of all x in S (T0, ω) such that $(HB(x) − T ) ≤ 0 for all

P 1 $ ∈ ∆b B. Recall the following deﬁnitions from [A2], section 6. Let FS (x, T ) be the

P characteristic function of the set of all x ∈ S(A) such that δx belongs to S (T0, T, ω) for some δ ∈ P (F ). For a pair of parabolic subgroups P1 ⊂ P2 of S deﬁne

2 X d(P3)−d(P2) P3 σ (X) = (−1) τ (X)τP (X). 1 P1 b 3 {P3:P3⊃P2}

For parabolic subgroups P1 ⊂ P2, set

X d(P )−d(S) φ1,2(h) = (−1) φP (h).

{P1⊂P ⊂P2} With these notations, as in [A3], section 1, we have

T,P X X 1 2 (Λm φ)(h) = FS (δh, T )σ1(HB(δh) − T )φ1,2(δh).

{P1,P2:P ⊂P1⊂P2} δ∈P1(F )\S(F )

116 The argument in [A3], pp. 96-97, applied to the group S, gives that there exist constants C1 and N1 > 0, such that

X 1 2 N1 FS (δh, T )σ1(HB(δh) − T ) ≤ C1||h||

δ∈P1(F )\S(F ) for all h ∈ S(A). From [A3], pp. 92-96, for the group G we have that for any N > 0

−N 1 there is a C > 0 such that |φ1,2(δg)| ≤ C||h|| for all δ ∈ G(F ) and g ∈ G(A) with

1 2 FS (δg, T )σ1(HB(δg) − T ) = 1. This completes the proof.

Deﬁnition of R #(following [JLR], pp. 178-185). Let V be a real vector space of ﬁnite dimension n. Let V ∗ be the space of complex valued linear forms on V and

V ∗ the space of real valued linear forms. An exponential polynomial function on V , R expoly for short, is a function f = f({λi,Pi}) of the form

X hλi,xi f(x) = e Pi(x), 1≤i≤r

∗ where λi are diﬀerent elements of V and Pi(x) are nonzero polynomials. By a cone in V we mean a closed subset of the form

C = {x ∈ V : hµi, xi ≥ 0}, where {µ } is a basis of V ∗. Let {e } be a basis of V dual to {µ }. i R j i The integral Z −hλ,xi IC (λ; f) := f(x)e dx, C with the expoly f = f({λi,Pi}), converges absolutely for λ in the open set

∗ U = {λ ∈ V : Rehλi − λ, eji < 0 for all 1 ≤ i ≤ r, 1 ≤ j ≤ n}.

117 Fix an index k. Let Ck be the intersection of C and the hyperplane Vk = {x ∈ V : hµk, xi = 0}. In the integral we can write x = hµk, xiek + y with y ∈ Vk and then m X X 1 Z ∂ I (λ; f) = ehλ−λi,yi P (y)dy. C hλ − λ , e im+1 ∂y i 1≤i≤r m≥0 i k Ck k

This formula gives the analytic continuation of IC (λ; f) to the tube domain

∗ Uk = {λ ∈ V : Rehλi − λ, eki < 0 for all 1 ≤ i ≤ r}.

The analytic continuation is a meromorphic function with hyperplane singularities, the singular hyperplanes being:

∗ Hk,i = {λ ∈ V : hλ, eki = hλi, eki for all 1 ≤ i ≤ r}.

∗ Therefore the function IC (λ; f) has analytic continuation to V by Hartogs’ lemma (see [JLR], pp. 180). Lemma 3 of [JLR] asserts:

Lemma 42. The function IC (λ; f) is holomorphic at λ = 0 if and only if hλi, eki= 6 0 for every pair (i, k).

∗ Denote the characteristic function of C by χC . Let T ∈ V . For λ ∈ V such that

Rehλi − λ, eji < 0 for all i and j set Z −hλ,xi fb(λ; C,T ) = f(x)χC (x − T )e dx V

Z X −hλ−λi,xi = Pi(x)χC (x − T )e dx 1≤i≤r V

X hλ−λi,T i hλi,xi = e IC (λ; Pi(x + T )e ). 1≤i≤r

118 The integrals are absolutely convergent and fb(λ; C,T ) extends to a meromorphic function on V ∗.

Deﬁnition 11. The function f(x)χC (x − T ) is called #integrable if fb(λ; C,T ) is regular at λ = 0. In this case, we deﬁne the #integral

Z # f(x)χC (x − T )dx (3.2.2) V to be the value fb(0; C,T ).

From [MW], I.3.2, any automorphic form φ ∈ AP (G) has the decomposition

l X hλi+ρP ,HP (a)i φ(namk) = αi(HP (a))ψi(mk)e . (3.2.3) i=1

1 Here n ∈ N(A), a ∈ AP , m ∈ M(F )\M(A) ∩ G(A) and k ∈ K. The αi(X)

∗ are polynomials, ψi ∈ AP (G) and λi ∈ aP are distinct. The decomposition is not unique. The set {λi} is uniquely determined by φ. For a function φ ∈ AP (G) with a decomposition as above denote the set {λi} by EP (φ). Let φ ∈ A(G) be an automorphic form on G and φP ∈ AP (G) its constant term along P . From (3.2.3) the constant term φP has a decomposition

l X φP (namk) = αi(HP (a))ψi(amk) (3.2.4) i=1

1 for n ∈ N(A), a ∈ AP , m ∈ M(A) ∩ G(A) and k ∈ K. Here αi are polynomials, and

ψi ∈ AP (G) are automorphic forms of level P , which satisfy

hλi+ρP ,HP (a)i ψi(ag) = e ψi(g)

∗ for some λi = λi(ψi) ∈ aP , for all a ∈ AP .

119 K R Deﬁnition 12. Write ψ (m) for K ψ(mk)dk. For an automorphic form φ ∈ A(G) deﬁne

Z # T,P Λm (φP ) (h)τP (HP (h) − T )dh 1 b PS (F )\S(A)∩G(A) to be

l Z Z # X T,P K hλi,Xi Λ (ψi ) (m)dm αi(X) · e · τP (X − T )dX. 1 b i=1 MS (F )\MS (A)∩G(A) aP By (3.2.4) this integral is equal to

Z # Z Z T,P X −hρP ,Xi Λ (φP ) (e mk)dmdk e · τP (X − T )dX. 1 b aP K MS (F )\MS (A)∩G(A) Here ΛT,P is Arthur’s truncation for the group M, where M is the standard Levi of

T,P P . The functions m 7→ (Λ ψi)(mk) are rapidly decreasing. Hence the functions

1 ψi(mk) are absolutely integrable over MS(F )\(MS(A) ∩ G(A) ) × K. Therefore the integral in the deﬁnition is convergent if and only if the integral R # is. As shown aP ∨ ∨ ∨ in [JLR], page 198, this is the case if and only if hλi, $ i= 6 0 for all $ ∈ ∆b P and

λi ∈ EP (φP ). A similar deﬁnition holds with τcP replaced by τP throughout.

Deﬁnition 13. Recall that S = GL(n − 1) × GL(1). For an automorphic form

φ ∈ A(G), deﬁne the regularized integral

Z ∗ φ(h)dh 1 S(F )\S(A)∩G(A) to be Z # X T,P (Λm (φP ))(h)τP (HP (h) − T )dh. 1 b P ⊂G PS (F )\S(A)∩G(A)

120 The integral is convergent for φ ∈ A(G) such that for all proper parabolic subgroups

∨ ∨ ∨ P we have that hλ, $ i= 6 0 for all $ ∈ ∆b P and λ ∈ EP (φP ) (see [JLR], page 198). It will also be denoted by ΠG/S(φ), when a reference to the pair (G, S) is needed.

Let P = MN be a standard parabolic subgroup of G corresponding to the par- Qr tition (n1, n2, . . . , nr). Let M be the group i=1 GL(ni). Let MS be the group Qr−1 i=1 GL(ni) × Snr . Embed Snr = GL(nr − 1) × GL(1) in GL(nr) via the map

x 0 (x, a) 7→ ( 0 a ). Then MS is naturally a subgroup of M. We note that the regularized integral deﬁned above can be deﬁned similarly for the pair of groups (M,MS) and will be denoted by ΠM/MS (·).

