A Survey on the Arthur-Selberg Trace Formula ∗

Takuya KONNO †

Contents

1 Introduction 2

2 Spectral decomposition 4 2.1Siegeldomains...... 5 2.2Cuspforms...... 6 2.3Decompositionbycuspidaldata...... 7 2.4CuspidalEisensteinseries...... 7 2.5 Residual Eisenstein series and the spectral decomposition ...... 9 2.6Spectralkernel...... 10

3 Convergence 12 3.1Truncatedkernel...... 12 3.2Truncationoperatorandthebasicidentity...... 15

4 The ﬁne O-expansion 17 4.1 J T (f)asapolynomial...... 17 4.2ReductionbytheJordandecomposition...... 20 4.3Weightedorbitalintegrals...... 22 4.4 The ﬁne O-expansion...... 25

5Fineχ-expansion 26 5.1AnapplicationofthePaley-Wienertheorem...... 27 5.2 Logarithmic derivatives ...... 31 5.3 Normalization and estimation of intertwining operators ...... 35 5.4Weightedcharacters...... 38

6The invariant trace formula 40 6.1Non-invariance...... 40 6.2ApplicationofthetracePaley-Wienertheorem...... 42 ∗Talk at the symposium “Automorphic Forms and Representations on Algebraic Groups and Auto- morphic L-functions, 29 June, 2000. †Graduate School of Math., Kyushu University 812-8581, Hakozaki 6-10-1, Higashi-ku, Fukuoka city, Japan

1 1 Introduction

In this note, we present a brief survey on the Arthur-Selberg trace formula. Interested readers can consult more detailed expositions [1], [28], [29], and of course, the original papers [2] to [13]. See also [20] for some important ideas and several appropriate arguments in reduction theory. For the purpose of this introduction, it is suﬃcient to recall the original Selberg trace formula and give some words about arithmetic backgrounds. The Selberg trace formula was originally proved for a pair (G, Γ) of a semisimple Lie group and a cocompact discrete subgroup in it [37], [38]. If we exclude some exceptional cases, this is equivalent to the setting of anisotropic ad´ele groups. A A ||

Thus let F be a number ﬁeld and write = F for its ring of ad´eles. A denotes the A× A1 ||

id´ele norm on and set := Ker A . For a connected semisimple group G over F ,its group of adelic points G(A) is a locally compact topological group, in which the group G(F )ofF -rational points is a discrete subgroup. We assume that G is anisotropic over F ,thatis,G(F ) contains only semisimple elements. Then G(F )\G(A)iscompact bythe result of [18]. R denotes the right regular representation of G(A)onthespaceL2(G(F )\G(A)). ∞ A A Write Cc (G( )) for the space of functions with compact supports on G( )whichis smooth in the archimedean components and locally constant in the non-archimedean ∈ ∞ A components. For f Cc (G( )), the operator

[R(f)φ](x):= f(y)φ(xy) dy = f(x−1y)φ(y) dy A G( A ) G( )

= f(x−1γy)φ(y) dy \ A G(F ) G( ) γ∈G(F ) is an integral operator with the kernel K(x, y):= f(x−1γy). γ∈G(F )

Since G(F )\G(A)iscompact,thisoperatorisofHilbert-Schmidt class (i.e. K(x, x) is square integrable on G(F )\G(A)). In particular one can show that the representation R decomposes into a direct sum of irreducible representations where each irreducible representation occurs with ﬁnite multiplicity: R = π⊕m(π) (1.1)

π∈Π(G( A ))

Here, Π(G(A)) denotes the set of isomorphism classes of irreducible unitary representations of G(A) . Moreover an argument of Duﬂo-Labesse [21, I.1.11] shows that R(f)isof

2 trace class. That is, it admits a trace given by the integral of K(x, y) on the diagonal:

trR(f)= K(x, x) dx = f(x−1δ−1γδx) dx

\ A \ A G(F ) G( ) ∈ γ \ G(F ) G( ) {γ}∈ Ç(G) δ G (F ) G(F ) 1 = f(x−1γx) dy dx

[Gγ(F ):G (F )]

\ A \ A

γ Gγ ( A ) G( ) Gγ (F ) Gγ ( ) { }∈ Ç γ (G) G = a (γ)IG(γ,f).

{γ}∈ Ç(G)

Here O(G) is the set of (semisimple) conjugacy classes in G(F ), Gγ := Cent(γ,G)isthe 0 centralizer of γ in G, Gγ := Cent(γ,G) is its identity component, and meas(G (F )\G (A)) aG(γ):= γ γ ,I(γ,f):= f(x−1γx) dx.

[Gγ(F ):G (F )] G \ A γ Gγ ( A ) G( ) This combined with (1.1) yields the Selberg trace formula: G G

a (γ)IG(γ,f)= a (π)IG(π,f), (1.2) ∈ A {γ}∈ Ç(G) π Π(G( ))

G where a (π):=m(π)andIG(π,f):=trπ(f) is the distribution character of π.Theleft and right sides are called the geometric and the spectral side, respectively. The Selberg trace formula has many variations reflecting its applications to wide variety of fields in mathematics. But the Arthur-Selberg trace formula is the only extension to the case when G has the F -rank greater than one. It is one of the important ingredients of the Langlands program. Perhaps it will be helpful to explain the program briefly. It is a program to understand the deep relationship between automorphic forms and Galois representations and motives. Two main processes are

(1) To describe the automorphic representations of reductive groups by means of their associated automorphic L-functions (or their Langlands parameters);

(2) To construct the correspondence between automorphic representations of arithmetic type and -adic representations of Galois groups of the ﬁeld of deﬁnition of such forms. This correspondence should be characterized by the equality between automorphic and Artin L-functions.

Both of them are very hard but will provide a lot of fruitful arithmetic informations. As for (1), the case of inner forms of GL(n) was successfully treated [25], [26], [35] and [15]. The relevant L-functions are the Rankin product L-functions. Then one expects to reduce the case of general reductive (or at least classical) groups to the GL(n)case.This divides into two steps. First we relate the automorphic representations of a reductive group G to those of its quasisplit inner form G∗. This is suggested by the experience in the GL(n) case [24], [15]. Then relate the automorphic representation of G∗ with those of some GL(n). In [14, § 8] the detailed framework of the second part for classical G are explained. We need to compare the trace formula for G with that of G∗ in the ﬁrst

3 problem, while a twisted trace formula for GL(n) need to be compared with the ordinary one for G∗ in the second. The problem (2) contains so many aspects that we cannot explain them in any detail here. But the starting point is to construct the Galois representations associated to an arithmetic automorphic representations in the -adic cohomology of certain Shimura variety. Here the trace formula with special geometric test functions will be compared with the Lefschetz-Verdier trace formula of the Shimura variety. For these purposes, the trace formula must be of the arithmetic form, i.e. must be stabilized. In fact, already in the spectral decomposition of the L2-automorphic spectrum, the normalization of intertwining operators by L-functions and the precise analytic properties of those L-functions must be established. The analytic trace formula alone yields little arithmetic information ! In this note, however, we deal only with the analytic aspects in the construction of the Arthur-Selberg trace formula. Some parts of the stabilization process will be found in the article of Hiraga in this volume. The contents of this note is as follows. We start with a brief review of Langlands’ theory of spectral decomposition of the automorphic spectrum by Eisenstein series § 2. In the higher rank case, the residual Eisenstein series appears which makes the construction in what follows much more complicated. Anyway this allows us to express the kernel of the right translation operator in two forms, one geometric and the other spectral. § 3 explains the construction of [2] and [3]. We define the truncated kernel and prove that its integral over the diagonal converges. We obtain the coarse trace formula. § 4 is devoted to the fine O-expansion. We write the geometric terms in terms of the unipotent terms of certain reductive subgroups of G. Then they are expressed by means of weighted orbital integrals. The spectral counter part, the fine X-expansion, is explained in § 5. This is the heart of this note, because it is the most important part in applications. We use the fact that the coarse X-expansion is a polynomial in the truncation parameter T to deduce the precise expression of the X-expansion from the asymptotic formula of the inner product of truncated Eisenstein series. Finally in § 6, we illustrate the rough idea of Arthur’s construction of the invariant trace formula [11], [12]. Because of the lack of time and volume, many important ideas are overlooked. In particular, we ignore many convergence/finiteness arguments, and also cannot refer to the works of Osborne-Warner [34]. Instead we add some examples in the simplest case G = GL(2) with some figures.

2 Spectral decomposition

We begin with some notation. For a finite set of places S of F , we write AS := {(av)v ∈ A | av =0, ∀v/∈ S}. The infinite and finite components of A are denoted by A∞ and Af , respectively. Let G be a connected reductive group over F . For brevity, we write G := G(A), GS := G(AS), etc. Let AG be the maximal F -split torus in the center Z(G)ofG, while the maximal R-vector subgroup in the center Z(G∞)ofG∞ is denoted by AG. Write aG 1 for its Lie algebra. We have a direct product decomposition G = G × AG such that

| | | | ∈ ∈ 1 A χ(ag) A = χ(a) ,aAG,g G

4 holds for any F -rational character of G.Set

1 proj log HG : G = G × AG −→ AG −→ aG.

For a parabolic subgroup P = MU, we write F(M), P(M)andL(M) for the set of parabolic subgroups containing M, the set of parabolic subgroups having M as a Levi factor and the set of Levi subgroups containing M. We ﬁx a minimal parabolic subgroup P0 = M0U0. F(P0)andL(P0) denotes the set of parabolic subgroups containing P0 and the set of Levi components of elements of F(P0) containing M0 (the set of standard parabolic and Levi subgroups). Fix a maximal compact subgroup K = v Kv of G such that the Iwasawa decomposition G = PK holds for any P ∈F(M0). Using this we extend the map HM : M → aM to HP : G = UMK umk −→ HM (m) ∈ aM .

G W = W denotes the (relative) Weyl group Norm(A0,G)/M0 of A0 = AM0 in G, where Norm(H, G) means the normalizer of a subgroup H in a group G.WeidentifyW with a ﬁxed system of representatives in Norm(A0,G). W acts by conjugation on F(M0) and the associated objects. We write the action w : P → w(P )=wP := wPw −1 etc.

2.1 Siegel domains

Fixing an invariant measure on G and a Lebesgue measure on AG, we can consider the space → C (i) φ(γag)=φ(g), ∀a ∈ A ,γ∈ G(F ) 2 \ φ : G G L (G(F )AG G):= | |2 ∞ measurable (ii) \ φ(g) dg < + G(F ) AG G

∞ as in the anisotropic case. Let Cc (G/AG) for the space of smooth functions on G which are compactly supported modulo AG. Similar calculation as in the anisotropic case shows that, the operator [R(f)φ](x):= f(y)φ(xy) dy \ AG G 2 on L (G(F )AG\G) is an integral operator with the kernel K(x, y):= f(x−1γy) (2.1) γ∈G(F ) ∈ ∞ H for f Cc (G/AG). Later we shall use the subspace (G/AG)ofelementswhichare ∞ K-ﬁnite on both sides in Cc (G/AG) as the space of test functions. The spectral decomposition of the right regular representation R = RG of G on 2 L (G(F )AG\G) is more complicated than in the anisotropic case, because G(F )AG\G = G(F )\G1 is no longer compact. The form of the non-compactness is described as follows.

5 Since M0 is anisotropic modulo center by deﬁnition, we can choose a compact subset 1 1 ω1 of M0 satisfying M0 = M0(F )ω1.AsU0 is a multiple extension of additive groups, there exists a compact subset ω2 ⊂ U0 such that U0 = U0(F )ω2.ForT ∈ a0,weset

A0(T ):={a ∈ A0 | α(H0(a) − T ) > 0, ∀α ∈ ∆},

wherewehaveabbreviatedAM0 , HP0 as A0, H0, respectively, and ∆ is the set of simple ∗ roots of A0 := AM0 in P0, considered as a subset of a0.

Proposition 2.1 ([23]). We can choose T0 ∈ a0 suﬃciently negative so that

G = G(F )S(T0), S(T0):=ω2ω1A0(T0)K

Thus the non-compactness comes from that of A0(T0), a shifted positive cone.

2.2 Cusp forms For an F -parabolic subgroup P = MU of G and a measurable function φ on U(F )\G, we can deﬁne its constant term along P by

φP (g):= φ(ug) du. U(F )\U

2 2 \ ∈ 2 \ The space of L -cusp forms L0(G(F )AG G) consists of φ L (G(F )AG G) such that φP vanishes almost everywhere for any P ∈F(P0). Of course this is not the space of cusp forms A0(G(F )AG\G) in the usual sense [33, Chapt. I], but A0(G(F )AG\G) is a dense 2 \ subspace of L0(G(F )AG G). The following lemma is fundamental in our estimation arguments.

