数理解析研究所講究録
1173 巻 2000 年 243-290

243

A Survey on the Arthur-Selberg Trace Formula*

Takuya KONNO \dag er

Contents
1

Introduction22Spectral decomposition. .
2.1 Siegel domains. . .
.
2.2 Cusp forms
2.3 Decomposition by cuspidal data
. .
. .
2.4 Cuspidal Eisenstein series
2.5 Residual Eisenstein series and the spectral decomposition.
2.6 Spectral kernel

4
5
6

7
7
9
10

3 Convergence
3.1 Ikuncated kernel
3.2 Truncation operator and the basic identity

12
12

4 The fine -expansion.
4.1 $J^{T}(f)$ as a polynomial
decomposition
4.2 Reduction by the Jordan
4.3 Weighted orbital integrals.4.4 The fine -expansion.

17
17

5 Fine -expansion
5.1 An application of the Paley-Wiener theorem. .
. .
5.2 Logarithmic derivatives.
5.3 Normalization and estimation of intertwining operators
5.4 Weighted characters

26

15$\mathrm{O}$

20
22

25

$\mathrm{J}\supset$$\chi$

.

6

27
31
35
38

$\cdot$

The invariant trace formula
6.1 Non-invariance
6.2 Application of the trace Paley-Wiener theorem.

.

.

40
40
42

*Talk at the symposium “Automorphic Forms and Representations on Algebraic Groups and Automorphic -functions, 29 June, 2000.
\dagger Graduate School of Math., Kyushu University
812-8581, Hakozaki 6-10-1, Higashi-ku, Fukuoka city, Japan
$L$

1

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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 theoriginal Selberg trace formula and give some words about arithmetic backgrounds.
of a semisimple Lie
The Selberg trace formula was originally proved for a pairgroup 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.
for its ring of ad\’eles.
denotes theThus let $F$ be a number field and write$G$id\’ele norm on. For a connected semisimple groupover $F$ , its
and setis a locally compact topological group, in which the group
group of adelic points$G(F)$ of $F$ -rational points is a discrete subgroup. We assume that $G$ is anisotropic over$F$ , that is, $G(F)$ contains only semisimple elements. Thenis compact by the
result of [18].
$R$ denotes the right regular representation of.
on the space
which isfor the space of functions with compact supports on
Writesmooth in the archimedean components and locally constant in the non-archimedean, the operator
components. For$(\mathrm{G}, \Gamma)$$\mathrm{A}=\mathrm{A}_{F}$$\mathrm{A}^{\cross}$$||_{\mathrm{A}}$$\mathrm{A}^{1}:=\mathrm{K}\mathrm{e}\mathrm{r}||_{\mathrm{A}}$$G(\mathrm{A})$$G(F)\backslash G(\mathrm{A})$$L^{2}(G(F)\backslash G(\mathrm{A}))$$G(\mathrm{A})$$G(\mathrm{A})$$C_{c}^{\infty}(G(\mathrm{A}))$$f\in C_{c}^{\infty}(G(\mathrm{A}))$$[R(f) \phi](x):=\int_{G(\mathrm{A})}f(y)\phi(xy)dy=\int_{G(\mathrm{A})}f(x^{-1}y)\phi(y)dy$$= \int_{G(F)\backslash G(\mathrm{A})}\sum_{\gamma\in G(F)}f(x^{-1}\gamma y)\phi(y)dy$

is an integral operator with the kernel$K(x, y):= \sum_{\gamma\in G(F)}f(x^{-1}\gamma y)$.

is compact, this operator is of Hilbert-Schmidt class (i.e. $K(x, x)$
Since. In particular one can show that the representation
is square integrable on
decomposes into a direct sum of irreducible representations where each irreducible
representation occurs with Xinite multiplicity:$G(F)\backslash G(\mathrm{A})$$G(F)\backslash G(\mathrm{A}))$$R$$R= \bigoplus_{\pi\in\Pi(G(\mathrm{A}))}\pi^{\oplus m(\pi)}$Here,denotes the set of isomorphism classes of irreducible unitary representa. Moreover an argument of Duflo-Labesse [21, I. 1.11] shows that $R(f)$ is of$\Pi(G(\mathrm{A}))$

tions of$G(\mathrm{A})$

(1.1)