For φ ∈ AP (G) with decomposition

l X φ(namk) = αi(HP (a))ψi(amk) i=1 deﬁne the regularized integral

Z ∗ φ(h)τP (HP (h) − T )dh 1 b PS (F )\S(A)∩G(A) to be l Z # X M/MS K hλi,Xi Π ψi (m) αi(X)e τbP (X − T )dX. i=1 aP

K R M/MS K Here ψi (m) denotes K ψi(mk)dk and Π (ψi ) is the regularized integral of Def- inition 4 for the pair of groups (M,MS). The integral is well deﬁned if and only if the following conditions are satisﬁed (see

[JLR], page 198):

∨ ∨ ∨ P 1. hµ, $ i= 6 0 for all Q ⊂ P , $ ∈ (∆b )Q and µ ∈ E(φP );

121 ∨ 2. hλ, α i= 6 0 for all α ∈ ∆P , and λ ∈ E(φP ).

Theorem 43. Let φ ∈ A(G) be such that the conditions (1) and (2) are satisﬁed.

T 1 Then the integral of (Λmφ)(h) over S(F )\S(A) ∩ G(A) is equal to Z ∗ X d(P )−d(G) (−1) φP (h)τP (HP (h) − T )dh. 1 b P ⊂G PS (F )\S(A)∩G(A) To prove this we ﬁrst prove the following lemma.

Lemma 44. Let P be a proper parabolic subgroup of G. Then

Z # T,P (Λm φP )(h)τP (HP (h) − T )dh (3.2.5) 1 PS (F )\S(A)∩G(A) is equal to

Z ∗ X d(R)−d(P ) P (−1) φR(h)τR (HR(h) − T )τP (HP (h) − T )dh. (3.2.6) 1 b R⊂P RS (F )\S(A)∩G(A)

Proof. As in (3.2.4) write the constant term φP as

l X hλi+ρP ,HP (a)i φP (mnak) = αi(HP (a))e ψi(mk). i=1

1 Here n ∈ N(A), a ∈ AP , m ∈ M(A) ∩ G(A) and k ∈ K, αi are polynomials and

∗ λi ∈ aP are exponents. Then by Deﬁnition 12 with τbP replaced by τP , the integral (3.2.5) is equal to

l Z Z # X T,P K hλi,Xi (Λm ψi )(m)dm αi(X)e τP (X − T )dX . 1 i=1 M(F )\M(A)∩G(A) aP Using induction on the rank of M we can write it as the sum over i (1 ≤ i ≤ `) and over the parabolic subgroups R ⊂ P , of the product of (−1)d(R)−d(P ),

Z ∗ (ψ )K (m)τ P (H (m) − T )dm (3.2.7) i RM R R 1 b RM (F )\M(A)∩G(A)

122 where RM = R ∩ M, and

Z # hλi,Xi αi(X)e τP (X − T )dX. aP

Decompose the constant term (ψi)RM as

s X P hλi,j +ρR,HR(a)i (ψi)RM (mnak) = αi,j(HR(a))e ψi,j(mk), j=1

1 1 where n ∈ (NR ∩ M)(A), a ∈ AR ∩ M(A) ∩ G(A) , m ∈ MR(A) ∩ G(A) and k ∈ K.

P ∗ Here αi,j are polynomials and λi,j ∈ (aR) are exponents. Then we can write (3.2.7) as s Z ∗ Z # X K hλi.j ,Xi P ψi,j(m)dm αi,j(X)e τR (X − T )dX. 1 P b j=1 MR(F )\MR(A)∩G(A) aR R # R # Write µi,j for λi + λi,j. Combining the integrals P and to a single integral over aR aP aR, we have that

Z ∗ Z # (ψ )K (m)τ P (H (m) − T )dm α (X)ehλi,Xiτ (X − T )dx i RM R R i P 1 b RM (F )\M(A)∩G(A) aP is equal to the sum over j(1 ≤ j ≤ s) of

Z ∗ Z # K hµi,j ,Xi P ψi,j(m)dm αi,j(X)αi(X)e τR (X − T )τP (X − T )dX. 1 b MR(F )\MR(A)∩G(A) aR Since the regularized integrals are convergent we can sum over j inside the integral sign of the last formula. Using the decomposition formula (3.2.4) for (ψi)R, the last expression is equal to

Z ∗ P αi(HR(g)) · (ψi)R(g) · τR (HR(g) − T ) · τP (HP (g) − T )dg. 1 b RS (F )\S(A)∩G(A)

123 Summing over i, and using the decomposition formula (3.2.4) for the constant term

φR, this is equal to Z ∗ P φR(g)τR (HR(g) − T )τP (HP (g) − T )dg. 1 b RS (F )\S(A)∩G(A) Therefore (3.2.5) is equal to (3.2.6), as required.

Using Lemma 44 we now prove Theorem 43.

Proof. Summing both sides of the equality in Lemma 44 over P , and using Deﬁnition

4, we have that Z ∗ Z T φ(h)dh − (Λmφ)(h)dh 1 1 S(F )\S(A)∩G(A) S(F )\S(A)∩G(A) is equal to Z ∗ X d(R)−d(P ) P (−1) φR(h)τR (HR(h) − T )τP (HP (h) − T )dh. (3.2.8) 1 b RS (F )\S(A)∩G(A) R⊂P $G For R 6= G, Langlands’ Combinatorial Lemma (see [A1], section 2) gives

X d(R)−d(P ) P (−1) τbR (HR(h) − T )τP (HP (h) − T ) {P : R⊂P $G}

d(R)−d(P ) = −(−1) τbR(HR(h) − T ) and so the theorem will follow if we check that the summation can be taken inside the integral in (3.2.8).

Consider the three cones

CbR = {X ∈ aR : τbR(X) = 1} ,

CP = {X ∈ aP : τP (X) = 1} ,

P P P CbR = X ∈ aR : τbR (X) = 1 .

124 P The product CP ×CbR is contained in CbR. Indeed, CbR is the positive span of the coroots

∨ ∨ P {α : α ∈ ∆R}, CP is the positive span of the coweights in ∆b P and CbR is the positive

∨ P span of the coroots {α : α ∈ ∆R}, so the assertion follows from the fact that all

∨ ∨ coweights in ∆b P are non-negative linear combinations of coroots in ∆P . Now let

P λ ∈ E(φR) and for P containing R, write λ = λR + λP relative to the decomposition

P ∨ ∨ ∨ aR = aR ⊗ aP . By our hypothesis, hλP , $ i= 6 0 for all $ ∈ ∆b P . Hence λP is

P ∨ ∨ P nondegenerate with respect to CP . Similarly, hλR, α i= 6 0 for all α ∈ ∆R. Hence

P P P λR is nondegenerate with respect to CbR . Since CP × CbR ⊂ CbR for all P , we may and do apply Lemma 6 of [JLR] to conclude that for any polynomial α(X),

Z # d(R)−d(P ) hλ,Xi −(−1) α(X)e τbR(X − T )dX aR is equal to

Z # X d(R)−d(P ) hλ,Xi P (−1) α(X)e τbR (X − T )τP (X − T )dX. aR {P : R⊂P $G}

This implies that (3.2.8) equals

Z ∗ X d(R)−d(G) − (−1) φR(h)τR(SR(h) − T )dh, 1 b RS (F )\S(A)∩G(A) R$G and this proves the theorem.

Remark 7. This proof follows closely the proof of Theorem 10 in [JLR].