Lemma 2.2. Suppose φ : G(F )\G → C is slowly increasing and suﬃciently smooth relative to dim U0. Then the alternating sum aG cφ(g):= (−1) M φP (g)

P =MU∈F(P0)

2 is rapidly decreasing. Moreover c extends to a projection on L (G(F )AG\G) whose re- 2 \ G striction to L0(G(F )AG G) is the identity. Here aM := dim aM /aG. The proof is a simple extension of the argument showing that the classical holomorphic cusp forms are rapidly decreasing. From this, we see that the kernel of the restriction 2 \ R0(f)ofR(f)toL0(G(F )AG G)iscyK(x, y)(cy means the operator c applied in y) which is rapidly decreasing. Hence R0(f) is of Hilbert-Schmidt class and R0 decomposes discretely. Moreover each irreducible component has ﬁnite multiplicity: ⊕ 2 \ mcusp(π) L0(G(F )AG G)= π . π∈Π(G)

6 2.3 Decomposition by cuspidal data ∈L 2 \ Apair(M, ρ)ofM (P0) and an irreducible component ρ of L0(M(F )AM M)is 2 \ called a cuspidal pair for G. We write L0(M(F )AM M)ρ for the ρ-isotypic subspace 2 \ A \ A \ in L0(M(F )AM M)and 0(M(F )AM M)ρ for its intersection with 0(M(F )AM M). This is the underlying admissible (g∞, K∞) × Gf -module of the unitary representation 2 \ L0(M(F )AM M)ρ.AG(F )-conjugacy class of cuspidal pairs is a cuspidal datum for G. We write X(G) for the set of cuspidal data for G.

∈ ∈ F Fix a cuspidal pair (M, ρ) X X(G) and a ﬁnite set of K-types F. Write P(M,ρ) for the space of smooth functions φ : UM(F )AG\G → C such that (1) M(F )AM \M m → φ(mg) ∈ C belongs to A0(M(F )AM \M)ρ for any g ∈ G; (2) AM /AG a −→ φ(ag) ∈ C is compactly supported for any g ∈ G;

(3) K k −→ φ(gk) ∈ C belongs to the linear span of the matrix coeﬃcients of K-types in F for any g ∈ G.

∈ F Noting that φ P(M,ρ) is rapidly decreasing, we deduce the following.

∈ F Lemma 2.3. (i) For φ P(M,ρ), θφ(g):= φ(γg) γ∈P (F )\G(F )

2 \ converges absolutely and belongs to L (G(F )AG G). 2 \ { | ∈

(ii) If we write L (G(F )AG G)X for the closure of the span of θφ φ ∈ X F (M,ρ)

F } P(M,ρ) , then we have the orthogonal decomposition 2 \ 2 \

L (G(F )AG G)= L (G(F )AG G)X . ∈ X X (G)

2.4 Cuspidal Eisenstein series

Next we set P F for the space of functions φ :(aG )∗ × UM(F )A \G → C satisfying (M,ρ) M C M (1) (aG )∗ λ → φ(λ; g) ∈ C is of Paley-Wiener type for any g ∈ M1K; M C

(2) M(F )AM \M m → φ(λ; mk) ∈ C belongs to A0(M(F )AM \M)ρ for any λ ∈ (aG )∗ , k ∈ K; M C (3) K k −→ φ(λ; mk) ∈ C belongs to the linear span of the matrix coeﬃcients of K-types in F for any λ ∈ (aG )∗ , m ∈ M1. M C

7 ∈ F Obviously, any φ P(M,ρ) is the Fourier transform φ(uamk):= φ(λ; mk)aλ dλ (2.2) G ∗

i( aM ) F ∈ F ∈ of some φ P(M,ρ). On the other hand, associated to each φ P(M,ρ) is a “Paley-Wiener section” ∗ (aG ) λ −→ [g → φ (g):=e λ+ρP ,HP (g) φ(λ; g)] ∈IG(ρ ) M C λ P λ IG G λ ⊗ ⊗ → of the bundle of induced representations P (ρλ):=indP [(e ρ) 1U] λ.Here ∈ ∗ ρP aM denotes the half of the sum of positive roots of AM in P .

∈ F For φ P(M,ρ), the associated cuspidal Eisenstein series is deﬁned by EP (x, φλ):= φλ(δx). (2.3) δ∈P (F )\G(F )

P = MU, P = M U ∈F(P0) are said to be associated if the set W (M, M ):={w ∈ W | w(M)=M } is not empty. Obviously the Weyl group W M of M acts on this set by the right translation. A system of representatives of W M -orbits in W (M, M )isgivenby

{ ∈ | M ⊂ } ∈ WM,M := w W (M, M ) w(P0 ) P0 .Forw WM,M we deﬁne the intertwining operator by the integral −1 − w(λ)+ρ ,H (x) [M(w, ρλ)φ](x):= φλ(w ux) du · e P P . (U∩w(U))\U

Using the theory of resolvent, Langlands established the following properties [31], [33, Chapt. II, IV].

(1) Convergence. EP (x, φλ)andM(w, ρλ)φ absolutely converge for Re(λ) >> 0. At such λ, EP (x, φλ) deﬁnes and automorphic form on G(F )AG\G,andφλ → (M(w, ρλ)φ)w(λ) IG →IG deﬁnes an intertwining operator P (ρλ) P (w(ρλ)). Moreover the following holds. IG ∈H (i) Equivariance. EP (x, P (ρλ,f)φλ)=R(f)EP (x, φλ) for any f (G/AG).

(ii) Constant terms. The constant term of EP (x, φλ)alongQ = LV ∈F(P0)is given by L EP (x, φλ)Q = E L (x, (M(w, ρλ)φ)w(λ)). Pw w∈W (L) M L Here WM (L):= ∈LL L WM,M and P denotes the unique element of M (P0 ) w F L L (P0 )havingMw := w(M) as its Levi component.

(iii) Functional equations. EP (x, (M(w, ρλ)φ)wλ)=EP (x, φλ)forw ∈ WM (G).

Also for w ∈ WM,M, w ∈ WM ,M ,wehaveM(w ,w(ρλ))M(w, ρλ)φ = M(w w, ρλ)φ at λ where both operators converge absolutely.

(iv) Fourier transform. For λ0 whose real part is suﬃciently positive, we have

λ+ρP θφ(uamk)= EP (umk, φλ)a dλ. G ∗ λ0+i( aM )

8 (2) Analytic continuation. EP (x, φλ)andM(w, ρλ)φ extend meromorphically to the whole (aG )∗ . The properties (i) – (iv) of (1) still hold as equalities of meromorphic M C functions.

(3) Singularities. The set of poles of EP (x, φλ) (hence those of M(w, ρλ)φ by (1-ii)) is a union of locally ﬁnite collection of aﬃne hyperplanes whose vector parts are zeroes of some coroots.

2.5 Residual Eisenstein series and the spectral decomposition Essentially by the Perseval formula, we deduced from (ii), (iv) the L2-inner product formula:

θφ,θφ G = M(w, ρλ)φ, φ M dλ. (2.4) G ∗ λ0+i( a ) ∈ M w WM,M

2 Here , G denotes the hermitian inner product on L (G(F )AG\G). That is, the inner G product between two θφ’s is the integral over the Pontrjagin dual of AM of the Petersson inner product of M(w, ρλ)φ and φ . Certain residue analysis transforms (2.4) into

θφ,θφ =T M(w, ρλ)φ, φ M dλ G ∗ i( a ) ∈ M w WM,M (2.5)

+ ResË M(w, ρλ)φ, φ M dλ.

G ∗ a o( Ë)+i( ) ∈ Ë M w W Ë M,M

=T denotes a certain equivalence relation, the sum on the right runs over a ﬁnite set of intersections S of singular hyperplanes of EP (x, φλ)ando(S) is a certain “origin” of

∈L G ∗ Ë S. MË (M0) is such that i(a ) equals the vector part of S.Res means the

MË

iterated residue along S.Notingthat ∈ ResË M(w, ρλ)φ belongs to the discrete w WM,M

2 \ Ë

spectrum of L (MË (F )A M ) and, at the same time, equals to the constant term of MË certain residual Eisenstein series, we arrive at the following theorem [31, Chapt. 7], [33, Chapt. 6]

Theorem 2.4. We call a pair (M, π) consisting of M ∈L(P0) and an irreducible sub- 2 representation π of L (M(F )AM \M) a discrete pair.AG(F )-conjugacy class [M, π] of

discrete pairs is a discrete datum.Write[P ] for the associated class of P ∈F(P0). F (1) For a discrete pair (M, π) and a ﬁnite set of K-types F,wedeﬁnethespaceP(M,π) in

F A \ A \ the same manner as P(M,ρ) with 0(M(F )AM M)ρ replaced by 2(M(F )AM M)π,the intersection of the space of square-integrable automorphic forms A2(M(F )AM \M) with 2 the π-isotypic subspace L (M(F )AM \M)π. Then the residual Eisenstein series EP (x, φλ)

∈ F deﬁned by (2.3) with φ P(M,π) satisﬁes the properties (1), (2) and (3) of 2.4 (But this time, the formula (1-ii) holds only for Q ⊃ M.). (2) Let L[M,π] for the Hilbert space of families of functions F = {FP }P ∈[P ] such that ∗ G → 2 \ ∈ (i) FP : i(aM ) L (U M (F )AM M )π is a measurable function. Here (M ,π) [M, π].

9 (ii) FP (w(λ)) = M(w, πλ)FP (λ), w ∈ WM,M. (iii) The norm 2 1 2 F := FP (λ ) M dλ |W (G)| G ∗ M i( a ) (M ,π )∈[M,π] M is ﬁnite.

2 Then there exists a G-equivariant unitary injection L[M,π] 5→ L (G(F )AG\G), whose restriction to the automorphic spectrum valued Paley-Wiener sections is given by 1 F −→ EP (x, FP (λ )) dλ . |W (G)| G ∗ M i( a ) (M ,π )∈[M,π] M

2 If we write L (G(F )AG\G)[π] for the image of this map, we have the orthogonal decomposition 2 2 L (G(F )AG\G)= L (G(F )AG\G)[π]. [M,π]

Note that the properties of residual Eisenstein series are deduced from their realization as residues of cuspidal Eisenstein series. In particular, various growth properties of cuspidal Eisenstein series are no longer assured for residual ones. This makes the construction of the trace formula much harder in the case when the F -rank is more than one.

2.6Spectral kernel Th. 2.4 allows us to deduce the spectral expression of the kernel K(x, y) (2.1). Choose a discrete datum [M, π], a cuspidal datum X ∈ X(G) and a ﬁnite set of K-types F.Set 2 A \ A \ ∩ \ M

M X M M 2(M(F )AM )π, := 2(M(F )AM )π L (M(F )AM )X ,

∈ X ∩ X X (M)

M F M and write A2(M(F )AM\M) for the subspace of functions which transform under F , π,X the set of irreducible components of the restrictions to KM := K∩M of elements in F, un- M 2 der K . This is a ﬁnite dimensional subspace of L (M(F )AM \M), so that we can choose a

M M M

F F

ortho-normal basis BF of it satisfying B ⊂ B if F ⊂ F .Similarly we have an ortho-

X X π,X π, π,

F F F K

normal basis B of the induced space A2(UM(F )AM \G) =Ind M A2(M(F )AM \M) . X X

π,X π, K π, F Associated to each ϕ ∈A2(UM(F )AM \G) is the “constant section” π,X

λ+ρP ϕλ(uamk):=a ϕ(mk) of the bundle IG(π ) → λ ∈ (aG )∗ . P λ M C

2 F Now the space L (G(F )AG\G) for functions transforming according to F under K [π], X in 2 \ ∩ 2 \

L (G(F )AG G)[π] L (G(F )AG G)X

10 is spanned by

ΦF (x):= E(x, F (λ)λ) dλ, G ∗ i( aM ) where F : i(aG )∗ →A(UM(F )A \G)F is a Paley-Wiener section. Using the expan- M 2 M π,X

sion F (λ)= F F (λ)λ,ϕλϕλ,wehave ϕ∈ B

π,X [R(f)ΦF ](x)=R(f) F (λ)λ,ϕλEP (x, ϕλ) dλ G ∗

i( a ) M F

ϕ∈ B

π,X = R(f)EP (x, ϕλ)F (λ)λ,ϕλ dλ, G ∗

i( a ) M F

ϕ∈ B

π,X for f ∈H(G/AG)F . Now recall the H(G/AG)-equivariance IG R(f)EP (x, ϕλ)=EP (x, P (πλ,f)ϕλ) and the following formula obtained by the Fourier inversion formula applied to AM /AG × G ∗ (aM ) :

F (λ)λ,ϕλ = EP (F (µ)µ) dµ, EP (ϕλ)G. G ∗ i( aM )

We see that [R(f)ΦF ](x)equalsto IG EP (x, P (πλ,f)ϕλ) EP (F (µ)µ) dµ, EP (ϕλ) G dλ

G ∗ G ∗ a i( a ) i( ) M F M

ϕ∈ B

π,X IG = EP (x, P (πλ,f)ϕλ)EP (y,ϕλ) dλΦF (y) dy.