2

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trace class. That is, it admits a trace given by the integral of $K(x, y)$ on the diagonal:$\mathrm{t}\mathrm{r}R(f)=\int_{G(F)\backslash G(\mathrm{A})}K(x, x)dx=\sum_{\{\gamma\}\in \mathfrak{O}(G)}\sum_{\delta\in G^{\gamma}(F)\backslash G(F)}\int_{G(F)\backslash G(\mathrm{A})}f(x^{-1}\delta^{-1}\gamma\delta x)dx$$= \sum_{\{\gamma\}\in \mathfrak{O}(G)}\frac{1}{[G^{\gamma}(F).G_{\gamma}(F)]}.\int_{G_{\gamma}(\mathrm{A})\backslash G(\mathrm{A})}\int_{G_{\gamma}(F)\backslash G_{\gamma}(\mathrm{A})}f(x^{-1}\gamma x)dydx$$= \sum_{\{\gamma\}\in 1\supset(G)}a^{G}(\gamma)I_{G}(\gamma, f)$.

Hereis the set of (semisimple) conjugacy classes in $G(F),$centralizer of in $G,$
is its identity component, and$\mathrm{O}(G)$$G^{\gamma}:=\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}(\gamma, G)$

is the$G_{\gamma}:=\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}(\gamma, G)^{0}$$\gamma$$a^{G}( \gamma):=\frac{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(G_{\gamma}(F)\backslash G_{\gamma}(\mathrm{A}))}{[G^{\gamma}(F).G_{\gamma}(F)]}.$’$I_{G}( \gamma, f):=\int_{G_{\gamma}(\mathrm{A})\backslash G(\mathrm{A})}f(x^{-1}\gamma x)dx$.

This combined with (1.1) yields the Selberg trace formula:$\sum_{\{\gamma\}\in \mathfrak{O}(G)}a^{G}(\gamma)I_{G}(\gamma)f)=\sum_{\pi\in\Pi(G(\mathrm{A}))}a^{G}(\pi)I_{G}(\pi, f)$

,

(1.2)

where $a^{G}(\pi):=m(\pi)$ andis the distribution character of . The left
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 ingredientsof the Langlands program. Perhaps it will be helpful to explain the program briefly.
It is a program to understand the deep relationship between automorp. hic forms andGalois representations and motives. Two main processes are$I_{G}(\pi, f):=\mathrm{t}\mathrm{r}\pi(f)$$\pi$(1) To describe the automorphic representations of reductive groups by means of theirassociated automorphic -functions (or their Langlands parameters);$L$(2) To construct the correspondence between automorphic representations of arithmetic
type and -adic representations of Galois groups of the field of definition of suchforms. This correspondence should be characterized by the equality between automorphic and Artin L-functions.
$\ell$

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 -functions are the Rankin product -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
. This is suggested by the experience in
the $GL(n)$ case [24], [15]. Then relate the automorphic representation ofwith thoseof some $GL(n)$ . In [14, \S 8] the detailed framework of the second part for classical $G$
are explained. We need to compare the trace formula for $G$ with that ofin the first
$L$

$L$

$G^{*}$$G^{*}$$G^{*}$

3

246
problem, while a twisted trace formula for $GL(n)$ need to be compared with the ordinary
one forin the second.
The problem (2) contains so many aspects that we cannot explain them in any detailhere. But the starting point is to construct the Galois representations associated toan arithmetic automorphic representations in the -adic cohomology of certain Shimuravariety. Here the trace formula with special geometric test functions will be comparedwith 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 -automorphic spectrum,
the normalization of intertwining operators by -functions and the precise analytic properties of those -functions must be established. The analytic trace formula alone yields$G^{*}$$p$$L^{2}$$L$$L$