125 3.3 The Main Distribution

1 Recall from section 2 that G(A) is the kernel of the Harish-Chandra map HG, and

∞ 1 the group N is the unipotent group deﬁned in the introduction. For f ∈ Cc (G(A) ), let

X −1 Kf (x, y) = f(x γy) γ∈G(F ) be the kernel of the convolution operator r(f), where r is the right regular representation of G(A)1 on L2(G(F )\G(A)1). A choice of a Tamagawa measure is implicit. We want to ﬁnd the spectral expansion of the integral Z Z J(f) = Kf (h, n)ψ(n)dndh. (3.3.1) 1 N(F )\N(A) S(F )\S(A)∩G(A)

We recall the spectral formula for the kernel Kf (x, y) from [A2]. Consider pairs of the form χ = (M, π), where M is a standard Levi subgroup of G and π a discrete spectrum representation of M(A) ∩ G(A)1. Such a pair will be called a discrete pair. Two such pairs (M, π) and (M 0, π0) are called equivalent if there is an element ω in the Weyl group Ω such that M = ω−1M 0ω and πω is unitarily equivalent to π0. Here

πω is deﬁned by πω(m) = π(ω−1mω).

The spectral formula for the automorphic kernel is

X Kf (x, y) = Kf,χ(x, y). χ∈X Here the sum over χ = (M, π) ranges over the set X of equivalence classes of discrete pairs, and Kf,χ(x, y) is given by the formula

X Z X n(P )−1 E(x, I(f, λ)φ, λ)E(y, φ, λ)dλ. ia∗ (M,π) P φ∈BP (π)

126 Here BP (π) is an orthonormal basis of the (normalizedly) induced representation

I(π). The constant n(P ) is the number of chambers in aP , where P is any parabolic subgroup with Levi M. The E(g, φ, λ) are Eisenstein series; see the next section.

Set Z Z Jχ(f) = Kf,χ(h, n)ψ(n)dndh. 1 N(F )\N(A) S(F )\S(A)∩G(A) Then (3.3.1) becomes X J(f) = Jχ(f). (3.3.2) χ ∗ The test function f will be assumed to be the convolution of two functions f = f1 ∗f2 ,

∞ 1 ∗ for fi = ⊗vfiv ∈ Cc (G(A) ). Here we denote by f (g) the function given by g 7→

−1 f(g ). Then the formula for the kernel Kf (x, y) becomes Z X −1 X n(P ) E(x, I(f1, π)φ, λ)E(y, I(f2, λ)φ, π, λ)dλ. ia∗ (M,π) P φ∈BP (π) Recall the following majorization ([A2], Lemma 4.4), whose notations we use.

Proposition 45. For any X,Y in the universal enveloping algebra of G∞, and for any left and right K-ﬁnite function f, there is a continuous seminorm ν on the space of smooth compactly supported functions on G∞, and an integer N, such that Z X −1 X n(P ) R(X)E(x, I(f, λ)φ, λ)R(Y ∗)E(y, φ, λ) dλ ia∗ (M,π) P φ∈BP (π) is bounded by

N N ν(X ∗ f∞ ∗ Y )||x|| ||y|| .

It follows that if y is in a compact set the function x 7→ Kf (x, y) is of moderate growth

T T and therefore the truncated function (Λ1,mKf )(x, y) is rapidly decreasing. Here Λ1,m is truncation with respect to the ﬁrst variable.

127 Lemma 46. Let C be a compact subset of G(A). Then for T suﬃciently regular (thus

α(T ) >> 0 for all α ∈ ∆), y ∈ C, x ∈ S(A) and any χ,

T T (Λ1,mKf )(x, y) = Kf (x, y), (Λ1,mKf,χ)(x, y) = Kf,χ(x, y).

Since N(F )\N(A) is compact, from this lemma it follows that Jχ(f) is equal to Z Z T (Λ1,mKf,χ)(h, n)ψ(n)dndh. 1 S(F )\S(A)∩G(A) N(F )\N(A) With the notations as above set Z Wψ(λ, φ) = E(n, φ, λ)ψ(n)dn. N(F )\N(A) Then we have the following:

∗ ∗ −1 ∞ Proposition 47. If f = f1 ∗ f2 , where f2 (g) = f2(g ), and fi = fi,∞fi , then the

−1 distribution Jχ(f) equals the sum over (M, π) of n(P ) times Z Z X T (ΛmE)(h, I(f1, λ)φ, λ)dh · Wψ(λ, I(f2, λ)φ)dλ. ∗ 1 iaP φ S(F )\S(A)∩G(A)

3.4 Regularized Periods of Eisenstein series

In this section we compute the regularized period of a cuspidal Eisenstein series for

G = GL(n) relative to S = GL(n − 1) × GL(1).

Let P = MN be a proper parabolic subgroup and σ a cuspidal representation of

2 1 ∗ L (M(F )\M(A)∩G(A) ). For φ ∈ AP (G)σ and λ ∈ aP deﬁne the cuspidal Eisenstein series by X E(g, φ, λ) = φ(γg)ehλ+ρP ,HP (γg)i. (3.4.1) γ∈P (F )\G(F )

128 Also for a pair of parabolic subgroups P ⊂ Q deﬁne the Eisenstein series induced from MP to MQ by

X EQ(g, φ, λ) = φ(γg)ehλ+ρP ,HP (γg)i. (3.4.2) γ∈P (F )\Q(F )

For a pair of associate parabolic subgroups P and Q deﬁne the standard intertwining operator by

Z −1 (M(s, λ)φ)(g) = e−hsλ+ρQ,HQ(g)i φ(s−1ng)ehsλ+ρP ,HP (s ng)idn. Ns(F )\NQ(A) Proposition 48. If E(g, φ, λ) is a cuspidal Eisenstein series then the period integral R ∗ 1 E(h, φ, λ)dh is a meromorphic function of λ with hyperplane singu- S(F )\S(A)∩G(A) larities.

Proof. Same as Proposition 12 in [JLR].

For a standard Levi subgroup M of G denote by Ω(M) the set of Weyl group elements

−1 s ∈ Ω such that M1 = sMs is a standard Levi subgroup and s is of minimal length in the class sΩM , where ΩM is the Weyl group of M. For a pair of standard Levi subgroups M and M1 set

−1 Ω(M,M1) = {s ∈ Ω(M): sMs = M1} and

ΩM = {s ∈ Ω: sMs−1 = M}.

Conjugation by the order two element γ = diag(1,..., 1, −1) is an involution of G deﬁned over F . We denote it by θ.

129 We recall the following definitions from [MW]. Let P(M,σ) be the space of vector valued holomorphic Paley-Wiener functions on (aG )∗. For any φ ∈ P define F M,C (M,σ) φ by Z Fφ(g) = (φ(λ)) (g)dλ. G ∗ i(aM ) Define the pseudo-Eisenstein series

X θφ(g) = Fφ(γg). γ∈P (F )\G(F )

By the proposition on page 95 of [MW], the series is absolutely convergent and rapidly decreasing. As in [MW], page 99, we have Z θφ(g) = E(g, φ(λ), λ)dλ. Reλ=λ0

Here λ0 is in the domain of convergence of the Eisenstein series.

Deﬁnition 14. For a standard parabolic subgroup P of G with Levi decomposition MN and a representative η of the double coset P (F )\G(F )/S(F ), deﬁne the subgroups of G

−1 −1 Sη = S ∩ ηP η ,Pη = ηSη ∩ P,Mη = M ∩ Pη,Nη = N ∩ Pη.

Following [JLR], page 211, we will call a representative η admissible if ηθ(η) lies in

M the Weyl group Ω . Let σ be a cuspidal representation of M(A) trivial on AP ,

φ ∈ AP (G)σ. For a double coset ω in P (F )\G(F )/S(F ) deﬁne ! X Z Z J(ω, φ, λ) = ehλ+ρP ,HP (ηh)i φ(mηh)dm dh. 1 η Sη(A)\S(A) Mη(F )\Mη(A)∩G(A) The summation ranges over the admissible representatives η of the double coset ω.

130 Proposition 49. For generic λ ∈ (aG )∗ we have the equality M,C Z ∗ E(h, φ, λ)dh = J(ω, φ, λ). (3.4.3) 1 S(F )\S(A)∩G(A) To prove this proposition, we require two lemmas ﬁrst.