G ∗ \ a G(F ) A G i( ) G M F

ϕ∈ B

π,X F F 2 In other words, R(f)(f ∈H(G/AG) ) restricted to L (G(F )AG\G) has the kernel [π], X

F G K (x, y):= EP (x, I (πλ,f)ϕλ)EP (y,ϕλ) dλ. [π], X P G ∗

i( a ) M F

ϕ∈ B

π,X It was shown in [2] that 1

KX (x, y):= | | 1 WM (G) P ∈F(P0) π∈Π(M ) (2.6) IG EP (x, P (πλ,f)ϕλ)EP (y,ϕλ) dλ, G ∗

i( a )

M ∈ B ϕ π,X

deﬁned as a certain limit, exists. Here Bπ,X := B is empty if π does not appear X F π, 2 in the discrete spectrum of L (M(F )AM \M). Also the factor 1/|WM (G)| is necessary because we take the sum over all the standard parabolics, not over their associated classes. We obtain the spectral kernel:

K(x, y)= KX (x, y). (2.7) ∈ X X (G)

11 3Convergence

3.1 Truncated kernel One can easily see that the operator R(f) is no longer of trace class and K(x, x) dx \ G(F ) AG G diverges. The ﬁrst step towards the trace formula is to truncate the kernel so that its integral over the diagonal converges. To see how this works, we need the following combinatorial functions. ∨ We write ∆ 0 and ∆0 for the set of simple roots of A0 in P0 and the set of corresponding ∈F { | | ∈ \ M } simple coroots, respectively. For P = MU (P0), set ∆P := α AM α ∆0 ∆0 and ∨ { ∨ | ∈ \ M } M ⊕ ∈ ∆P := (α )M α ∆0 ∆0 ,whereXM denotes the aM component of X = X XM M ⊕ a0 under the W -invariant decomposition a0 = a0 aM . Using this, we deﬁne the positive chamber + { ∈ | ∀ ∈ } aP := X a0 α(XM ) > 0, α ∆P , ⊂ G ∗ whose characteristic function is denoted by τP . Next let ∆P (aM ) be the dual basis of ∨ ⊂ G the basis ∆P aM . Then the convex open cone

+ aP := {X ∈ a0 | :(X)=:(XM ) > 0, ∀: ∈ ∆P }

∨ is spanned by ∆P (Strictly speaking, this is the direct sum of aG with an open cone.). Write τP for the characteristic function of this cone. Let O(G) be the set of semisimple classes in G(F ). For each o ∈ O(G), we write o forthesetofγ ∈ G(F ) whose semisimple part γs under the Jordan decomposition γ = γsγu = γuγs belongs to o.Wehave

−1 Ó

K(x, y)= KÓ (x, y),K(x, y):= f(x γy).

∈ Ó ∈ Ç Ó (G) γ

Next, for P = MU ∈F(P0) write

−1 Ó KP (x, y)= KP,Ó (x, y),KP, (x, y):= f(x γuy) du

U(F )\U

∈ Ç ∈ ∩ Ó Ó (G) γ P (F )

2 for the kernel of the right translation operator RP (f)onL (UM(F )AG\G) deﬁned similarly as R(f). Choosing T ∈ a0, deﬁne the truncated kernel

kT (x, f):= kT (x, f),

Ó ∈ Ç Ó (G) G T − aM −

k (x, f):= ( 1) K Ó (δx,δx)τ (H (δx) T ).

Ó P, P 0

P =MU∈F(P0) δ∈P (F )\G(F )

12 Example 3.1. We present the explicit formulae in the case of G = GL(2) [22]. The

minimal parabolic subgroup will be denoted by B = TU.ThekT (x, f) is given as follows. Ó (i) If o is elliptic, then f(x−1γx). (3.1)

γ∈ Ó

α (ii) If o ( β ) is hyperbolic regular, then −1 −1 −1 α f(x γx) − f(x δ uδx)τB(H0(δx) − T ) du U β γ∈ Ó δ∈B(F )\G(F ) (3.2) − − β − f(x 1δ 1 uδx)τ (H (δx) − T ) du. α B 0 δ∈B(F )\G(F ) U

(iii) If o = {ζ · 1}, ζ ∈ F ×,then −1 −1 −1 f(ζx γx) − f(ζx δ uδx)τB(H0(δx) − T ) du. (3.3) U

γ∈ Ó δ∈B(F )\G(F ) Note that this last term uniﬁes the unipotent and identity terms.

Theorem 3.2 ([2] Th. 7.1). If we choose T ∈ a0 suﬃciently positive with respect to suppf,then

J T (f):= J T (f),JT (f):= kT (x, f) dx

Ó Ó Ó \

G(F ) AG G ∈ Ç Ó (G) converges absolutely. Let us sketch the proof. In the higher rank case, the following combinatorial argument is fundamental. § 2.1 combined with the Iwasawa decomposition yields G = P (F )SP (T0), where M P • SP (T0)=ω2ω1A0 (T0)AM K. If we write FQ ( ,T) for the characteristic function of

{ ∈ | − ≤ ∀ ∈ L} Q(F ) x SP (T0) :(H0(x) T ) 0, : ∆0 P P P − and JQ(δx,T):=FQ (x, T )τQ (H0(x) T ), then we have [2, Lem. 6.4]: P JQ(δx,T)=1. (3.4)

Q=LV ∈F(P0) δ∈Q(F )\P (F ) Q⊂P

P P P − P Next we put KQ(x, T ):=FQ (x, T )σQ(H0(x) T ), where σQ is the characteristic function of   ∀ ∈  (1) α(XL) > 0, α ∆QM ,  ≤ ∀ ∈ \ X ∈ a0 (2) α(XL) 0, α ∆Q ∆QM ,   (3) :(X) > 0, ∀: ∈ ∆Q.

13 Example 3.3. If G = SL(3), the chambers in a0 are illustrated as

❏❏ ✡✡ α2 =0 ❏ ✡ ❏ ✡ ❏✡ ✡❏ α1 =0 ✡ ❏ ✡ ❏ ✡✡ ❏❏

P and σQ are the characteristic functions of the regions

❏❏ ✡✡ ❏❏ ✡✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏✡ ❏✡ σG : ◗ σP1 : ◗ 0 ✡❏ ◗ 0 ✡❏ ◗ ✡ ❏ ◗ ✡ ❏ ◗ ✡ ❏ ◗◗ ✡ ❏ ◗◗ ✡✡ ❏❏ ✡✡ ❏❏

❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏✡ ❏✡ σP2 : ◗ τ : ◗ 0 ✡❏ ◗ 0 ✡❏ ◗ ✡ ❏ ◗ ✡ ❏ ◗ ✡ ❏ ◗◗ ✡ ❏ ◗◗ ✡✡ ❏❏ ✡✡ ❏❏

❏❏ ✡✡ ❏❏ ✡✡ ❏ ✡ ◗◗ ❏ ✡ ◗ ❏ ✡ ◗❏ ✡ G ❏✡ G ◗❏✡ τP = σ : ◗ τP = σ : ◗ 1 P1 ✡❏ ◗ 2 P2 ✡❏ ◗ ✡ ❏◗ ✡ ❏ ◗ ✡ ❏ ◗◗ ✡ ❏ ◗◗ ✡✡ ❏❏ ✡✡ ❏❏

∈F { } where Pi = MiUi (P0) are such that ∆ Mi = αi ,(i =1, 2). P0 R As these illustrate, we have R⊃P σQ = τQM τP and hence R P KQ(x, T )=JQ(x, T ). (3.5)

R∈F(P0) R⊃P

14 Applying these combinatorial formulae to our integrand, we have (3.4) G T − aM P −

k (x, f) = ( 1) K Ó (δx,δx)J (δx,T)τ (H (δx) T ) Ó P, Q P 0 Q⊂P ∈F(P ) δ∈Q(F )\G(F ) 0 (3.5) G R − aM = KQ(δx,T) ( 1) KP,Ó (δx,δx).

Q⊂R∈F(P0) δ∈Q(F )\G(F ) P ∈F(P0) Q⊂P ⊂R Thus it suﬃces to prove the convergence of G | R − aM | KQ(x, T ) ( 1) KP (x, x) dx. \ Q(F ) AG G P ∈F(P0) Q⊂P ⊂R

R KQ(x, T ) cut oﬀ the domain of integration to a product of a compact set and a transported MR positive cone in aL . But on such cone the alternating sum in the integrand is rapidly decreasing by Lem. 2.2.

3.2 Truncation operator and the basic identity Next we turn to the convergence of the spectral side [3]. Already it follows from Th. 3.2 that the integral of the total kernel kT (x, f) dx \ X

G(F ) AG G ∈ X X (G) converges absolutely. Here we have written G T − aM −

k (x, f):= ( 1) K X (δx,δx)τ (H (δx) T ),

X P, P 0

P =MU∈F(P0) δ∈P (F )\G(F ) where 1

K X (x, y)= P, |W M (M)| 1 L Q=LV ∈F(P0) τ ∈Π(L ) Q⊂P P IM P EQ (x, QM (τλ,f)ϕλ)EQ (y,ϕλ) dλ G ∗

i( a )

L ∈ B

ϕ τ,X 2 \ is the kernel of RP (f) restricted to L (UM(F )AG G)X . Thus the problem is the com- mutativity of the integration and the summation. T 2 For T ∈ a0, deﬁne the truncation operator ∧ on L (G(F )AG\G)by T aG (∧ φ)(x):= (−1) M φP (δx)τP (H0(δx) − T ).

P =MU∈F(P0) δ∈P (F )\G(F ) As in the proof of Th. 3.2, we can rewrite this as P G ∧T 1 − aM ( φ)(x)= KQ (δx,T) ( 1) φP (δx),

Q⊂P1∈F(P0) δ∈Q(F )\G(F ) P ;Q⊂P ⊂P1 and obtain the following properties:

15 (1) ∧T is an orthogonal projection.

∧T ∈ 2 \ (2) φ = φ for φ L0(G(F )AG G).

T (3) If φ is slowly increasing and suﬃciently smooth relative to dim U0,then∧ φ is rapidly decreasing.

T,P 2 We have similar operators ∧ on L (UM(F )AG\G). Example 3.4. Consider the simplest example, the GL(2) case. Note that if T is suﬃ- ciently positive the sum on δ contains only one term. (Recall the classical situation where any element of SL2(Z) which preserves the region

y = t

✬✩

✫✪

with suﬃciently large t is upper triangular.) Then ∧T is the operator which cuts oﬀ the constant term in the neighborhood {x | τP (H0(x) − T ) > 0} of cusps, the complement of a compact set in the Siegel domain. The properties stated above are rather obvious in this example.

A similar combinatorics as in the geometric side yield kT (x, f)= σR(H (δx) − T ) X Q 0 ⊂ ∈F ∈ \ Q R (P0) δ Q(F ) G(F ) G T,Q × − aM ∧ ( 1) 2 KP,X (δx,δx).

P ∈F(P0) Q⊂P ⊂R

∧T Here 2 means the truncation operator is applied in the second variable. This combined with (3) above shows that the integration and the summation commute. Moreover the alternating sum on the right gives

J T (f):= kT (x, f) dx X X \ G(F ) A G G G R − − aM ∧T,Q = σQ(H0(x) T ) ( 1) KP,X (x, x) dx \ (3.6) Q(F ) AG G Q⊂R∈F(P0) P ∈F(P0) Q⊂P ⊂R ∧T

= KX (x, x) dx. \ G(F ) AG G That is, the term associated to Q, R vanishes unless Q = R = G.

16 We now come to the coarse Arthur-Selberg trace formula

J T (f)= J T (f), (3.7)

Ó X

∈ Ç X∈ X Ó (G) (G) for suﬃciently positive T ∈ a0.

4 The ﬁne O-expansion

Next we need to calculate each terms J T (f)andJ T (f) in (3.7). Recall that, in the GL(2)

Ó X case [24], [22], certain part of the hyperbolic terms and Eisenstein terms cancel each other. The resulting equality has a meaning when T →∞. Since such an explicit cancellation is not available in the present case, taking the limit under T →∞is not allowed. Instead we calculate the special value at “T = 0”.

4.1 J T (f) as a polynomial

For Q ⊂ P ∈F(P0), consider the function P aM M1 Γ (X, Y ):= (−1) τ M1 (X)τ M (X − Y ) Q Q P1 P1; Q⊂P1⊂P

aL aL on a0 × a0. Noting that the matrices ((−1) τQM )Q,P and ((−1) τQM )Q,P are inverse to each other, one sees that this function expresses the variation of τQM : M aM P M − − 1 M τQ (X Y )= ( 1) τQ 1 (X)ΓP1 (X, Y ). (4.1) P1; Q⊂P1⊂P Some calculation shows that T +X M,T G J• (f)= J•M (f P ) ΓP (H, X) dH,

G a P ∈F(P0) M where f is the descent P

ρP −1 f P (m):=m f(k muk) du dk K U G of f to M. Since the integral of ΓP (H, X)inH is a polynomial function in X,thesame T is true for J• (f) [4, Prop. 2.3]. Recall that the distributions in (3.7) depend on the choice of the set of representatives W ∈ Norm(A0,G), which can be chosen in G(F ) ∩ K if G is split. In general W ⊂ ∩ ∈ KM0 G(F ) and, for α ∆0, the representative wrα of the reflection rα attached to α −1 ∨ ∈ R ∈ satisfies H0((wrα ) )=hαα for some hα . We define the (analytic) origin T1 a0 by ∨ T1 := hα:α . α∈∆0 ∨ ∗ { } G ∈ G T1

Here : ∈ is the basis of a dual to ∆ (a ) . We deﬁne JÓ (f):=J (f), α α ∆0 0 0 0 Ó T1 §

JX (f):=J (f)[4, 1, 2]. X

Example 4.1. Let us compare this to the classical GL(2) case. The distributions J T (f) Ó are calculated as follows [22]. (1) If o is elliptic, we have J T (f)=meas(G (F )A \G ) f(x−1γx) dx (4.2)

Ó γ G γ Gγ \G

γ as in the anisotropic case. Note that we have G = Gγ in the special case of GL(2). α (2) If o is the hyperbolic regular class of ( β ), (3.2) equals the sum over δ ∈ B(F )\G(F ) and w ∈ W of 1 − α − α f Ad(wνδx) 1 − f Ad(wuδx) 1 duτ (H (δx) − T ). 2 β β B 0 ν∈U(F ) U