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 thearticle of Hiraga in this volume. The contents of this note is as follows. We start with abrief review of Langlands’ theory of spectral decomposition of the automorphic spectrum
by Eisenstein series \S 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 andthe other spectral. \S 3 explains the construction of [2] and [3]. We define the truncatedkernel and prove that its integral over the diagonal converges. We obtain the coarse trace
formula. \S 4 is devoted to the fine -expansion. We write the geometric terms in termsof the unipotent terms of certain reductive subgroups of . Then they are expressedby means of weighted orbital integrals. The spectral counter part, the fine X-expansion,is explained in \S 5. This is the heart of this note, because it is the most importantpart in applications. We use the fact that the coarse -expansion is a polynomial in thetruncation parameterto deduce the precise expression of the -expansion from theasymptotic formula of the inner product of truncated Eisenstein series. Finally in \S 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 manyarguments, and also cannot refer tothe works of Osborne-Warner [34]. Instead we add some examples in the simplest case
$G=GL(2)$ with some figures.
$\mathfrak{O}$

$G$

$X$

$T$

$X$

$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}/\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$2Spectral decomposition

We begin with some notation. For a finite set of places of $F$ , we write
A $|a_{v}=0,$}. The infinite and finite components of A are denoted by,andrespectively.
Let $G$ be a connected reductive group over $F$ . For brevity, we write,, etc. Let
be the maximal $F$ -split torus in the center $Z(G)$ of $G$ , while
the maximal -vector subgroup in the centerofis denoted by. Writefor its Lie algebra. We have a direct product decomposition
such that$S$$\mathrm{A}_{S}:=\{(a_{v})_{v}\in$$\forall v\not\in S$$\mathrm{A}_{\infty}$$\mathrm{A}_{f}$$\mathrm{G}:=G(\mathrm{A})$$\mathrm{G}_{S}:=G(\mathrm{A}_{S})$$A_{G}$$\mathbb{R}$$Z(\mathrm{G}_{\infty})$$\mathrm{G}_{\infty}$$\mathfrak{U}_{G}$$\mathrm{G}=\mathrm{G}^{1}\cross \mathfrak{U}_{G}$$|\chi(ag)|_{\mathrm{A}}=|\chi(a)|_{\mathrm{A}}$,4$a\in \mathfrak{U}_{G},$$g\in \mathrm{G}^{1}$$a_{G}$

247
holds for any $F$ -rational character of$H_{G}$

:

$G$

. Set

$\mathrm{G}=\mathrm{G}^{1}\cross \mathfrak{U}_{G}arrow \mathfrak{U}_{G}arrow a_{G}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{l}\mathrm{o}\mathrm{g}$.

for the set of
For a parabolic subgroup $P=MU$ , we write $F(M),$ $P(M)$ andparabolic subgroups containing $M$ , the set of parabolic subgroups having $M$ as a Levifactor and the set of Levi subgroups containing $M$ . We fix a minimal parabolic subgroup$P_{0}=M_{0}U_{0}$ .
denotes the set of parabolic subgroups containing
and(the set of standard
containing
and the set of Levi components of elements ofparabolic and Levi subgroups).
ofsuch that the Iwasawa decompoFix a maximal compact subgroup
Using
.
this
we
extend the map
:
any
holds
for
sitionto$\mathcal{L}(M)$$P_{0}$$\mathcal{L}(P_{0})$$\mathcal{F}(P_{0})$$M_{0}$$\mathcal{F}(P_{0})$$\mathrm{K}=\prod_{v}\mathrm{K}_{v}$$\mathrm{G}$$H_{M}$$P\in \mathcal{F}(M_{0})$$\mathrm{G}=\mathrm{P}\mathrm{K}$$H_{P}$$\mathrm{M}arrow a_{M}$: $\mathrm{G}=\mathrm{U}\mathrm{M}\mathrm{K}\ni umk-H_{M}(m)\in a_{M}$ .