Lemma 50. For λ0 in the domain of absolute convergence of J(ω, φ, λ) we have Z Z θφ(h)dh = J(ω, φ, λ)dλ. 1 G ∗ S(F )\S(A)∩G(A) λ0+i(aM ) Proof. Since the series X |Fφ(γg)| γ∈P (F )\G(F ) is rapidly decreasing, in particular it is integrable over S(F )\S(A) ∩ G(A)1, we can write Z X Z θφ(h)dh = Fφ(ηh)dh, 1 1 S(F )\S(A)∩G(A) {η} Sη(F )\S(A)∩G(A) where {η} is a set of coset representatives for P (F )\G(F )/S(F ). For each η, we can write the integral corresponding to η in the formula above as

ZZZ −hρP ,HP (p)i e Fφ(xpηh)dxdpdh. 1 Pη(F )\Pη(A)∩G(A)

1 1 Here the ﬁrst integral is over (Sη(A) ∩ S(A) ∩ G(A) )\(S(A) ∩ G(A) ) and the second

1 1 −1 over (Pη(A) ∩ G(A) )\(Pη(A) ∩ η(S(A) ∩ G(A) )η ). From the decomposition Pη =

1 1 MηNη, we have that Pη ∩G(A) = (Mη ∩G(A) )Nη. From the fact that Fφ is invariant under N(A), we can write the inner integral as an integral over Mη(F )\(Mη(A) ∩

1 G(A) ). Since Sη contains the center of S we can write the outer integral as an integral

131 1 1 −1 ∼ G over Sη(A)\S(A). Using (Mη(A) ∩ G(A) )\(Mη(A) ∩ η(S(A) ∩ G(A) )η ) = aM we

G can write the middle integral over aM . Thus the integral above is equal to Z Z Z −hρP ,ξi ξ e Fφ(e mηh)dξdmdh. 1 G Sη(A)\S(A) Mη(F )\Mη(A)∩G(A) aM Applying the Fourier inversion formula to the inner integral we get Z Z Z (φ(λ))(mηh)ehλ,SM (ηh)idλdmdh. 1 G ∗ Sη(A)\S(A) Mη(F )\Mη(A)∩G(A) λ0+i(aM ) For λ in the domain of absolute convergence of J(α, φ, λ) we can interchange the inner integral with the outer ones and sum over η to obtain that the above equals Z J(ω, φ, λ)dλ. G ∗ λ0+i(aM ) This completes the proof of the lemma.

Recall that P = MN is a standard parabolic subgroup of G, σ a cuspidal representation of M(A), E(g, φ, λ) the corresponding Eisenstein series and Q is a standard parabolic subgroup of G containing P . We write QS for Q ∩ S.

∨ Lemma 51. Suppose that φ(λ) vanishes at any λ with hsλ, $ i = 0 for all s ∈ MQ ΩM

and $ ∈ ∆b MQ for any Levi subgroups MQ of S, where Q ⊂ G contains P . Then for

G ∗ λ0 suﬃciently regular in the positive Weyl chamber in (aM ) , we have the equality

Z Z Z ∗ θφ(h)dh = E(h, φ(λ), λ)dh dλ. (3.4.4) 1 1 S(F )\S(A)∩G(A) Re λ=λ0 S(F )\S(A)∩G(A)

Proof. From the deﬁnition of θφ the left hand side is equal to Z Z E(h, φ(λ), λ)dλ dh. 1 S(F )\S(A)∩G(A) Reλ=λ0

132 From formula (3.2.1), we have

X X T,Q E(h, φ(λ), λ) = (Λm E)(δh, φ(λ), λ) · τQ(HQ(δh) − T ).

Q δ∈QS (F )\S(F ) Hence the integral above is equal to the sum over Q of Z Z X T,Q (Λm E)(δh, φ(λ), λ) · τQ(HQ(δh) − T )dλdh. 1 S(F )\S( )∩G( ) Reλ=λ0 A A δ∈QS (F )\S(F ) For a fixed h the sum over δ is finite. Hence we can interchange the summation and the inner integration and combine the summation with the outer integration to obtain Z Z X T,Q (Λm E)(h, φ(λ), λ) · τQ(HQ(h) − T )dλ dh. 1 Q QS (F )\S(A)∩G(A) Reλ=λ0 From the definition of the regularized integral R ∗, the right hand side of the equality in the lemma is equal to Z Z # X T,Q (Λm E)(h, φ(λ), λ) · τQ(HQ(h) − T )dh dλ. 1 Q Reλ=λ0 QS (F )\S(A)∩G(A)

For the constant term EQ of the Eisenstein series E(h, φ, λ) along Q we have

X Q EQ(h, φ, λ) = E (h, M(s, λ)φ, sλ), s where EQ(h, φ, λ) is deﬁned in (3.4.2). Hence for the truncated Eisenstein series we have

T,Q X T,Q Q (Λm EQ)(h, φ, λ) = (Λm E )(h, M(s, λ)φ, sλ). s Therefore it is enough to show that for each s we have the following equality Z Z T,Q (Λm E)(h, φ(λ), λ) · τQ(HQ(h) − T )dλ dh (3.4.5) 1 QS (F )\S(A)∩G(A) Reλ=λ0 Z Z # T,Q = (Λm E)(h, φ(λ), λ) · τQ(HQ(h) − T )dh dλ, (3.4.6) 1 Reλ=λ0 QS (F )\S(A)∩G(A)

133 where each side converges.

Using the deﬁnition of the R # we can write the right hand side as

Z Z Z # ! hsλ,Xi e τQ(X + HQ(k) − T )dX I(k, λ)dkdλ, (3.4.7) Reλ=λ0 K aQ where I(k, λ) is the integral

Z T,Q Q (Λm E )(mk, M(s, λ)φ(λ), sλ)dm. 1 MQ(F )\MQ(A)∩G(A) From [JLR], formula 15, we have

# h(sλ) ,T i Z e MQ ehsλ,Xiτ (X − T )dX = ν , Q Q Q hsλ, $∨i aQ $∈∆b Q which is bounded for Reλ = λ0. Hence (3.4.7) converges.

If λ1 is in the domain of absolute convergence of (3.4.7) we can shift the contour of integration of (3.4.7) to Reλ = λ1. We can do that since φ(λ) vanishes on the hyperplane singularities of R #. Thus we obtain the absolutely convergent integral

Z Z Z # ! hsλ,Xi e τQ(X + HQ(k) − T )dX I(k, λ)dkdλ. Reλ=λ1 K aQ

Therefore we can interchange the order of integration. Thus the last integral is equal to Z Z T,Q (Λm E)(h, φ(λ), λ)τQ(HQ(h) − T )dλ dh, 1 QS (F )\S(A)∩G(A) Reλ=λ0 which is (3.4.5) once we shift the contour of integration back to Reλ = λ0. This proves the lemma.

Now we prove Proposition 49.

134 Proof. From lemma 50 and 51 we have that

Z Z ∗ Z E(h, φ, λ)dh dλ = J(ω, φ, λ)dλ G ∗ 1 G ∗ λ0+i(aM ) S(F )\S(A)∩G(A) λ0+i(aM ) for any suﬃciently positive λ0 and for every φ satisfying the conditions of Lemma 51. Since this is true for all such functions φ, we conclude that we have equality of the integrands.

Let us now record the statement of Bernstein’s Principle from Lemma 16 of [JLR].

It will be used in the next section.

Lemma 52. (Bernstein’s principle) Let G be GL(m) and P = MN a proper parabolic

2 subgroup. Let σ be an irreducible cuspidal representation in L (M(F )\M(A)∩G(A)1).

Let E(g, φ, λ) be the corresponding Eisenstein series with φ ∈ AP (G(A))σ. Then

Z ∗ E(g, φ, λ)dg = 0 1 G(F )\G(A) for all λ for which E(g, φ, λ) and the integral are deﬁned.