One sees that J T (f) equals Ó 1 − α f Ad(wνx) 1

\ β A 2 ∈ B(F ) G G ∈ w W ν U(F ) −1 α − f Ad(wux) duτB(H0(x) − T ) dx U β −1 α 1 = f Ad(wux) − τB(H0(x) − T ) du dx \ β 2 UT (F ) A G U w∈W G −1 −1 α 1 = f w x xw − τB (H0(x) − T ) dx \ β T (F ) A G 2 w∈W G −1 α = f x x 1 − τB (H0(x) − T ) − τB (H0(wx) − T ) dx \ β T (F ) AG G using the Iwasawa decomposition, − − α =meas(T (F )A \T1) f(k 1u 1 uk) G β K U

1 − τB (H0(a) − T ) − τB (H0(wau) − T ) da du dk. G AT

Noting H0(wau)=w(H0(a)) + H0(wu), the inner integral becomes

∨ α(T )α /2 α(2T − H (wu)) dH = √ 0 . ∨ α(H0(wu)−T )α /2 2 Here α denotes the unique positive root of T in G.Weobtain − − α α(H (wuk)) J T (f)=−meas(T (F )A \T1) f k 1u 1 uk 0√ du dk (4.3) Ó G β K U 2 √ α + 2α(T )meas(T (F )A \T1)f . (4.4) G P β

18 (3) The case of central o is easily reduced to the case o = {1}.Ifo = {1}, o = −1 δ∈B(F )\G(F ) Ad(δ) U(F ) allows us to write (3.3) as −1 −1 −1 −1 f(1) + f(x δ νδx) − f(x δ uδx) duτB(H0(δx) − T ) . δ∈B(F )\G(F ) ν∈U(F )\{1} U

T Thus, by the Iwasawa decomposition, J{1}(f) equals the sum of

meas(G(F )AG\G)f(1) (4.5)

and    −1 −1  −2ρB f(t νt) − f(t ut) duτB(H0(t) − T ) t dt, (4.6) \ T (F ) AG T ν∈U(F )\{1} U where f(x):= f(k−1xk) dk. K Fix a non-trivial character ψ of F \A.Writeu for the Lie algebra of U and u∨ be its dual. Then the Poisson summation formula for the Fourier transformation ∨ ∨ ∨

(Ad(t)f ◦ exp)(X ):= (Ad(t)f)(exp X)ψ(X ,X) dX = t2ρB (f ◦ exp)(α(t)X ) A Ù( ) implies that (4.6) equals √ · ×\A1 ◦ −1 | |−1 − ◦ | | ×

2 meas(F ) (f exp)(x ξ) x (f exp)(0)τ (log x A ) dx A >α(T ) × × F \ A \{ } ξ∈ Ù(F ) 0 √ = 2 · meas(F ×\A1) (f ◦ exp)(x−1ξ)|x|−1 dx×

× A \ A F <1 \{ } ξ∈ Ù(F ) 0 ∨ −1 ×

+ (f ◦ exp)(xξ ) − f(1)|x| − (f ◦ exp)(0)τ (log |x|A) dx A >α(T ) × F \ A ≥1 ∨ ∨

ξ ∈ Ù (F ) √ ×\A1 ◦ | | ×

= 2meas(F ) (f exp)(xξ) x A dx

× \ A F >1 \{ } ξ∈ Ù(F ) 0 + (f ◦ exp)(xξ∨) − f(1)|x|−1 dx×

× A \ A F ≥1 ∨ ∨ \{ } ξ ∈ Ù (F ) 0 ◦ − | | ×

+ (f exp)(0) 1 τ>α(T )(log x A ) dx . × F \ A ≥1

Here τ>A is the characteristic function of R>A. Consider the integral Z(f,s):= (f ◦ exp)(x)|x|s dx×.

× A A

19 This converges absolutely for Re(s) > 1 and extends to a meromorphic function in s. It turns out that the sum of the ﬁrst two integrals in the above equals the regular part ×\A1 regs=1 Z(f,s) of this “zeta function” at s =1, while the last equals meas(F )α(T ). ◦ Noting that (f exp)(0) = f P (1), we conclude that J T (f)=meas(G(F )A \G)f(1) + meas(Z(F )A \Z)reg Z(f,s) (4.7) {1} √ G G s=1 \ 1 + 2α(T )meas(T (F )AG T )f P (1). (4.8)

We choose T1 =0. Then taking the special value at T1 is equivalent to throw away the sum of (4.4) and (4.8): √ α 2α(T )meas(T (F )A \T1)f . G P β α,β∈F × This is precisely the term which cancels the analogous term in the spectral side [22, p. 289]. Thus we can say that the present construction is a generalization of the GL(2) case. Note also that this term can be viewed as the geometric side of the “trace formula” for the Levi subgroup T applied to f P .

4.2 Reduction by the Jordan decomposition

Calculate the geometric terms. Our goal is to express JÓ (f)asasumofterms,eachof which is a product of a global constant and an Euler product of local distributions. First we replace kT (x, f)with Ó G T − aM −

j (x, f):= ( 1) j Ó (δx)τ (H (δx) T ),

Ó P, P 0

P ∈F(P0) δ∈P (F )\G(F ) −1 −1

jP,Ó (x):= φ(x ν γuνx) du. γs ∈ γs \ U γ∈M(F )∩ Ó ν U (F ) U(F )

We have Ó KP,Ó (x, x)= jP, (ux) du U(F )\U and hence

J T (f)= jT (x, f) dx. Ó Ó \ G(F ) AG G

Take a representative σ ∈ o which is elliptic in some P1 = M1U1 ∈F(P0), that is, σ ∈ M1(F ) but it is not contained in any proper parabolic subgroup of M1.ThenP1,σ is a ∈ U minimal parabolic subgroup of Gσ.Eachγ o is conjugate to some element of σ Gσ (F ), U where Gσ is the unipotent variety of Gσ, the connected centralizer of σ. Thus writing G σ ι (σ):=[G (F ):Gσ(F )], we have G −1

jQ,Ó (x)=ι (σ)

w∈WM (L,Gσ) 1 f(x−1π−1w(σνv)πx) dv. −1 ∩ ∈ w ∩ \ ∈ −1 ∩U w (V) Gσ π ( Gσ Q)(F ) Q(F ) ν w (L)(F ) Gσ(F )

20 Here −1 ∀ ∈ (i) w (α) > 0, α ∆wP L , W (L, G ):= w ∈ W (L) 1 , M1 σ M1 ∀ ∈ (ii) w(β) > 0, β ΣP1,σ

ΣP being the set of positive roots of P . If we write R for the standard parabolic subgroup w−1 Q ∩ Gσ of Gσ and F(M1)R := {P ∈F(M1) | Pσ = R},theabovegives J T (f)=ιG(σ)−1 Ó \ Gσ\G Gσ(F ) AG Gσ R∈F(P1,σ ) δ∈R(F )\Gσ(F ) −1 −1 −1 f(y x δ σνvδxy) dv (4.9) ν∈U (F ) UR MR G − − aM − 1 − − ( 1) τP (H0(δxy) wP (T T1) T1) dx dy.

P ∈F(M1)R

wP wP ∈ W is such that P ∈F(P0). R = MRUR denotes the standard Levi decomposition MS,T of R. We shall write this in terms of the unipotent terms J{1} of the trace formula for the Levi subgroups of Gσ. ∈ As in the case of G,wehavetheoriginT1,σ aM1 for Gσ. We write Tσ for the image − of T T1 + T1,σ in aM1 and set T − −1 − − ∈F YP (x, y):= HP (kPσ (x)y)+wP (T T1) Tσ + T1,P (M1)R. YT { T | ∈F } Then the family P (x, y):= YP (x, y) P (M1)R is compatible in the sense of [9, § 4]. Generalizing (4.1), we see that the last line in (4.9) can be written as

M a S G T − MR − − Y ( 1) τRMS (HR(δx) Tσ)ΓS (HS (δx) Tσ, S (δx,y)), Gσ S∈F (M1,σ ) S⊃R where   G G Y  − aL −  ΓR(X, R):= τRMS (X) ( 1) τQ(X YQ) . Gσ ∈F S∈F (M1,σ ) Q (M1)S S⊃R This combined with the Iwasawa decomposition yields J T (f)=ιG(σ)−1 J MS,Tσ (ΦT ) da dk dy, (4.10) Ó {1} S,a,k,y

G \ A Gσ G G Kσ S∈F σ (P1,σ ) S with − − T ρS 1 1 G − YT ΦS,a,k,y(m):=m f(y σk muky) duΓS (HS (a) Tσ, S (k, y)). US

G YT1 G − Finally if we specialize this to T = T1,ΓS (X, S (k, y)) reduces to ΓQ(X, HQ(ky)+ T1 − T1,σ) for any Q ∈F(M1)S. Applying this to the integral over a in (4.10), we obtain G −1 M S − JÓ (f)=ι (σ) J{1} (ΦS,y,T1 T1,σ ) dy \ Gσ G G S∈F σ (P1,σ ) (4.11) | MR | G −1 W M R − = ι (σ) G J{1} (ΦR,y,T1 T1,σ ) dy, \ |W σ | Gσ G G R∈F σ (M1,σ )

21 where

ρS −1 −1 ΦS,y,T (m):=m f(y σk muky)vS(ky, T) du dk, K U σ S G − vS (ky, T):= ΓQ(X, HQ(ky)+T ) dX.

G a Q∈F(M1)S L

4.3 Weighted orbital integrals Our goal here is to express (4.11) as a linear combination of weighted orbital integrals. We must ﬁrst recall the notion of (G, M)-family. For Q ⊂ P ∈F(M0), we deﬁne two denominator functions 1 θP (λ):= α∨(λ), Q M Z ∨ meas(aL / [∆QM ]) ∈ α ∆QM 1 θP (λ):= :∨(λ). Q M Z ∨ meas(a / [∆ M ]) ∨ ∨ L Q 1 ∈∆b QM

∨ G ⊂ G ∗ Here ∆P denotes the basis of aM dual to ∆P (aM ) . These functions ﬁrst appeared in G the Fourier transform of ΓQ(X, Y )inX [4, Lem. 2.2]: λM ,Y M e G λ(X) − aL ΓQ(X, Y )e dX = ( 1) , G P G aL P ⊃Q θQ(λ)θP (λ) but its real nature is to produce certain “diﬀerence”

M a 1 cP1 (λ) c (λ):= (−1) L (4.12) Q P θ 1 (λ)θG (λ) P1⊃Q Q P1

of a smooth function c (λ)onia∗ , Q ⊃ P .Herec (λ):=c (λ ), λ being the ia∗ P M P1 P M1 M1 M1 ∗ M1 ∗ ⊕ ∗ component of λ under iaM = i(aM ) iaM1 . Example 4.2. InthecaseofGL(2), we have √ √ 2 2 c (λ)= (c (λ) − c (λ)) = (c (λ) − c (λ )). B α∨(λ) B G α∨(λ) B B G √ ∨ Since 2 is the length of α , this is literally the diﬀerence of cB(λ) around λG. { } ∗ A family cP (λ) P ∈P(M) of smooth functions on iaM is a (G, M)-family if cP (λ)= cP (λ) for any P , P ∈P(M) whose associated chambers share a wall and λ in that wall. For a (G, M)-family {cP (λ)}P ∈P(M), the function (4.12) is well-deﬁned because it is independent of the choice of P ∈PP1 (M).

22 Lemma 4.3 ([4] Lem. 6.1, 6.2, 6.3). Let {cP (λ)}P ∈P(M) be a (G, M)-family. ∗ (1) cQ(λ) extends to a smooth function on iaL. (2) The function c (λ) cQ (λ):= P M θQ(λ) P ∈P(M) P P ⊂Q ∗ extends to a smooth function on iaM . (3) If {dP (λ)}P(M) is another (G, M)-family, (cd)P (λ):=cP (λ)dP (λ) form a (G, M)- family and we have Q (cd)M (λ)= cM (λ)dQ(λ). Q∈F(M)

−λ(HQ(x)) An example of (G, L)-family is the family {vQ(λ, x):=e }Q∈P(L) for x ∈ G.If P = MU ∈F(L), then we have the bijection

P P (L) Q −→ QM := Q ∩ M ∈PM (L).

Thus, from a (G, L)-family {cQ(λ)}Q∈P(L), one deduces a (M, L)-family {cQM (λ):= } { } cQ(λ) QM ∈PM(L). Applying this to vQ(λ, x) Q∈P(L), we have the smooth function v (λ, x) vP (λ, x):= Q L θP (λ) QM ∈PM(L) Q by Lem. 4.3 (2). This depends on P ∈P(M) (as the notation suggests) because the M P P function HQ depends not only Q but also Q.SetvL (x) := limλ→0 vL (λ, x). Let S be a finite set of places of F . We now define the weighted orbital integral JM (γ,f)forM ∈L(M0)andγ ∈ MS . Write γv = γv,sγv,u for the Jordan decomposition ⊂ ∈ of γv.IfGγv,s Mγv,s at any v S, we define G 1/2 −1 JM (γ,f)=|D (γ)| f(x µx)vM (x) dµ dx A S

M \ Ó MS GS S (γ) = |DG(γ)|1/2 f(x−1γx)v (x) dx. A S M GS,γ \GS

G G G − | Here D (γ)=(D (γv))v∈S is given by D (γ):=det(1 Ad(γv,s) g(Fv)/gγv,s(Fv)), gγv,s M being the ﬁxed part of Ad(γv,s)ingv,theLiealgebraofG ⊗F Fv.Alsoo S (γ)istheMS - orbit of γ. The convergence of JM (γ,f)isassuredby[10,Lem.2.1]andtheDeligne-Rao theorem [36]. In the general case, we deﬁne L JM (γ,f) := lim rM (γ,a)JL(aγ, f). (4.13) a→1 L∈L(M)

L Q ∈P Here rM (γ,a)=rM (γ,a)(Q (L)) is constructed from the (G, M)-family − ∨ | β − β|ρ(β,γv,u)λ(β )/2 ∈P rP (λ, γ, a)= a a v ,P (M) v∈S β∈Σr Pγv,s

23 by the process explained above. The non-negative integers ρ(β,u) are deﬁned in [10, § 3], and the existence of the limit in (4.13) is assured by [10, Th.5.2]. In this case the resulting function is independent of Q ∈P(L) [10, Lem. 5.1]. Note that, if M = G this reduces to the ordinary orbital integral G 1/2 −1 JG(γ,f)=|D (γ)| f(x γx) dx. A S GS,γ \GS We need the following descent formula for weighted orbital integrals.