denotes the (relative) Weyl group Norm $(A_{0}, G)/M_{0}$ of $A_{0}=A_{M_{0}}$ in $G$ ,where Norm $(H, G)$ means the normalizer of a subgroup $H$ in a group $G$ . We identify $W$
with a fixed system of representatives in Norm $(A_{0}, G)$ . $W$ acts by conjugation on $F(M_{0})$and the associated objects. We write the action $w:P-\rangle$ $w(P)=^{w}P:=wPw^{-1}$ etc.
$W=W^{G}$2.1

Siegel domains

Fixing an invariant measure on
space$\mathrm{G}$and a Lebesgue measure on$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G}):=\{\mathrm{m}\mathrm{e}.\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\phi.\mathrm{G}arrow \mathbb{C}$as in the anisotropic case. Let
are compactly supported modulo$|(\mathrm{i}\mathrm{i})(\mathrm{i})$$\mathfrak{U}_{G}$$\phi(\gamma ag)=\phi(g),\forall a\in \mathfrak{U}_{G},\gamma\in G(F)\int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}|\phi(g)|^{2}dg<+\infty\}$which
for the space of smooth functions on. Similar calculation as in the anisotropic case shows$\mathrm{G}$$C_{C}^{\infty}(\mathrm{G}/\mathfrak{U}_{G})$$\mathfrak{U}_{G}$, we can consider thethat, the operator$[R(f) \phi](x):=\int_{\mathfrak{U}_{G}\backslash \mathrm{G}}f(y)\phi(xy)dy$on$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$is an integral operator with the kernel$K(x, y):= \sum_{\gamma\in G(F)}f(x^{-1}\gamma y)$(2.1)

of elements which are. Later we shall use the subspaceforfunctions.
space
of
test
as
the
in
-finite on both sideson
The spectral decomposition of the right regular representation $R=R_{G}$ ofis more complicated than in the anisotropic case, because
is no longer compact. The form of the non-compactness is described as follows.
$\mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$$f\in C_{C}^{\infty}(\mathrm{G}/\mathfrak{U}_{G})$$\mathrm{K}$$C_{c}^{\infty}(\mathrm{G}/\mathfrak{U}_{G})$$\mathrm{G}$$G(F)\mathfrak{U}_{G}\backslash \mathrm{G}=$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$G(F)\backslash \mathrm{G}^{1}$5

248
is anisotropic modulo center by definition, we can choose a compact subset
Since
. As
is a multiple extension of additive groups,
satisfyingof. For $T\in a_{0}$ , we set
such that
there exists a compact subset$M_{0}$$\mathrm{M}_{0}^{1}=M_{0}(F)\omega_{1}$$\mathrm{M}_{0}^{1}$$\omega_{1}$$U_{0}$$\mathrm{U}_{0}=U_{0}(F)\omega_{2}$$\omega_{2}\subset \mathrm{U}_{0}$$\mathfrak{U}_{0}(T)$, respectively, andaswhere we have abbreviated$A_{0}:=A_{M_{0}}$, considered as a subset of .
inroots of$\mathfrak{U}_{M_{0}},$$H_{P_{0}}$$\mathfrak{U}_{0},$$H_{0}$$P_{0}$$\triangle$

is the set of simple$\alpha_{0}^{*}$

Proposition 2.1 ([23]). We can choose$\mathrm{G}=G(F)\mathfrak{S}(T_{0})$,sufficiently negative so that

$T_{0}\in\alpha_{0}$$\mathfrak{S}(T_{0}):=\omega_{2}\omega_{1}\mathfrak{U}_{0}(T_{0})\mathrm{K}$Thus the non-compactness comes from that of

2.2,$:=\{a\in \mathfrak{U}_{0}|\alpha(H_{0}(a)-T)>0, \forall\alpha\in\triangle\}$$\mathfrak{U}_{0}(T_{0})$, a shifted positive cone.