Proof. Suppose that the Eisenstein series and the regularized integral are deﬁned at λ0. Then, by continuity, there exists an open set U containing λ0 such that the Eisenstein series and the regularized integral are deﬁned for all λ ∈ U. The map

R ∗ 1 G λ φ 7→ 1 E(g, φ, λ)dg deﬁnes a G( ) -invariant functional on Ind (σ ⊗ e ) G(F )\G(A) A P for λ ∈ U. Since there does not exist such a functional for generic values of λ, the R ∗ function 1 E(g, φ, λ)dg must vanish identically. G(F )\G(A)

135 3.5 Periods of Truncated Eisenstein Series

Let G = GL(n) and S = GL(n − 1) × GL(1) be as in the introduction. Let P = MN

∗ be a standard parabolic subgroup of G. For λ ∈ aP , a cuspidal representation σ of

1 M(A) ∩ G(A) and φ ∈ AP (G)σ, let E(g, φ, λ) be the corresponding Eisenstein series. Let P 0 be an associate of P . Let Q be a standard parabolic subgroup of G containing

0 −1 Q P . Let s be an element of Ω(MP ,MP 0 ) such that s α > 0 for all α ∈ ∆P 0 . Denote by G(P ) the set of pairs (Q, s).

Proposition 53. With the notations as above we have, Z T (ΛmE)(h, φ, λ)dh 1 S(F )\S(A)∩G(A) X eh(sλ)Q,T i = ν J((M(s, λ)φ)K, λ). Q Q ∨ h(sλ)Q, α i (Q,s)∈G(P ) α∈∆Q Proof. By Theorem 43 we have Z T (ΛmE)(h, φ, λ)dh 1 S(F )\S(A)∩G(A) Z ∗ X d(Q)−d(G) = (−1) EQ(g, φ, λ)τQ(HQ(g) − T )dg. 1 b Q QS (F )\S(A)∩G(A)

0 The constant term EQ(g, φ, λ) is zero unless Q contains an associate of P , say P . When this is the case the formula for the constant term is given by

X Q EQ(g, φ, λ) = E (g, M(s, λ)φ, sλ). s∈Ω(P,Q) Therefore the integral above is equal to

Z ∗ X d(Q)−d(G) X Q (−1) E (g, M(s, λ)φ, sλ)τQ(HQ(g) − T )dg. 1 b Q s∈Ω(P,Q) QS (F )\S(A)∩G(A)

136 For each Q and s, by the deﬁnition of the regularized integrals, we have that the integral Z ∗ Q E (g, M(s, λ)φ, sλ)τQ(HQ(g) − T )dg 1 b QS (F )\S(A)∩G(A) is equal to

Z ∗ Z # Q K Q h(sλ)Q,Xi E (g, (M(s, λ)φ) , (sλ)P 0 )dm · e τbQ(X − T )dX. 1 a MQS (F )\MQS (A)∩G(A) Q

Here MQ is the standard Levi of Q and MQS the standard Levi of QS.

Applying Lemma 52 and Proposition 49 to the factors of the group MQ we obtain that the ﬁrst integral in the product above is zero unless Q ∈ G(P ). When Q ∈ G(P ) the integral is equal to J((M(s, λ)φ)K, λ). Applying [JLR], formula 15 to the second integral in the product displayed above we have

Z # eh(sλ)Q,T i eh(sλ)Q,Xiτ (X − T )dX = ν , bQ Q Q h(sλ) , α∨i aQ α∈∆Q Q which completes the proof.

3.6 Limit Formula for J T (f)

In order to compute the contribution to the spectral side of the trace formula from the proper parabolic subgroups we recall, in the beginning of this section, the basic deﬁnitions and theorems about (G, M)-families as introduced by Arthur in [A4] (see also [G] and [Lap2]). These will be used in the computation of the limit formula for the distribution J T (f).

Let M be a ﬁxed Levi subgroup of G. Denote by P(M) the set of parabolic subgroups

137 ∗ of G for which M is a Levi component. In this section we denote by aM the vector

∗ ∗ space X (M) ⊗Z R. Here X (M) is the group of rational characters of M.

∗ Deﬁnition 15. A collection {cP (λ): P ∈ P(M)} of smooth functions cP (λ) on iaM is called a (G, M)-family if for any two elements P1 and P2 of P(M) contained in a

∗ given parabolic subgroup Q, and λ belonging to iaQ, we have

cP1 (λ) = cP2 (λ).

Let νM be the covolume of the lattice spanned by the coroots in aM . Deﬁne

−1 Y ∨ θP (λ) = νM λ(α ). α∈∆P

M1 For a parabolic subgroup Q, containing P , with Levi M1 denote by νM the covolume

M1 of the lattice spanned by coroots in aM . Deﬁne

−1 M1 Y ∨ θP (λ) = νM λ(α ). Q α∈∆P The following is Lemma 6.2 of [A4].

Lemma 54. For a (G, M)-family {cP (λ): P ∈ P(M)}, the function

X cP (λ) cM (λ) = , θP (λ) P ∈P(M)

∗ ∨ initially deﬁned away from the hyperplanes {λ ∈ iaM : hλ, α i = 0}, extends to a

∗ smooth function on iaM .

Denote by F(M) the set of parabolic subgroups of G whose Levi component contains

M. Fix Q in F(M). Let L be its Levi component. Denote by PL(M) the set

138 of parabolic subgroups of L whose Levi component is M. For each R ∈ PL(M), set Q(R) to be the unique group in P(M) which is contained in Q and such that

Q(R) ∩ L = R. Then the family of functions

Q L {cR(λ) = cQ(R)(λ): R ∈ P (M)}

Q is an (L, M)-family. For this (L, M)-family denote by cM (λ) the function

X Q Q −1 cR(λ)θR (λ) . R∈PL(M)

Suppose {cP (λ)} and {dP (λ)} are two (G, M)-families. Put eP (λ) = cP (λ)dP (λ).

Then {eP (λ)} is obviously a (G, M)-family. For a pair of parabolic subgroups Q ⊂ R, deﬁne

R R −1 Y ∨ θbQ(λ) = (ˆνM ) λ($ ). R $∈∆Q

R R ∨ R Hereν ˆM is the covolume of the lattice in aP spanned by the elements α for α ∈ ∆P .

∨ R The vector $ in aQ is deﬁned by

∨ ∨ R α($ ) = $(α ), α ∈ ∆Q.

Deﬁne the functions

0 dQ(λ),Q ⊃ P by

0 X d(Q)−d(P ) R −1 −1 dQ(λ) = (−1) θbQ(λ) dR(λ)θR(λ) . {R:R⊃P } The following is Lemma 6.3 of [A4].

139 Proposition 55. With the notations as above we have

X Q 0 eM (λ) = cM (λ)dQ(λ). Q∈F(M)

Let P = MN be a parabolic subgroup of G, π a cuspidal representation of M(A) ∩

G(A)1 and χ = (M, π) the corresponding cuspidal datum. Assume that the test

∞ 1 function f ∈ Cc (G(A) ) is the convolution f1 ∗ f2 of two K-ﬁnite functions and that each of the functions fi is also the convolution of two K-ﬁnite functions, say

1 2 T fi = fi ∗ fi . In this section we compute the limit as T goes to inﬁnity of Jχ (f) and thus obtain a formula for Jχ(f). Combining the results from Proposition 47 and Proposition 53 we can write the

T Q,T distribution Jχ (f) as the sum over Q ∈ F(M) of Jχ (f), which is

1 Z X eh(sλ)Q,T i ν J((M(s, λ)I(f , λ)φ)K, λ) · W (λ, I(f , λ)φ). Q Q ∨ 1 ψ 2 n(P ) ∗ h(sλ)Q, α i iaP φ α∈∆Q Set

2 0 Bφ,φ0 (λ) = hI(f1 , λ)φ, φ i,

1 0 Aφ,φ0 (λ) = J M(s, λ)I(f1 , λ)φ , λ · Wψ(λ, I(f2, λ)φ).