Lemma 4.4 ([10] Cor. 8.7). Suppose that γ ∈ GS satisﬁes

(i) (γv,s)v∈S ∈ G(F );

(ii) aMγs = aM (γs is elliptic in M(F )). Then we have   G 1/2  MR  JM (γ,f)=|D (γs)| J (γu, ΦR,y,T ) dy, A Mγ S \ s Gγs,S GS ∈FGγ R s (Mγs ) where

ρR −1 −1 ΦR,y,T (m):=m f(y γsk muky)vR(ky, T) du dk Kγs,S UR,S is the local analogue of ΦR,y,T in (4.11). In particular the right hand side does not depend on T . We now go back to the calculation of (4.11). Take a ﬁnite set S of places of F suﬃciently large with respect to G and f so that   |W MR | G −1  MR  − JÓ f ι σ J dy ( )= ( ) G {1} (ΦR,y,T1 T1,σ ) (4.14) \ |W σ | Gσ,S GS G R∈F σ (M1,σ )

We have following formulae on the variation of J{1}(f)andJM (γ,f) under conjugations:

| L| −1 W L J{1}(Ad(g )f)= | | J{1}(f Q,g), ∈F W Q (M0) −1 L JM (γ,Ad(g )f)= JM (γ,f Q,g),

Q∈F(M0) where

ρQ −1 f Q,g()= f(k vk)uQ(k, y) dv dk K V G − uQ(k, y)= ΓQ(X, H0(ky)) dX. G aL

24 From this Arthur deduced [8, Cor. 8.3] that there exists a family of complex numbers L {a (S, u) | L ∈L(M0),u∈UL(F )} such that

| L| W L J{ }(f)= a (S, u)J (u, f ). 1 |W | L S L∈L(M0) u∈UL(F )/Ad(LS )

Using this (4.14) becomes | L| G −1 W

JÓ f ι σ ( )= ( ) G \ |W σ | Gσ,S GS ∈LG ∈FG L σ (M1,σ ) R σ (L) × L MR a (S, u)JL (u, ΦR,y,T1−T1,σ ) dy

u∈UL(F )/Ad(LS)

L0 { ∈L | } writing σ(M1):= M (M1) aM = aMσ , |W Mσ | =ιG(σ)−1 |W Gσ | ∈L0 ∈U M σ(M1) u Mσ (F )/Ad(Mσ,S ) Mσ G 1/2 MR a (S, u)|D (σ)| J (u, ΦR,y,T −T ) dy. A Mσ 1 1,σ S \ Gσ,S GS G R∈F σ (Mσ)

Going back from Gσ to G byLem.4.4,weobtain

| Mσ | G −1 W Mσ

JÓ (f)=ι (σ) a (S, u)JM (σu,f). (4.15) |W Gσ | ∈L0 ∈U M σ(M1) u Mσ (F )/Ad(Mσ,S )

4.4 The ﬁne O-expansion Our ﬁnal step is to get rid of σ from (4.15). We say that γ, γ ∈ M(F )is(M, S)-equivalent (notated as γ ∼ γ ) if there exists δ ∈ M(F ) such that (M,S)

• −1 γs = δγsδ and • −1 γu and δγuδ are conjugate in MS .

We write (M(F ) ∩ o)M,S for the (ﬁnite) set of (M, S)-equivalence classes in M(F ) ∩ o. ∈ ∈L0 Noting that σ M(F ) is elliptic if and only if M σ(M1), we set

M J (γs) aM S, γ aMγs S, u , ( ):= M ( ) ι (γs) ∈U u Mγs (F )/Ad(Mγs ) γsu ∼ γ (M,S) where 1ifσ is elliptic in M(F ), JM (σ):= 0otherwise.

25 This allows us to write

| M | | |·| Mσ |· M W W W ι (σ) M

JÓ (f)= a (S, γ)JM (γ,f) |W | |W M |·|W Gσ |·ιG(σ)

M∈L(M1) γ∈(M(F )∩ Ó)M,S γs=σ |W M | = aM (S, γ)J (γ,f)

|W | M

∈ Ç ∈ ∩ Ó M∈L(M0) ÓM (M) γ (M(F ) )M,S

γ =σ Ó Ad(G(F ))ÓM = s |W M | = aM (S, γ)J (γ,f). |W | M

M∈L(M0) γ∈(M(F )∩ Ó)M,S

Theorem 4.5 ([9] Th.9.2). For each compact neighborhood Ω ⊂ AG\G of 1, there exists a ﬁnite set of places SΩ such that we have

|W M | J(f)= aM (S, γ)J (γ,f) |W | M M∈L(M0) γ∈M(F )M,S ⊃ ∈ ∞ for S SΩ and f Cc (G/AG) supported on Ω. We must remark that aM (S, γ) are constants which are not well-understood. The only information for these is the following theorem.

Theorem 4.6([9] Th. 8.2). For a semisimple γ ∈ G(F ) and suﬃciently large S,we have  meas(G (F )A \G ) γ G γ if γ is elliptic in G F , G G ( ) a (S, γ)= ι (γ) 0 otherwise.

5Fineχ-expansion

Next we calculate the spectral terms. Let us recall the deﬁnition of J T (f). We have the X A \ § T

induced space (UM(F )A G) X 2.6. Deﬁne the linear transformation Ω (P,λ) 2 M π, π,X on this space by [ΩT (P,λ)φ](mk)φ (mk) dm dk = ∧T E (x, φ )∧T E (x, φ ) dx,

π,X P λ P λ

\ A \ K M(F ) AM M G(F ) G G ∈A \

for any φ, φ 2(UM(F )AM G)π,X . From (2.6), we know that 1

∧T ∧T X 2 KX (x, y)= K (x, y)= |P | 1 (M) P ∈F(P0) π∈Π(M ) ∧T IG ∧T EP (x, P (πλ,f)ϕλ) EP (y,ϕλ) dλ. G ∗

i( a )

M ∈ B ϕ π,X

26 Replacing this in (3.6), we have 1

J T (f)= tr ΩT (P,λ)IG (π ,f) dλ, (5.1)

X X X π, P, λ G ∗ |P(M)| 1 i( a ) P ∈F(P0) π∈Π(M ) M

IG A \

where (π ) denotes the representation of G on (UM(F )A G) X . We would like P,X λ 2 M π, to obtain a more explicit expression for this.

5.1 An application of the Paley-Wiener theorem We put T G Z ∨ ω (λ, λ ,φ,φ):= meas(aL / [∆Q]) ∈F ∈ ∈ Q=LV (P0) w WM,L w WM,L (w(λ)−w(λ))(T ) M(w, π )φ, M(w ,π )φ e × λ λ , (w(λ) − w (λ ))(α∨) α∈∆Q ∗ ∗

∈A \ ∈A \ ∈ ∈ X for φ 2(UM(F )AM G)π,X , φ 2(U M (F )AM G)π , , λ iaM , λ iaM .Ex- tending the GL(2) case, Langlands proved that if φ and φ belong to the space of (induced

2 ∧T ∧T from) cusp forms, the L -inner product of truncated Eisenstein series EP (φλ), EP (φλ ) is given by ωT (λ, λ ,φ,φ ) [30, § 9]. For general φ, φ we have the following weaker result. Proposition 5.1 ([5] Th. 9.1). There exist J>0 and a locally bounded function ρ(λ, λ ) ∗ × ∗ on iaM iaM such that − |∧T ∧T − T |≤ · 3 T EP (φλ), EP (φλ ) ω (λ, λ ,φ,φ) ρ(λ, λ ) φ φ e . Here the norms on the right are given by φ2 := |φ(mk)|2 dm dk. \ K M(F ) AM M

Example 5.2. In the GL(2) case, we write χ, χ for π and π ∈ Π(T1). Then we have ∧T ∧T ∧T EB (φλ), EB(φλ ) = EB(φλ), EB (φλ ) − − = EB(x, φλ) φλ (x) EB (x, φλ)BτB (H0(x) T ) dx \ B(F ) AG G − − = EB (x, φλ)B φλ(x) EB(x, φλ )BτB(H0(x) T ) dx \ UT (F ) AG G using § 2.4 (1-ii),

= (φλ(x)+M(w, χλ)φλ(x)) \ UT (F ) AG G × − − − − (φλ (x)(1 τB(H0(x) T )) M(w, χλ )φλ (x)τB (H0(x) T )) dx.

27 Using the Iwasawa decomposition this becomes, for Reα∨(λ ) > ±Reα∨(λ), α(T ) tα∨(λ+λ )/2 tα∨(w(λ)+λ )/2 dt φ, φ e + M(w, χλ)φ, φ e √ −∞ 2 ∞ − tα∨(λ+w(λ ))/2 tα∨(w(λ+λ))/2 √dt φ, M(w, χλ )φ e + M(w, χλ)φ, M(w, χλ )φ e α(T ) 2 √ √ (λ+λ )(T ) (w(λ)+λ )(T ) 2e 2e = φ, φ + M(w, χλ)φ, φ α∨(λ + λ ) α∨(w(λ)+λ ) √ √ (λ+w(λ ))(T ) w(λ+λ )(T ) 2e 2e + φ, M(w, χλ )φ + M(w, χλ)φ, M(w, χλ)φ . α∨(λ + w(λ )) α∨(w(λ + λ ))

∈ ∗ × ∗ T If we restrict the last formula to (λ, λ ) iaT iaT ,weobtainω (λ, λ ,φ,φ). In the higher rank cases, π can be in the residual spectrum, an irreducible quotient of an induced from cuspidal representation. Since it throw away some submodules which might contribute to the inner product formula, we can control the inner product only asymptotically.

We would like to replace ωT (λ, λ ,φ,φ ) for the inner product of truncated Eisenstein series in (5.1). But this is not allowed because the domain of the integral is not compact. We bypass this diﬃculty in the following way. Fix an R-minimal parabolic subgroup P∞ = M∞U∞ of G∞ which is contained in the global minimal parabolic subgroup P0,∞. Write a∞ for the Lie algebra of the R- split component A∞ of Z(M∞). Choose a Cartan subalgebra hK∞ in the Lie algebra of ∩ ⊕

K∞ M∞ and set h := ihK∞ a∞, a Cartan subalgebra of g∞. Write W (hC )forthe h

C E W ( C ) ∞ C Weyl group of hC in g ( )andlet (h) bethespaceofW (h )-invariant compactly supported distributions on h. The following is a corollary of the Paley-Wiener theorem

for real reductive groups.

h W ( C ) Proposition 5.3 (Multiplier theorem, [7] Th. 4.2). For γ ∈E(h) and f∞ ∈ H(G∞), we can ﬁnd f∞,γ ∈H(G∞) such that

π∞(f∞,γ)=γ(χπ∞ )π∞(f∞), ∀π∞ ∈ Π(G∞).

∗

Here γ denotes the Fourier transform of γ and χ ∞ ∈ h is a representative of the in- π C ﬁnitesimal character of π∞.

G exp HG Write h for the kernel of h → G∞ → aG. Applying this proposition to the inﬁnite

component of f ∈H(G/AG), we have:

h G W ( C ) Corollary 5.4. For γ ∈E(h ) , we have a linear map f → fγ on H(G/AG) such that 1 π(fγ)=γ(χπ∞ )π(f), ∀π ∈ Π(G ). We use the abbreviation 1

ΨT (λ, f):= tr(ΩT (P,λ)IG (π ,f)).