T Then for T >> 0 the distribution Jχ (f) is equal to

Z h(sλ)Q,T i −1 X X e n(P ) ν A 0 (λ)B 0 (λ)dλ. Q Q ∨ φ,φ φ,φ ia∗ h(sλ)Q, α i φ,φ0 P {(Q,s)∈G(P )} α∈∆Q

Q We proceed to compute the limit formula for Jχ (f). Let MQ be the standard Levi of Q

0 and P(MQ) the set of parabolic subgroups whose Levi is MQ. For Q ∈ P(MQ) there

0 T h(sλ)Q,T i exists an s ∈ Ω such that sQ = Q. Deﬁne cQ0 (λ) = e and dQ0 (λ) = Aφ,φ0 (λ).

140 T Lemma 56. The families {cQ0 (λ)} and {dQ0 (λ)} are (G, MQ)-families.

Proof. One has to show that if s is a simple reﬂection and s1, s2 are related by ss1 = s2 and if λ is such that s1λ = s2λ, then

T T c −1 (λ) = c −1 (λ) and ds−1Q(λ) = ds−1Q(λ). s1 Q s2 Q 1 2

Since M(s2, λ) = M(s, s1λ)M(s1, λ) it is enough to show the relations

T T cs−1Q(λ) = cQ(λ) and ds−1Q(λ) = dQ(λ) whenever sλ = λ. For the ﬁrst family this is obvious. For the second, it follows from the fact that the values of the intertwining operators at λ and sλ depend only on the

∗ projections of λ and sλ in iaM (see (1.3) in [A5], page 1293).

Recall that Q is a parabolic subgroup with Levi containing the Levi of P . With these notations we can write   Z Q,T −1 X X T −1 Jχ (f) = n(P ) νQ  cQ0 (λ)dQ0 (λ)θQ0 (λ)  b(λ)dλ. ∗ 0 ia 0 φ,φ P Q ∈P(MQ)

−1 Q ∨ Here θQ0 (λ) = ν λ(α ) and we have set b(λ) = Bφ,φ0 (λ) to simplify nota- Q α∈∆Q0 tions. Using the splitting formula of [A4], Lemma 6.3, or Proposition 55 above, we have   Z Q,T −1 X X Q0,T 0 J (f) = n(P ) ν c (λ)d 0 (λ) b(λ)dλ. χ Q  MQ Q  ∗ 0 ia 0 φ,φ P Q ∈F(MQ)

Q0,T 0 We recall the deﬁnitions of c (λ) and d 0 (λ) from [A4]. Let L be the standard Levi MQ Q 0 L of Q . Denote by P (MQ) the set of parabolic subgroups of L whose Levi is MQ. For

L Q0,T T 0 R ∈ P (MQ) deﬁne cR (λ) = cQ0(R)(λ), where Q (R) is the unique group in P(MQ)

141 0 0 Q0 Q0 −1 Q ∨ contained in Q such that Q (R) ∩ L = R. Set θR (λ) = (νR ) Q0 hλ, α i. Then α∈∆R we have 0 X 0 0 cQ ,T (λ) = cQ ,T (λ)θQ (λ)−1. (3.6.1) MQ R R R∈PL(M)

0 S S −1 Q ∨ For two parabolic subgroups S ⊃ Q set θbQ0 (λ) = (νQ0 ) ω∈∆S hλ, ω i. Then b b Q0

0 X d(Q0)−d(S) S −1 dQ0 (λ) = (−1) θbQ0 (λ)dS(λ)θS(λ) . {S:S⊃Q0}

Fix a Q1 ∈ F(MQ). Consider the integral Z cQ1,T (λ)d0 (λ)b(λ)dλ. (3.6.2) MQ Q1 ∗ iaP

∗ ∗ Q ∗ Using the decomposition aP = aQ ⊕ (aP ) , write λ = (λ1, λ2), and write the integral (3.6.2) as a double integral Z Z ! cQ1,T (λ )d0 (λ , λ )b(λ , λ )dλ dλ . MQ 1 Q1 1 2 1 2 1 2 Q ∗ ∗ i(aP ) iaQ

Note that this integral depends on T through the function cQ1,T (λ ). MQ 1

Proposition 57. For Q1 ∈ F(MQ), the limit Z lim cQ1,T (λ )d0 (λ , λ )b(λ , λ )dλ MQ 1 Q1 1 2 1 2 1 T →∞ ∗ iaQ

0 is zero unless Q1 = G. If Q1 = G the limit is equal to dG(0, λ2)b(0, λ2).

Proof. Suppose that Q = G. Following [A7], denote by YG (T ) the convex hull of 1 MQ −1 the set {s T for all s ∈ Ω(aG, aMQ )}. It forms a positive (MQG,MQ)-orthogonal set in a in the sense of [A7]. Here M = G. Let ψG,T (x) be the characteristic MQ G MQ function of the set YG (T ). Then by Lemma 17.2 in [A7] the function cG,T (λ ) is the MQ MQ 1

142 Fourier transform of ψG,T (x). Therefore, the limit of the proposition can be written MQ as Z lim cG,T (λ )d0 (λ , λ )b(λ , λ )dλ MQ 1 G 1 2 1 2 1 T →∞ ∗ iaQ Z = lim ψ[G,T (λ )(d0 b)(λ , λ )dλ . MQ 1 G 1 2 1 T →∞ ∗ iaQ Since taking Fourier transform is an isometry we can write the last integral as

Z [ lim ψ[G,T (λ )([d0 b)(λ , λ )dλ MQ 1 G 1 2 1 T →∞ ∗ iaQ Z = lim ψG,T (λ ) d0 b (λ , λ )dλ = d0 (0, λ )b(0, λ ), MQ 1 dG 1 2 1 G 2 2 T →∞ ∗ iaQ by Fourier inversion.

Now suppose that Q1 ∈ F(MQ) is a proper subgroup of G, with Levi M1 ⊃ MQ. The functions cQ1,T (λ ) form an (M ,M )-family having hyperplane singularities along MQ 1 1 Q

∨ Q1 the hyperplanes hλ, ω i = 0 in aMQ for ω ∈ ∆b R . These singularities come from pairs R and R0, in the sum deﬁning cQ1,T , with ω and ω0 such that ω0 = −ω. Consider MQ a∗ a∗ aQ1 ∗ such a pair. Decompose the space i Q as the direct sum i Q1 ⊕ i( Q ) . Further a∗ ∗ ∗ ∗ ∨ decompose i Q1 as i(VR) ⊕ i(Rω) , where i(VR) is the hyperplane orthogonal to ω a∗ a in i Q1 . Similarly, for the dual spaces we have that i Q1 decomposes as iVR ⊕ iRω. a∗ a∗ ∗ ∗ Write λ1 ∈ i Q1 as (x, y) according to the decomposition i Q1 = i(VR) ⊕ i(Rω) . Let

T1 be the restriction of T to iaQ1 and write T1 as (t, t0) according to the decomposition iaQ1 = iVR ⊕ iRω. Then it is enough to show that the limit Z lim cQ1,T1 (λ )d0 (λ , λ )b(λ , λ )dλ MQ 1 Q1 1 2 1 2 1 T1→∞ ia∗ Q1 is zero.

143 From equation (3.6.1), the deﬁnitions preceding it, and the decompositions above,

Q1,T1 Q1,(t,t0) M the function c (λ ) = c (x, y) is equal to the sum over R ∈ P Q1 (M ) of MQ 1 MQ Q terms of the form

A(x, t, ω∨) · u(x, y)e−hy,t0i, (3.6.3) where ehx,ti e−hx,ti A(x, t, ω∨) = − hx, ω∨i hx, ω∨i and u(x, y) is the smooth function

Q1 −1 Y ∨ −1 (νR ) h(x, y), α i . α6=ω

Therefore, for each of the terms (3.6.3), we have to evaluate the limit

Z Z ∨ 0 −hy,t0i lim A(x, t, ω ) · u(x, y)dQ1 (x, y)b(x, y)dx e dy. (3.6.4) T1→∞ ∗ ∗ i(Rω) i(VR)

The inner integral is a function of y and t, and is independent of t0 . The outer integral is the integral of that function against the character e−hy,t0i, thus it can be viewed as a Fourier transform. The function A(x, t, ω∨) is the Fourier transform of the characteristic function of an interval (depending on t) on the line along ω. Denote that function by χt. Then the inner integral above is equal to Z Z χ (x) u(x, y)d0 (x, y)b(x, y) dx = χ (x) u(x, y)d0 (x, y)b(x, y)bdx, bt Q1 t Q1

∗ the integration is over i(VR) . For the L1-norm of the last integral, viewed as a function of y, we have

Z Z 0 b 0 b χt(x) u(x, y)dQ1 (x, y)b(x, y) dx ≤ u(x, y)dQ1 (x, y)b(x, y) dx , 1 1

144 ∗ here the integrals are over i(VR) . Thus the inner integral in (3.6.4) represents an L1 function of y depending on t, whose

L1-norms are bounded by a quantity independent of t. Therefore the integral over i(Rω)∗ in (3.6.4) above is the Fourier transform of the function, represented by the inner integral, evaluated at t0. Hence, by the Riemann-Lebesgue lemma, the integral

∗ over i(Rω) in the limit above tends to zero as T1(and hence t0) tends to inﬁnity. We obtain that the limit in the proposition is zero if Q1 is a proper subgroup of G.