X X X,π |P(M)| π, P, λ

28 There exists C0 > 0 such that if α(T ) >C0, ∀α ∈ ∆0 we have

J T (f )= ΨT (λ, f ) dλ X X γ ,π γ G ∗ 1 i( a ) P ∈F(P0) π∈Π(M ) M T = γ(χ ∞ + λ)Ψ (λ, f) dλ π X,π G ∗ 1 i( a ) P ∈F(P0) π∈Π(M ) M = ΨT (λ, f) γ(X)e(χπ∞ +λ)(X) dX dλ X,π

G ∗ G h 1 i( a ) P ∈F(P0) π∈Π(M ) M = ψT (X, f)γ(X)eχπ∞ (X) dX, X,π G h 1 P ∈F(P0) π∈Π(M ) where

ψT (X, f):= ΨT (λ, f)eλ(X) dλ. X X,π ,π G ∗

i( a ) M −1 | | −1

In particular, if we write γH := W (hC ) δw (H) (average of Dirac distribu-

∈ h w W ( C ) tions), then for T with α(T ) >C0, ∀α ∈ ∆0 we have 1 − −1

J T (f )= ψT (w 1(H),f)eχπ∞ (w (H)). (5.2) X X γH | | ,π

W (hC ) ∈F 1 ∈ h ∈

w W ( C ) P (P0) π Π(M ) Let pT (H) be the right hand side. This is a polynomial in T and is smooth in H.Wecan

recover J T (f)aspT (0) =“δ (pT )”. X 0 The last expression is incorrect because the right hand side of (5.2) has non zero real exponent, i.e. is not tempered. Instead, we look at the coeﬃcient 1 − −1 ψT (H):= ψT (w 1(H),f)eImχπ∞ (w (H)) λ | | X,π

W (hC ) ∈F × 1 P (P ) ∈ h

0 (w,π) W ( C ) Π(M ) w(Reχπ∞ )=λ T T T λ(H) of each real exponent λ of p (H): p (H)= λ ψλ (H)e , the sum is finite. We can asymptotically approximate these by polynomial functions: { T } Lemma 5.5 ([6] I, Prop. 5.1). There exists a unique (finite) family pλ (H) of polynomials in T which satisfies the following conditions for some C, J>0. For any differ- G ential operator D on h , we can find cD > 0 such that

| T − T |≤ − d0 ∈ G (1) D(ψλ (H) pλ (H)) cD exp( J infα∈∆0 α(T ))(1 + T ) ,forH h and suﬃ- ciently positive T ,

| T |≤ d0 d0 ∈ G ∈ (2) Dpλ (H) cD(1 + H ) (1 + T ) ,forH h and T a0. T Here d0 is the maximum of the degrees of pλ . By construction, these pT (H) are tempered and we can consider λ T T ∈S G pλ (β):= pλ (H)β(H) dH, β (h ),

G h a polynomial in T .Moreoverwehave

29 (1) For β ∈S(hG) satisfying β(H) dH =1,

G h G − T T − dim h 1 we have lim3→0 pλ (β3)=pλ (0) where β3(H):=J β(J H). (2) For β ∈S(hG), we have T − T → ψλ (H)β(H) dH pλ (β) 0

G h

as α(T ) →∞for any α ∈ ∆0. Now we are ready to calculate J T (f). Take B ∈ C∞(i(hG)∗)andwriteB for the X c M G ∗ ⊂ G ∗ ∈S G restriction of this to i(aM ) i(h ) .Thereisaβ (h ) such that B(λ)= β(H)eλ(H) dH.

G h T T If we put P (B):= λ pλ (β), then the theory of Fourier transformation and the above (1), (2) imply:

T T

(1) If B(0) = 1 we have lim → P (B )=J (f), where B (λ):=B(Jλ). 3 0 3 X 3 (2) P T (B) is the unique polynomial in T such that ΨT (λ, f)B (λ) dλ − P T (B) X,π M G ∗ 1 i( a ) P ∈F(P0) π∈Π(M ) M

goes to zero as α(T ) tends to inﬁnity for any α ∈ ∆0. Consider the linear transformation (w(λ)−w(λ))(T ) e −1

A X (λ, λ ):= M(w, π ) M(w ,π ) P,π, G − λ λ θQ(w (λ ) w(λ)) Q∈F(P0) w, w ∈WM,L A \

on 2(UM(F )AM G)π,X . Since the global intertwining operators are unitary on the imaginary axis, we have (w(λ)−w(λ))(T ) M(w ,λ)φ ,M(w, λ)φ e

A X (λ, λ )φ ,φ = P,π, G − θQ(w (λ ) w(λ)) Q∈F(P0) w, w ∈WM,L = ωT (λ ,λ,φ ,φ). T G ∗ × G ∗ It is shown that ω (λ ,λ,φ,φ) is holomorphic on i(aM ) i(aM ) [5, Cor .9.2], and we T ∈ G ∗

can deﬁne ω (P,λ):=A X (λ, λ)forλ i(a ) .SinceB are compactly supported, X,π P,π, M M we can apply Prop. 5.1 to see that P T (B) is the unique polynomial in T such that 1

tr[ωT (P,λ)IG (π ,f)]B (λ) dλ − P T (B) X X,π P, λ M |P(M)| G ∗ 1 i( a ) P ∈F(P0) π∈Π(M ) M goes to zero as α(T ) tends to inﬁnity for any α ∈ ∆0.

30 5.2 Logarithmic derivatives Yet we need to give an explicit expression for ωT (P,λ). We ﬁrst look at the simplest X,π case.

Example 5.6. Suppose G = GL(2). In the notation of Ex. 5.2, we have √ √ (λ−λ)(T ) (w(λ)−λ)(T ) 2e 2e

A X(λ, λ )= + M(w, χ ) B,χ, ∨ − ∨ − λ √ α (λ λ) α (w(λ )√ λ) (λ−w(λ))(T ) w(λ−λ)(T ) 2e −1 2e −1 + M(w, χ ) + M(w, χ ) M(w, χ ) ∨ − λ ∨ − λ λ α√(λ w(λ)) α (w(λ λ)) 2 (λ−λ)(T ) (λ−λ)(w(T )) −1 = e − e M (1,χ ) M (1,χ ) ∨ − B|B λ B|B λ α (λ √λ) 2 (w(λ)−λ)(T ) (w(λ)−λ)(w(T )) −1 + e M(w, χ ) − e M (1,χ ) M (w, χ ) , α∨(w(λ ) − λ) λ B|B λ B|B λ ◦ wherewehavewrittenMB|B (1,χλ):=w M(w, χλ) and MB|B(w, χλ):=w.Thisillus-

trates how AB,χ,X(λ, λ ) becomes holomorphic on λ = λ . Moreover, writing λ = λ+tα/2, the ﬁrst row in the right hand side restricts to λ = λ as √ 2 α(T )t/2 − −α(T )t/2 −1 lim e e MB|B(1,χλ) MB|B(1,χλ+tα/2) t→0 t √ 2 −α(T )t −1 = lim 1 − e M | (1,χλ) M | (1,χλ+tα/2) , t→0 t B B B B which gives rise to the logarithmic derivative term in the trace formula of GL(2).

To produce the higher dimensional analogue of the logarithmic derivative, we use two kinds of (G, M)-families. We need the following recapitulation of intertwining operators.

Take M, M ∈L(M0). For P ∈P(M), P ∈P(M )andw ∈ WM,M, we deﬁne −1 −1 λ+ρP ,HP (w ux) −(w(λ)+ρ ),H (x) [MP |P (w, πλ)φ](x)= φ(w ux)e du · e P P , (U∩wU)\U

A \ →A \ X a linear operator 2(UM(F )AM G)π, 2(U M (F )AM G)w(π), X . One can easily show that

−1 − (w(λ)+ρ ),T1−v (T1) v ◦ MP |P (w, πλ)=e P Mv(P )|P (vw,πλ) −1 −1 λ+ρP ,T1−v (T1) −1 MP |P (w, πλ) ◦ v = e MP |v(P )(wv ,v(πλ)).

∈F ∈ v1 v1 If we choose P1, P1 (P0)andv1, v1 W such that P = P1, P = P1,wemaywrite −1 w = v w v for some w ∈ W . Then the above formulae imply that 1 1 1 1 M1,M1

MP |P (w, πλ) − − 1 − 1 −1 −1 (5.3) w1v1 (λ)+ρP ,T1 v1 (T1) − v (λ)+ρ ,T −v (T ) −1 −1 = e 1 e 1 P1 1 1 1 v ◦ M | (w ,v (λ)) ◦ v . 1 P1 P1 1 1 1

31 This relation allows us to translate the basic properties of M(w, πλ)tothoseofMP |P (w, πλ). In particular the latter operator absolutely converges if the real part of λ belongs to a cone, and extends meromorphically to a∗ . This deﬁnes an intertwining operator at λ M,C where it is holomorphic. Finally this ﬁts into the functional equations:

EP (x, MP |P (w, πλ)φλ)=EP (x, φλ), (5.4)

MP |P (w ,w(πλ))MP |P (w, πλ)=MP |P (w w, πλ),P∈P(M ),w ∈ WM ,M . (5.5)

These results allow us to write AP,π,X (λ, λ )as:

(w(λ)−w(λ))(T ) e −1 | | A X (λ, λ )= M (w, π ) M (w ,π ) P,π, θG (w (λ ) − w(λ)) P1 P λ P1 P λ ⊃ ∈ P1 P1 P0 w, w WM,M1 v(w(λ)−λ)(T ) e −1 = M | (v,π ) M | (vw,π ) θG (v(w(λ ) − λ)) P1 P λ P1 P λ ⊃ ∈ P1 P1 P0 v WM,M1 w∈WM,M using (5.5)

− − − −1 ev(w(λ ) λ)(T )e(w(λ ) λ)(T1 v (T1)) = G θ −1 (w(λ ) − λ) ⊃ ∈ v (P1) P1 P0 v WM,M1 w∈WM,M −1 × −1 −1 Mv (P1)|P (1,πλ) Mv (P1)|P (w, πλ)

−1 putting Q := v (P1) ∈P(M)

w(λ )−λ,YQ(T ) e −1 = M | (1,π ) M | (w, π ). G − Q P λ Q P λ θQ(w(λ ) λ) Q∈P(M) w∈WM,M

−1 − −1 ∈ Here we have written YQ(T ):=T1 + v (T T1)forQ = v (P1), v WM,M1.To

calculate tr[ωT (P,λ)IG (π ,f)], which equals the restriction of X π,X P, λ

(w(λ )−λ)(YQ(T )) e −1 G tr(M | (1,π ) M | (w, π )I (π ,f)) (5.6) G − Q P λ Q P λ P,X λ θQ(w(λ ) λ) w∈WM,M Q∈P(M) to λ = λ ,wenotethat

Λ(Y (T )) w −1 G c (T,Λ) := e Q ,d(Λ) := tr(M | (1,π ) M | (w, π )I (π ,f)) Q Q Q P λ Q P λ+Λ P,X λ form (G, M)-families. Then we know from Lem. 4.3 that (5.6) is smooth at any (λ, λ ) and

tr[ωT (P,λ)IG (π ,f)] = cP1 (T,w(λ) − λ)dw (λ ,w(λ) − λ). X π,X P, λ M P1 Lw w∈WM,M P1∈F(M)

32 ∈L { ∈ | } Here Lw (M) is such that aLw = H aM w(H)=H and

w −1 G d (λ , Λ) := tr(M | (1,π ) M | (w, π )I (π ,f)), Q Lw Q P λ Q P λ+ζ P,X λ − ∈ ∗ if Λ = w(λ) λ + ζ, ζ iaM . (Notice that λ and ζ can be recovered from Λ and λLw .) We now combine the above with the result of § 5.1 to see that 1 |P(M)| 1 ∈ P ∈F(P0) π∈Π(M ) w WM,M (5.7)

P1 − w − cM (T,w(λ) λ)dP1 (λLw ,w(λ) λ)BM (λ) dλ G ∗

i( a ) M P1∈F(M)

T P1 • is asymptotic to P (B). Write χM (T, ) for the characteristic function of the convex hull { | ∈P ⊂ } P1 § of YQ(T ) Q (M),Q P1 ,thencM (T ) is its Fourier transform [4, 6]:

P1 P1 µ(X) cM (T,µ)= χM (T,X)e dX. M1 (YQ(T ))M1+aM

G ∗ Then the integral over i(aM ) can be calculated as 1 P1 L χM (T,H) | − | w | M1 det(w 1 aM ) ∈F (YQ(T ))M +a P1 (M) 1 M µ(H) w − −1 e dP1 (λ, µ)BM ((w 1) (µ)+λ) dλ dµ dH

Lw ∗ G ∗ a i( a ) i( ) M Lw

In this, the terms associated to P1 +⊃ Lw goesto0asα(T ) →∞, ∀α ∈ ∆0.Those associated to P1 ⊃ Lw becomes 1 cP1 (T,0) dw (λ, 0)B (λ) dλ. L Lw P1 M | − | w | G ∗

det(w 1 a ) i( a ) M Lw We conclude that (5.7) equals 1 |P | 1 (M) P ∈F(P0) π∈Π(M ) w∈WM,M (5.8) 1 cP1 (T,0)dw (λ, 0)B (λ) dλ. L Lw P1 M | − | w | G ∗

det(w 1 a ) i( a ) M Lw P1∈F(Lw) Final step towards the ﬁne X-expansion is to look at the nature of the “logarithmic derivative” w cQ(T,Λ)d (λ, Λ) cP1 (T,0)dw (λ, 0) = Q . (5.9) Lw P1 G θQ(Λ) P1∈F(Lw) Q∈P(Lw) Λ=0

w As in Ex. 5.6, we divide dQ(λ, Λ) as w −1 G d (λ, Λ) = tr[ M | (1,π ) M | (1,π ) ◦ M | (w, π )I (π ,f) ] Q Q P λ Q P λ+Λ P P λ+Λ P,X λ

33 The latter simply restricts to Λ = 0. But in the former,

−1 MQ(P,π,λ,Λ) := MQ|P (1,πλ) MQ|P (1,πλ+Λ),Q∈P(M)

is a (G, M)-family in Λ and so is

MT M Q(P,π,λ,Λ) := cQ(T,λ) Q(P,π,λ,Λ).

This combined with Lem. 4.3 justiﬁes to write the specialization of (5.9) to Λ = 0 as 1 T G tr[ M (P,π,λ,Λ) M | (w, π)I (π ,f)]. G Q P P P,X λ θQ(Λ) Λ=0 Q∈P(Lw) ∈ ∗ Note that MP |P (w, πΛ+λ)=MP |P (w, π)forΛ+λ iaL. Since this is a linear combination Λ(Y (T )) of certain derivatives at Λ = 0 of the exponential functions cQ(T,Λ) = e Q ,the exponents being linear in T , this is a polynomial function in T . Since (5.8) and P T (B) are both polynomials and asymptotic to each other, they must coincide: 1 1 P T (B)= |P(M)| | det(w − 1|aL )| ∈F 1 ∈L L,reg M P (P0) π∈Π(M ) L (M) w∈W M,M (5.10) × MT I tr( L(P,π,λ,0)MP |P (w, π) P,X (πλ,f))BM (λ) dλ. G ∗ i( aL )

L,reg { ∈ L | − | L + } Here WM,M := w WM,M det(w 1 aM ) =0. Oncewehaveanequality,wecan specialize it to T = T1.SinceYQ(T1)=T1,wehave

MT1 Λ(T1)M | M L (P,π,λ,0) = e L(P,π,λ,Λ) Λ=0 = L(P,π,λ,0).

Using p. 30 (1), we conclude from (5.10) that 1 1

tr( L(P,π,λ,0)MP |P (w, π) P,X (πλ,f))(B3)M (λ) dλ G ∗ i( aL ) |W M | 1 = lim (5.11) 3→0 |W | | det(w − 1|aL )| 1 L,reg M M∈L(M0) L∈L(M) π∈Π(M ) ∈ w WM,M 1 G × tr(M (P,π,λ,0)M | (w, π)I (π ,f))(B ) (λ) dλ. L P P P,X λ 3 M G ∗ |P(M)| i( a ) L P ∈P(M)

∈ ∞ G ∗ Here B Cc (i(h ) ) is such that B(0) = 1.

34 5.3 Normalization and estimation of intertwining operators

We hope to get rid of the limit J → 0 and the factor B3 from (5.11). For this we have to estimate the integrand and show that it converges without the factor B3. For such an estimation, we adopt the usual approach of normalizing intertwining operators. The normalization is constructed locally. Consider a connected reductive group G over alocalﬁeldF of characteristic zero. We adopt the local analogue of various notation pre- sented above. In particular we write G = G(F ) and ﬁx its maximal compact subgroup K so that the Iwasawa decomposition G = PK holds for any P ∈F(M0). For M ∈L(M0), write Πadm(M) for the set of isomorphic classes of irreducible admissible representations M ∈ ∈P

(irreducible (mC , K )-modules if F is archimedean) of M.Forπ Π(M)andP (M), write VP (π) for the space of smooth right K-ﬁnite functions on G satisfying

φ(umg)=π(m)φ(g),u∈ U,m∈ M,g∈ G.

The parabolically induced representation IG(π ), λ ∈ a∗ is deﬁned by P λ M,C

− IG λ+ρP ,HP (xg) λ+ρP ,HP (x) ∈ ∈V [ P (πλ,g)φ](x):=φ(xg)e e ,gG,φ(x) P (π).

This is isomorphic to the usual parabolically induced representation by

IG −→ λ+ρP ,HP (x) ∈ G ⊗ P (πλ) φ(x) φ(x)e indP [πλ 1U].

As in the global case, we deﬁne the intertwining integral MP |P (w, πλ), (P = MU, P =

M U ∈F(M0), w ∈ WM,M)by −1 −1 λ+ρP ,HP (w ux) −w(λ)+ρ ),H (x) [MP |P (w, πλ)φ](x):= φ(w ux)e du · e P P . (U∩wU)\U

Proposition 5.7 ([27], [39] § 2.2). (i) [MP |P (w, πλ)φ](x) converges absolutely if Re(λ) belongs to some open cone in a∗ , and meromorphically continued to the whole a∗ . M M,C (ii) If we write P (w) for the length function on WM (G) with respect to P ∈P(M)

[33, I.1.7], then for w ∈ WM,M and w ∈ WM ,M with P (w w)=P (w)+P (w ),the functional equation

MP |P (w ,w(πλ))MP |P (w, πλ)=MP |P (w w, πλ) holds.

Consider ﬁrst the following two special cases.

(1) F is archimedean.

(2) G is quasisplit over F and π is generic with respect to some non-degenerate character of a maximal unipotent subgroup of M.

35 (Recall that a splitting of G is a triple of a Borel subgroup B, a maximal torus T in B and a system of root vectors {X}⊂g for the simple roots of T in B. G is quasisplit if it { } admits a splitting splG =(B,T, X ) which is stable under Gal(F/F). Then a character θ of the unipotent radical N of B is non-degenerate if its stabilizer in B equals the center ∈ θ → C of G.Forsuchθ,aθ-Whittaker functional on π Π(G)isalinearfunctionalΛπ : Vπ on a realization of π such that

θ θ ∈ ∈ Λπ(π(n)ξ)=θ(n)Λπ(ξ),nN,ξ Vπ.

We say that π is θ-generic if it admits a non-trivial θ-Whittaker functional.) In these cases, we have the automorphic L and ε-factors L(s, π, r)andε(s, π, r, ψ)ofπ attached to certain finite dimensional continuous representation r of the L-group LG of G [17], ψ being L a non-trivial character of F .(N.B.TheL-group should be the Weil form G = G ρG WF instead of the Galois form G ρG Gal(F/F) adopted in [17], since some important cocycles on Gal(F/F) does not split while its inflation to WF does.) In the case (1), these are defined in terms of the local Langlands correspondence established in [32]. In the case § ∈ 1 ∈ ∗ (2), the definition is given in [40, 7]. Now let P , P , π Πadm(M ), λ iaM beas w−1 above. Writing uw := u/ u ∩ u,set

L −→ ◦ | ∈ rw : M m w Ad(m) ρG(w) ubw GL(uw).

Deﬁne the normalization factor for MP |P (w, πλ)by

L(0,πλ,rw) rP |P (w, πλ,ψ):= . ε(0,πλ,rw,ψ)L(1,πλ,rw)

−1 The normalized operator NP |P (w, πλ):=rP |P (w, πλ,ψ) MP |P (w, πλ) enjoys the following properties [13, I, §§ 2,3], [40, Th. 7.9]:

IG IG ∈H (N1) NP |P (w, πλ) P (πλ,f)= P (w(πλ),f)NP |P (w, πλ), f (G). (N2) Without any length condition, the functional equation

NP |P (w ,w(πλ))NP |P (w, πλ)=NP |P (w w, πλ)

holds.

∗ ∈ (N3) For λ iaM , NP |P (w, πλ) is unitary. ∨ (N4) Inthecase(1),NP |P (w, πλ) is a rational function in (α (λ))α∈∆P .Inthecase(2) −α∨(λ) it is a rational function in (qF )α∈∆P ,whereqF is the cardinality of the residue ﬁeld of F .

(N5) Inthecase(1),andifπ is tempered, then rP |P (w, πλ,ψ) has no poles in the region ∨ Re(α (λ)) > 0, ∀α ∈ ∆P . IG 0 (N6) If G is unramiﬁed in case (2) and P (πλ) admits a ﬁxed vector φ under the 0 0 hyperspecial maximal compact subgroup K,thenNP |P (w, πλ)φ (k)=φ (k), k ∈ K.

36 The property (N5) should also hold in the case (2) [40, Conj. 7.1].

||λ1 ⊗ ||λ2 Example 5.8. ConsiderthecaseofGL(2).Ifχλ = χ1 F χ2 F , then the normalization factor equals − −1 L(λ1 λ2,χ1χ2 ) r | (w, π ,ψ)=r (1,π ,ψ)= . B B λ B|B λ − −1 − −1 ε(λ1 λ2,χ1χ2 ,ψ)L(1 + λ1 λ2,χ1χ2 ) (N5) is clearly valid in this case. In the general case, a normalization factor satisfying (N1) to (N6) was constructed by Langlands using the Plancherel measure [20, Lect.15]. We still use the notation rP |P (w, πλ,ψ) for this normalization factor, since these two essentially coincide in the above special cases. To obtain an expression of rP |P (w, πλ,ψ)intermsofL and J-factors as illustrated above is also important in the arithmetic applications. Going back to the global setting, we deﬁne rP |P (w, πλ):= rP |P (w, πv,λ,ψv),NP |P (w, πλ):= NP |P (w, πv,λ). v v A Here ψ = v ψv is a non-trivial character of /F and π = v πv.Weusetheseto estimate M (P,π,λ,0)IG (π ,f) dλ, L P,X λ G ∗

i( a ) π∈Π(M1) L where is the trace class norm. Deﬁne two (G, M)-families

−1 rQ|P (1,πλ+Λ) NQ(P,π,λ,Λ) := NQ|P (1,πλ) NQ|P (1,πλ+Λ),rQ(P,π,λ,Λ) := . rQ|P (1,πλ)

We apply Lem. 4.3 to MQ(P,π,λ,Λ) = rQ(P,π,λ,Λ)NQ(P,π,λ,Λ) to have

M (P,π,λ,0)IG (π ,f)= rP1 (P,π,λ,0)N (P,π,λ,0)IG (π ,f). X L P,X λ L P1 P, λ P1∈F(L) Since N (P,π,λ,0)IG (π ,f) is rapidly decreasing in λ, it suﬃces to show that P P,X λ 1 | P1 | −N rL (P,π,λ,0) (1 + λ ) dλ G ∗ i( aL ) converges absolutely for suﬃciently large N. Once we are reduced to the estimation of such a scalar valued function, we can deduce it from that of the inner product of two truncated Eisenstein series (using Prop. 5.1) . This is done in [6, II, § 9]. Finally we have the following.

Theorem 5.9 (The ﬁne X-expansion). For f ∈H(G/AG), we have |W M | 1

JX (f)= |W | | det(w − 1|aL )| 1 L,reg M M∈L(M0) L∈L(M) π∈Π(M ) ∈ w WM,M 1 G × tr(M (P,π,λ,0)M | (w, π)I (π ,f)) dλ. L P P P,X λ G ∗ |P(M)| i( a ) L P ∈P(M)

37 5.4 Weighted characters

To obtain an expression of JX (f) analogous to Th. 4.5, we need weighted characters, the spectral counter part of weighted orbital integrals [13]. Let S be a ﬁnite set of places of F . Consider the local analogue

N −1 ∈ ∈ ∗ Q(P,π,λ,Λ) = NQ|P (1,πλ) NQ|P (1,πλ+Λ),π Πadm(MS), Λ iaM

of the (G, M)-family deﬁned above. Since the singularity of NQ|P (1,πλ) as a function in λ is isolated, its “logarithmic derivative” at λ 1 NM (P,π,λ) := lim NQ(P,π,λ,Λ) Λ→0 θG(Λ) Q∈P(M) Q

can be defined. This is meromorphic in λ whose singularity set is a locally finite union of affine hyperplanes whose vector parts are the zeros of coroots. Define the weighted character to be N IG JM (πλ,f) := tr( M (P,π,λ) P (πλ,f)). ∈ ⊂ IM More generally, for τ Πadm(L)(L M), we define JM (τλ,f):=JM ( Q (τ)λ,f)with any Q ∈PM (L). This again is a meromorphic function having the same type singularities as NM (P,π,λ) has. Note that, by taking trace, this is independent of P ∈P(M)asthe notation suggests. We also need the distribution −λ(X) JM (π,X,f):= JM (πλ,f)e dλ, f ∈H(GS ). G∗ iaM,S

Here aM,S := HM (MS )isaM itself or a lattice in aM (according to either S contains an ∗ ∗ ∗ ∨ archimedean place or not), and we have written aM,S for aM or aM /(aM,S) accordingly. ∨ aM,S denotes the dual lattice of aM,S in the latter. We look at the discrete part (i.e. the term associated to L = G) of the ﬁne X-expansion Th. 5.9: | M | W 1 G tr(M | (w, 0)I (π ,f)). |W | | det(w − 1|aG )| P P P,X λ ∈L ∈ 1 ∈ reg M M (M0) π Π(M ) w WM,M

G G

Since tr(M | (w, 0)I (π ,f)) is an invariant distribution and I (π ) is admissible, we X P P P,X λ P, λ ﬁnd that this is a ﬁnite linear combination of characters: G adisc(π,X)trπ(f). π∈Π(G1)

Here we note that aG (π):= aG (π,X) are merely some scalars and are not the ∈ X disc X (G) disc multiplicity of π in the discrete spectrum (cf. Th. 2.4) 2 \ 2 \ Ldisc(G(F )AG G)= L (G(F )AG G)[π]. [G,π]

We now give an expression analogous to Th. 4.5 for the sum J(f)= JX (f). ∈ X X (G) Set ∈ IM L + π JH( Q (σµ)) (1) adisc(σ) =0, Πdisc(M):= ∈PM ∃ ∈ M,reg , Q (L) (2) w(σµ)=σµ, w WL,L ∗ | ∈ ∈ M1 Π(M1):= πλ π Πdisc(M),λ i(aM ) . M M∈L 1(M0)

Note again that Πdisc(M) is not the set of discrete automorphic representations of M. ∈ ∈ IM For πλ Π(M1), π JH( Q (σµ)) being as above, deﬁne r (Q ,π,λ,Λ) M1 L M1 M1 Q 1 a (πλ):=adisc(σ)rL (πλ),rL (πλ) := lim . Λ→0 θP1 (Λ) Q∈F(L) Q Q⊂P1

M1 Here rL (πλ) is independent of Q1 and P1.