Q ∗ Q Corrolary 58. Write µ for the projection of λ to (iaP ) . Then the distribution Jχ (f)

−1 is equal to n(P ) νQ times Z X 1 0 2 0 J(I(f1 , µ)φ ) · Wψ(µ, I(f2, µ)φ)hI(f1 , µ)φ, φ idµ Q ∗ φ,φ0 (iaP ) X Z = J(I(f1, µ)φ) · Wψ(µ, I(f2, µ)φ)dµ. Q ∗ φ (iaP ) Summarizing the results of this section we have the following:

Theorem 59. Let χ be a cuspidal datum represented by the pair (M, σ), where M is the standard Levi subgroup of a standard parabolic subgroup P . Then Z −1 X X Jχ(f) = n(P ) νQ J(I(f1, µ)φ) · Wψ(µ, I(f2, µ)φ)dµ. Q ∗ Q⊃P φ (iaP ) Summing over χ gives the ﬁne χ-expansion of the Fourier summation formula. In

[F5], Proposition 1, the geometric side is completely developed, also the spectral side in the case n = 3 and in the case n > 3 the distributions Jχ(f) are computed for special cases of χ. Thus this completes the computation of the Fourier summation formula for the symmetric space PGL(n)/GL(n − 1) initiated in [F5].

145 3.7 An application

In the section, we change our setting from that of S(A) ∩ G(A)1 inside G(A)1 (where G = GL(n) and S = GL(n − 1) × GL(1)), which is used by Arthur, to that of

S = GL(n − 1) in G = PGL(n), which is more suitable for our applications. There is no analytic difference between these two different settings. Let us first recall the expression for the geometric side of the summation formula from [F5].

Proposition 60. Put G = PGL(n), S = GL(n − 1) embedded in the top-left corner of G. Let N and ψ be as in the introduction. Then for n > 3 we have,

Z Z Kf (n, h)ψ(n)dndh N(F )\N(A) S(F )\S(A) X = Ψ(b; f; ψ). b∈F × Here Z Z −1 Ψ(b; f; ψ) = f(ngbh)ψ(n) dndh, −1 N(A)/N(A)∩gbS(A)gb S(A) Q and gb = diag(1,..., 1, b) ∈ G(F ). If f = ⊗fv we have Ψ(b; f; ψ) = v Ψ(b; fv; ψv) with Z Z −1 Ψ(b; fv; ψv) = fv(ngbh)ψv(n) dndh. −1 N(Fv)/N(Fv)∩gbS(Fv)gb S(Fv) The geometric side will be compared to the geometric side of a trace formula for the group PGL(2). This expression can be found in [F4]. We record it next.

Let G0 be the group PGL(2) as an algebraic group deﬁned over F , B0 the upper

146 triangular Borel subgroup. Let N 0 be the unipotent radical of B0. Similarly to the case of G, deﬁne for a test function f 0 the kernel function

X 0 −1 Kf 0 (x, y) = f (x γy). γ∈G0(F )

Proposition 61. The integral Z Z 0 −1 Kf 0 (x, y)ψ (y) dxdy 0 0 0 0 N (F )\N (A) N (F )\N (A) is equal to

X Ψ0(b; f 0; ψ0). b∈F × Here Z Z 0 0 0 0 −1 0 0 −1 Ψ (b; f ; ψ ) = f (x ωgby)ψ (y) dxdy, 0 0 N (A) N (A) 0 1 b 0 1 0 0 0 0 0 Q 0 0 0 and gb = ( 0 1 ) and ω = ( −1 0 ). If f = ⊗fv then Ψ (b; f ; ψ ) = v Ψ (b; fv; ψv) with Z Z 0 0 0 0 −1 0 0 −1 Ψ (b; fv; ψv) = fv(x ωgby)ψv(y) dxdy. 0 0 N (Fv) N (Fv)

∞ 0 Deﬁnition 16. The test function fv ∈ Cc (G(Fv)) and the test function fv ∈

∞ 0 × Cc (G (Fv)) are called matching if for all b ∈ Fv we have

−(n−2)/2 0 0 0 Ψ(b; fv; ψv) = |b|v Ψ (b; fv; ψv).

∞ 0 Conjecture 62. For every function fv ∈ Cc (G(Fv)) there exists a function fv ∈

∞ 0 0 0 Cc (G (Fv)), and for every fv there exists an fv, such that fv and fv are matching.

Let K0 a maximal compact subgroup of G0(A) as in section 2. For a ﬁnite set of places 0 0V 0 0V Q 0 0 Q 0 V write K = K K V , where K = v6∈V Kv and KV = v∈V Kv. Similarly ﬁx a

147 V V Q Q maximal compact subgroup K = K KV , with K = v6∈V Kv and KV = v∈V Kv, of G(A), where G is the group GL(n).

For complex numbers ξ1v and ξ2v, let Ξ be the character of the diagonal subgroup of

0 ξ1v ξ2v 0 G (Fv) given by Ξ(diag(a, b)) = |a|v |b|v . Extend Ξ to the Borel subgroup B (Fv)

0 0 of G (Fv) by letting Ξ be trivial on the unipotent radical of B (Fv). We will denote

0 G (Fv) by π 0 (ξ , ξ ) the irreducible subquotient of the induced representation I 0 (Ξ) G 1v 2v B (Fv)

(normalized induction). We similarly introduce πG(ξ1v, . . . , ξnv) for G.

0 0 Let us consider the case of an unramiﬁed local representation πv of G (Fv), which is the unramiﬁed constituent of the induced representation πG0 (ξ1v, ξ2v). It has pa-

0 ξ1v ξ2v rameter t(πv) = diag(|qv| , |qv| ). Let πv be the unramiﬁed local representation of

n−3 n−5 n−3 − ξ1v ξ2v GLn(Fv) with parameter diag(|qv| 2 , |qv| 2 ,..., |qv| 2 , |qv| , |qv| ). When the

0 0 representations πv and πv are related as above we will write πv 7→ πv.

0 0 0 Spherical functions fv in Hv, the Hecke algebra of Kv-biinvariant functions in the

∞ 0 0 0 space Cc (G (Fv)), and fv in Hv, are called corresponding if tr πv(fv) = tr πv(fv) for

0 00 0 0 all πv 7→ πv. The unit elements fv of Hv and fv of Hv are corresponding. The

00 0 Theorem of [F3] shows that fv and fv are matching.

0 Conjecture 63. Corresponding spherical functions fv and fv are matching.

Remark 8. Conjectures 62 and 63 are stated in [MR].

Consider the distribution

Z Z 0 0 0 −1 f 7→ J (f ) = Kf 0 (n1, n2)ψ(n1n2 )dn1dn2. 0 0 0 0 N (F )\N (A) N (F )\N (A) The equality of the geometric expressions for the distributions J(f) and J 0(f 0) for

148 matching test functions f and f 0 implies that the spectral expressions of these distributions are equal. Thus we have the following.

0 0 0 Proposition 64. Let f = ⊗fv and f = ⊗fv be test functions and suppose fv and fv are matching for all v. Then

J 0(f 0) = J(f).