Corollary 5.10 ([12]). For f ∈H(G/AG), we have |W M | M J J(f)= | | a (π) M (π,f) dπ. W Π(M) M∈L(M0)

Here we have written JM (π,f):=JM (π,0,f) and the measure dπ on Π(M) is such that |W L| φ(π) dπ = M φ(πλ) dλ |W | M ∗ Π(M) M i( a ) L∈L (M0) π∈Πdisc(L) L

holds.

M Proof. Write Rdisc for the right regular representation of M on the discrete spectrum 2 \ IG M Ldisc(M(F )AM M)ofM. We may consider the induced representation P (Rdisc,λ), which G

from Th. 2.4 is isomorphic to 1 I (π ). Th. 5.9 asserts that

X λ ∈ X ∈ X (G) π Π(M ) P,

|W M | 1 J(f)= |W | | det(w − 1|aL )| ∈L L L,reg M L (M0) M∈L (M0) w∈W M,M × 1 M IG M tr L(P,λ,0)MP |P (w, 0) P (Rdisc,λ,f) dλ. |P | G ∗

(M) i( a ) P ∈P(M) L

Here ML(P,λ,0), MP |P (w, 0) are deﬁned similarly as ML(P,π,λ,0), MP |P (w, π)withπ M replaced with Rdisc. M IG M One can easily see that the operator L(P,λ,0)MP |P (w, 0) P (Rdisc,λ,f)vanisheson IG ∈P the orthogonal complement of a subspace which is a direct sum of Q (πλ), (Q (L), ∈ IG M π Πdisc(L)). On the Q (πλ)-component, L(P,λ,0) equals 1 M N L(P,π,λ,0) = G rQ(P,π,λ,Λ) Q(P,π,λ,Λ) . θ (Λ) Λ=0 Q∈P(L) Q

39 P1 ∈P Noting that rL (P,π,λ,Λ) (cf. Lem. 4.3) depends only on M1 and not on P1 (M1), we can write this as M1 N M1 N rL (P,π,λ,Λ) M1 (P,π,λ,Λ) = rL (πλ) M1 (P,π,λ). Λ=0 M1∈L(L)

L This combined with the deﬁnition of adisc(σ) yields |W L| L M1 N IG J(f)= adisc(π)rL (πλ)tr M1 (Q, π, λ) Q (πλ,f) dλ |W | G ∗ i( a ) L∈L(M0) M1∈L(L) π∈Πdisc(L) L decomposing i(aG)∗ = i(aM1 )∗ ⊕ i(aG )∗, L L M1 |W L| L M1 J = adisc(π)rL (πλ) M1 (πλ,f) dλ

| | M1 ∗ W a ∈L M ∈ i( ) M1 (M0) L∈L 1 (M0) π Πdisc(L) L writing M for M1 and using the deﬁnition of the measure dπ, |W M | M J = | | a (π) M (π,f) dπ. W Π(M) M∈L(M0)

6The invariant trace formula

Recall, in many application of the trace formula, we need to compare the trace formulae of diﬀerent groups. The starting point of the comparison is a correspondence between the conjugacy classes of the relevant groups. Consequently, we need to express the trace formula in terms of invariant distributions, distributions which are invariant under the conjugation. Here we shall explain how Arthur achieved this [4], [11], [12].

6.1 Non-invariance

First we measure the non-invariance of the terms in the trace formula. Recall the geomet-

ric kernel Kf (x, y)= Kf (x, y) of the induced operator R (f)onL2(VL(F )A \G)

Ó Q G ∈ Ç Q Ó (G) Q, −1

§ 3.1. Since KAd(y )f (x, x)=Kf (xy−1,xy−1), we have Ó Q,Ó Q, G T − aL −

Ad(y)J (f)= ( 1) K Ó (δx,δx)τ (H (δxy) T ) dx Ó Q, Q Q \ G(F ) AG G Q∈F(P0) δ∈Q(F )\G(F )

40 using (4.1), aM − L M − = ( 1) KQ,Ó (δx,δx)τQ (HQ(δx) T ) \ G(F ) AG G Q⊂P ∈F(P0) δ∈Q(F )\G(F ) × G − − ΓP (HQ(δx) T, HQ(kQ(δx)y)) dx aM − L M − = ( 1) KQ,Ó (δx,δx)τQ (HQ(δx) T ) \ P (F ) AG G P ∈F(P0) Q∈F(P0) δ∈Q(F )\P (F ) Q⊂P × G − − ΓP (HP (δx) T, HP (kQ(δx)y)) dx.

Here we have written δx = q(δx)kQ(δx) for the Iwasawa decomposition of δx with respect ∈ \ ∈ ∈ ∈ G ∈ to G = QK. If we write x P (F )AG G as x = umak,(u U, m M, a AM , k K), then

k f fP K (δx,δx)=K M M (δma,δma),HM (δx)=H M (δm),

Ó Q Q Q, Q , Ó G − − G − − ΓP (HP (δx) T, HP (kQ(δx)y)) = ΓP (HP (a) T, HP (ky)), where − k ρP 1 ∈ fP (m):=m f(k muk) du, m M. U

Using the function f Q,g (see p. 24), we obtain the following. (The proof for the spectral formula is similar.)

Lemma 6.1 ([4] Th. 3.2). For f ∈H(G/AG), we have

| L| W L

Ad(y)JÓ (f)= J (f ), |W | Ó Q,y Q∈F(M0) | L| W L

Ad(y)JX (f)= J (f ). |W | X Q,y Q∈F(M0)

We now explain the rough idea of the combinatorial part of the construction. Suppose M H →I we are given a family of continuous linear maps φL : (M) (L) satisfying M −1 M1 (1) φ (Ad(y )f)= M φ (f ), L P1∈F (L) L P1,y M H →I (2) φM : (M) (M) is surjective, M H M M M ◦ (3) Any (Ad(M)-) invariant distribution I on (M) passes through φM : I = I M φM .

41 M M

Then we deﬁne a family of distributions {I ,I } ∈L by X Ó M (M0)

|W L|

I M (f M ):=J M (f M ) − IL(φM (f M )),

Ó Ó Ó |W M | L M L∈L (M0),=M |W L|

I M (f M ):=J M (f M ) − IL(φM (f M )).

X X X |W M | L M L∈L (M0),=M

Then it follows from Lem. 6.1 and (1) that (• is o or X)

| L| −1 −1 W L G −1 I•(Ad(y )f − f)=J•(Ad(y )f − f) − I (φ (Ad(y )f − f)) |W | • L L∈L(M0),=G |W M | |W L| M − L M = | | J• (f P,y) | | I• (φL (f P,y)) ∈F W ∈L W ∈F P (M0), =G  L (M0),=G P (L)  |W M | |W L| = J M (f ) − IL(φM (f )) |W | • P,y |W M | • L P,y M P ∈F(M0), =G L∈L (M0) =0.

M That is, I• are all invariant distributions. On the other hand we deduce from (3.7) the

invariant trace formula X

I Ó (f)= I (f).

∈ Ç X∈ X Ó (G) (G)

6.2 Application of the trace Paley-Wiener theorem

M Of course the most diﬃcult point is to construct the maps φL . Arthur used the distri- J § G bution M (π,X,f)( 5.4) for this. In fact, he deﬁned the “Fourier transform” φM (f)of f ∈H(GS )by

G × −→ J ∈ C φM (f):Πtemp(MS ) aM,S (π,X) M (π,X,f) .

Here Πtemp(MS) denotes the subset of tempered elements in Πadm(MS ). Of course this can be extended to Πadm(MS) × aM,S by analytic continuation. It was shown in [13, I. Lem. 6.2] that this satisﬁes (1) above: −1 L JM (π,X,Ad(y )f)= JM (π,X,f Q,y). Q∈F(M)

Here we overlook the technical imprecision that H(GS ) is not stable under Ad(GS). The I H G image (GS)of (GS ) under φG is described by the trace Paley-Wiener theorem [19], [16]. I G H + The next problem is that the image (MS) does not contain φM ( (GS )) if G = M. Then Arthur enlarged the range a little by relaxing the support condition in the direction of the center to obtain Iac(MS). To assure the surjectivity, he also enlarged the domain

42 H M H → to a little larger space ac(MS ). Finally we have the surjective maps φL : ac(MS ) Iac(LS) for any L ⊂ M ∈L(M0).

The ﬁnal problem is to show that the distributions I G, I G pass through φG in order Ó X G

to deﬁne IG, IG. More precisely, we need to establish the following inductive statement.

Ó X

∈L + M Problem 6.2. Suppose for any M (M0), = G, we are given the distributions IL (γ), M H M M M IL (π) on (M) which pass through φM , so that we can deﬁne IL (γ), IL (π), and satisfy J M (π,fM )=I M(π,fM )+ IM1 (γ,φM (f M )), L L L M1 ∈LM M1 (M0), =M M M M M M1 M M JL (γ,f )=IL (γ,f )+ IL (π,φM1(f )). M M1∈L (M0), =M

Then the distributions G G − M G IL (γ,f):=JL (γ,f) IL (γ,φM (f)), ∈L M (M0), =G G J G − M G IL (π,f):= L (π,f) IL (π,φM(f))

M∈L(M0), =G

G pass through φG. This was done at length in [11], [12]. We end this note by stating the resulting formula.

Theorem 6.3 (The invariant trace formula). If we take a ﬁnite set of places S suf- ﬁciently large for f ∈H(G/AG), then we have

| M | W M | | a (S, γ)IM (γ,f) ∈L W ∈ M (M0) γ (M(F ))M,S | M | W M = | | a (π)IM(π,f) dπ W Π(M) M∈L(M0)

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46 Index ∩ P (M(F ) o)M,S,25 vL (x), 23 (M, S)-equivalence, 25 vS (ky, T), 22 G AG, AG, aG,4 W = W the relative Weyl group, 5

AP,π,X (λ, λ ), 30 WM,M ,8 M T YT a (S, γ), 26 YP (x, y), P (x, y), 21 M a (π), 39 YQ(T ), 32 G A A A A aM ,6 , S , ∞, f ,4 G A A adisc(π), 38 G := G( ), GS := G( S), 4 ∞ 1 Cc (G/AG), 5 G ,4 cQ(λ), “diﬀerence” of cP (λ), 22 K = v Kv,5 P1 cQ(T,Λ), 32 χM (T,X), 33 G ∨ ∨ D (γ), 23 ∆0,∆P ,∆0 ,∆P , ∆P ,12 w w dQ(Λ), dQ(λLw , Λ), 32 A0(T ), 6 ∨ EP (x, φλ), 8 aM,S, aM,S,38 γ P G , Gγ ,3 JQ(δx,T), 13 P HG,5 KQ(x, T ), 13 HP ,5 O(G), 12 G G IL (γ,f), IL (π,f), 43 S(T0) a Siegel domain, 6 JM (γ,f), 23 X(G), 7 P JM (πλ,f), 38 ΓQ(X, Y ), 17

T h J (f), 13 E W ( C ) Ó (h) ,28 J T (f), 16 F F X (M), (P0), 5 jT (x, f), 20 Ó Hac(MS ), 43

KP,Ó (x, y), 12 I(MS), Iac(MS ), 43 G KP,X (x, y), 15 I (π ), 27 P,X λ G

KÓ (x, y), 12 I P (ρλ), 8

KX (x, y), 11 JM (π,f), 39 T k (x, f), 12 JM (π,X,f), 38 kT (x, f), 15 L L X (M), (P0), 5 2 2 L -cusp form, L (G(F )AG\G), 6 M MT 0 Q(P,π,λ,Λ), Q(P,π,λ,Λ), 34 2 \ L (G(F )AG G), 5 NM (P,π,λ), 38 2 \ L (G(F )AG G)[π],10 NQ(P,π,λ,Λ), 37 2 \ L (G(F )AG G)X ,7 P(M), 5 2 \ T L0(M(F )AM M)ρ,7 Ω (P,λ), 26 π,X M(w, ρλ) intertwining operator, 8 f (), 25 T Q,g

P (B), 30 ωT (λ, λ ,φ,φ ), 27 F P(M,ρ),8 T ΦS,a,k,y(m), 21 T T pλ (H), pλ (β), 29 ΦS,y,T (m), 22 R = RG,5 Π(G)=Π(G(A)), 2 rQ(P,π,λ,Λ), 37 Πadm(M), 35 T1 analytic origin, 17 Πdisc(M), Π(M), 39

T uQ(k, y), 25 Ψ (λ, f), 29 X,π

47 G φM ,42 φP the constant term along P ,6

∈ F φλ for φ P(M,ρ),8 T ψλ (H), 29 ψT (X, f), 29 X,π P σQ,13 τP , τP ,12 P P θQ(λ), θQ(λ), 22 θφ(g), 7 ∧T , truncation operator, 15

F P(M,ρ),7 ∨ ∆P ,22 {vQ(λ, x)}Q∈P(L),23 A0(G(F )AG\G) the space fo cusp forms, 6 A0(M(F )AM \M)ρ,7 A2(M(F )AM \M)π,9 A \

2(UM(F )AM G)π,X ,10 F A2(UM(F )AM \G) ,10 π,X H(G/AG)theHeckealgebra,5 cuspidal datum, 7 cuspidal pair, 7

disrete pair, datum, 9

Eisenstein series (cuspidal), 8

geometric side, 3

Hilbert-Schmidt class, 2

spectral side, 3

trace class, 3

weighted character, 38