Note that at each v where fv is spherical, by Conjecture 63, we may choose a cor-

0 responding fv . Using this, and Proposition 64, applying the standard generalized linear independence of characters arguments of section 11 of [L2] (see also Theorem

2 in [FK]) we conclude that the equality of the spectral sides of the two summation formulas implies an equality of the discretely occurring terms. Therefore we have the following.

Theorem 65. Fix a ﬁnite set V of F -places including the archimedean places. At

0 0 each v 6∈ V ﬁx an unramiﬁed G (Fv)-module πv = πG0 (ξ1v, ξ2v). Let πv be the unrami-

0 n−3 n−5 n−3 ﬁed constituent πv (ξ1v, ξ2v) of πG( 2 , 2 ,..., − 2 , ξ1v, ξ2v). Then for any matching 0 test functions fv and fv we have that

X X 0 0V 0 0 Wψ(π (f )φ )Wψ(φ ) 0 π φ0∈π0K0V is equal to the sum of

X X V J(φ)Wψ(π(f )φ) π φ∈πKV and

X νP X J(I(f V , 0)φ) · W (0,I(f V , 0)φ). n(P ) 1 ψ 2 P,π φ

149 The sum over π0 is over the cuspidal representations of PGL(2, A) which have a

0V 0 nonzero vector ﬁxed by K , such that for all v 6∈ V , πv is the unramiﬁed constituent

0 0 πv (ξ1v, ξ2v) of πG0 (ξ1v, ξ2v). The sum over φ ranges over an orthonormal basis of smooth vectors in the space π0K0V of K0V -ﬁxed vectors in π0.

The sums over π are over all automorphic representations of PGL(n, A) with πv =

0 πv (ξ1v, ξ2v) for all v 6∈ V (in particular the ﬁrst sum over π is empty). The sum over φ ranges over an orthonormal basis of smooth vectors in the space πKV of KV -ﬁxed vectors in π. The sum over P is over the standard proper parabolic subgroups of G.

The main representation theoretic application of this trace identity is the following.

Theorem 66. Let π0 be a cuspidal automorphic representation of PGL(2, A). Then

0 π = I(n−2,2)(1n−2 × π ) is an automorphic representation of PGL(n, A) which is S- cyclic and has a nonzero Fourier coeﬃcient Wψ(φ) for some φ ∈ π. Conversely every S-cyclic automorphic representation of PGL(n, A), which has a nonzero Fourier coeﬃcient Wψ(φ) for some φ ∈ π, has this form.

We note that Theorem 65 and Theorem 66 depend on Conjecture 62 and Conjecture

63.

0 Proof. Fix a set V of F -places containing the archimedean ones, such that πv is

0 0 unramiﬁed for v 6∈ V . For any matching test functions fv on Gv and fv on Gv the

0 identity from Theorem 65 holds. Namely it holds where the ﬁxed unramiﬁed Gv-

0 0 0 modules at all v 6∈ V are the components πv of π . In this case π parametrizes the

0 only term in the PGL(2)-side of the equality of Theorem 65. Note that tr πG0 (ξ1, ξ2; fv)

0 is equal to tr πv (ξ1v, ξ2v; fv).

150 Since π0 is cuspidal, it is generic. Hence there is a vector φ0 ∈ π0K0V such that the

0 0 value Wψ0 (φ ) of the Whittaker function of φ at the identity is not zero. Assuming Conjecture 62, that matching functions exist, we can use an arbitrary test function f 0V . Since the operators π0(f 0V ) span the endomorphism algebra of π0K0V , we may choose f 0V to have the property that π0(f 0V ) acts as 0 on each vector in π0 which is orthogonal to φ0, but ﬁxes φ0. With this choice of f 0V the left side of the identity of

Theorem 65 reduces to a single term, which is

0 0 Wψ0 (φ )Wψ0 (φ ).

This is nonzero, hence so is the PGL(2)-side, hence the PGL(n)-side of the identity is nonzero. Hence there exists an automorphic representation π of PGL(n, A), with

J(φ) 6= 0 and Wψ(φ1) 6= 0 for some φ1, φ2 in π.

0 This π has the property that tr πv(fv) = tr I(n−2,2)(1n−2 × πv; fv) for all v 6∈ V , namely the Gv-module πv is the unramiﬁed irreducible constituent of the represen-

0 0 tation I(n−2,2)(1n−2 × πv) of Gv, which is irreducible. Hence π = I(n−2,2)(1n−2 × π ), by rigidity theorem for GL(n), and the description of automorphic representations in

[L1].

Conversely, suppose that π is an automorphic representation of PGL(n) with J(φ1) 6=

0 and Wψ(φ2) 6= 0 for some φ1, φ2 in π. Then ﬁx a set V of F -places containing the archimedean ones, such that πv is unramiﬁed for v 6∈ V . For any matching test

0 0 V functions fv on Gv and fv on Gv the identity from Proposition 65 holds. Choose f to have the property that π(f V ) acts as 0 on each vector in π which is orthogonal to

φ1, but maps φ1 to φ2. Then the PGL(n)-side of the identity of Proposition 65 is a

151 single nonzero term, therefore the PGL(2)-side of the identity is nonzero, thus there

0 0 exists a π . As above from Theorem 65 it follows that π = I(n−2,2)(1n−2 × π ).

Recall the following results (Proposition 0 and Proposition 0.1 of [F5]).

Proposition 67. Let Fv be a local ﬁeld and πv a unitarizable irreducible admissible

representation of GL(n, Fv), such that HomGL(n−1,Fv)(πv, θ) 6= 0, where θ is a unitary

× character of Fv . Then πv = θ or there is an irreducible admissible unitarizable representation σv of GL(2,Fv) such that πv = I(n−2,2)(θ × σv).

Proposition 68. An irreducible representation πv of the group GL(n, Fv) of the form

± 1 I(n−1,1)(θν 2 × µ) or I(n−2,2)(θ × θσv), where µ is a character and σv is inﬁnite di-

n−2 ± 2 mensional or the character ν , has HomGL(n−1,Fv)(πv, θ) 6= 0. If dim σv = 1 and

n−2 ± 2 σv 6= ν , then HomGL(n−1,Fv)(πv, θ) = 0, where πv = I(n−2,2)(θ × θσv).

A multiplicity one theorem of A. Aizenbud ´etal. [AGRS] asserts that the dimension over C of HomSv (πv, τv) is less or equal than 1 for any irreducible representation τv of Sv = GL(n − 1,Fv) when v is nonarchimedean (this updates the last sentence on p. 39 of [F2]). When v is archimedean and τv = 1, the multiplicity one theorem is proven by A. Aizenbud, D. Gourevitch and E. Sayag [AGS]. This result with τv = 1, and Propositions 8.8 and 8.9 here for θ = 1, imply that there is a unique up to a scalar GL(n − 1,Fv)-invariant form on πv = I(n−2,2)(1n−2 × σv), provided that σv is

± n−2 inﬁnite dimensional or is the character ν 2 , and there is no invariant form if σv is any other one dimensional character.

Let π = ⊗πv be an S-cyclic automorphic representation of GL(n, A). For all nonarchimedean places v denote by Pv a GL(n−1,Fv)-invariant form on πv. For each v for

152 0 which πv is unramiﬁed ﬁx a nonzero Kv-invariant vector ξv . We normalize the form

0 Pv to take value 1 on ξv . Let {ξv} denote an orthonormal basis of πv, which includes

0 the chosen Kv-ﬁxed vector ξv if πv is unramiﬁed. Consider the distribution

X fv 7→ Lπv (fv) = Pv(πv(fv)ξv)Wψv (ξv).

ξv

It is independent of the choice of the basis {ξv}, and for a set of inequivalent representations πv, these distributions are linearly independent. Consider the global distribution occurring in the trace formula of Theorem 59:

X f 7→ Lπ(f) = J(π(f)φ)Wψ(φ). φ

From the uniqueness of the local forms it follows that there exists a nonzero constant c(π), such that for all f = ⊗fv we have the factorization

Y Lπ(f) = c(π) Lπv (fv). v all places

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