Cusp forms

For an $F$ -parabolic subgroup $P=MU$ of $G$ and a measurable function
we can define its constant term along $P$ by$\phi_{P}(g):=\int_{U(F)\backslash \mathrm{U}}\phi(ug)du$$\phi$

on$U(F)\backslash \mathrm{G}$,.

-cusp forms
The space ofconsists ofsuch thatvanishes almost everywhere for any. Of course this is not the space of cusp
forms
in the usual sense [33, Chapt. I], but
is a dense.
subspace of
The following lemma is fundamental in our estimationarguments.
$L^{2}$$\phi\in L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$P\in \mathcal{F}(P_{0})$$\phi_{P}$$A_{0}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$A_{0}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$Lemma 2.2. Suppose:
is slowly increasing and sufficiently smooth
relative to $\dim U_{0}$ . Then the alternating sum$G(F)\backslash \mathrm{G}arrow \mathbb{C}$$\phi$$c \phi(g):=\sum_{P=MU\in \mathcal{F}(P_{0})}(-1)^{a_{M}^{G}}\phi_{P}(g)$is rapidly decreasing. Moreover extends to a projection onstriction tois the identity. Here$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$c$$a_{M}^{G}:=\dim\alpha_{M}/\alpha_{G}$$L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$whose re-.

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$R_{0}(f)$ of $R(f)$ tois $c_{y}K(x, y)$ ( means the operator applied in )$R_{0}(f)$which is rapidly decreasing. Hence
is of Hilbert-Schmidt class anddecomposes
discretely. Moreover each irreducible component has finite multiplicity:$L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$c$$c_{y}$$R_{0}$$L_{0}^{2}(G(F) \mathfrak{U}_{G}\backslash \mathrm{G})=\bigoplus_{\pi\in\Pi(\mathrm{G})}\pi^{\oplus m_{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}(\pi)}$

6

.

$y$

249
Decomposition by cuspidal data

2.3

and an irreducible componentof
A pair $(M, \rho)$ ofis.
cuspidal
for
write
for
the
-isotypic
subspacepairWecalled a.
for its intersection withandin-module of the unitary representation
This is the underlying admissible$G(F)$-conjugacy class of cuspidal pairs is a cuspidal datum for .
. A
We write $X(G)$ for the set of cuspidal data for .
Fix a cuspidal pair $(M, \rho)\in X\in X(G)$ and a finite set of -types . Writeforsuch thatthe space of smooth functions$M\in \mathcal{L}(P_{0})$$G$$L_{0}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$L_{0}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$\rho$$L_{0}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$$\rho$$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$$(\mathrm{g}_{\infty}, \mathrm{K}_{\infty})\cross \mathrm{G}_{f}$$G$$L_{0}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$$G$$\mathrm{K}$$\hat{P}_{(M,\rho)}^{\mathfrak{F}}$$\mathfrak{F}$$\phi:\mathrm{U}M(F)\mathfrak{U}_{G}\backslash \mathrm{G}\wedgearrow \mathbb{C}$(1)$M(F)\mathfrak{U}_{M}\backslash \mathrm{M}\ni mrightarrow\phi(mg)\wedge\in \mathbb{C}$(2)$\mathfrak{U}_{M}/\mathfrak{U}_{G}\ni a-\phi(ag)\wedge\in \mathbb{C}$(3)$\mathrm{K}\ni k\mapsto\phi(gk)\wedge\in \mathbb{C}$in { for any$g\in \mathrm{G}$$\zeta$Noting that$\phi\wedge\in\hat{P}_{(M,\rho)}^{\mathrm{i}\mathrm{f}}$Lemma 2.3. (i) For

belongs to$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$

is compactly supported for any

for any$g\in \mathrm{G}$$g\in \mathrm{G}$;;belongs to the linear span of the matrix coefficients of K-types.

is rapidly decreasing, we deduce the following.
$\phi\wedge\in\hat{P}_{(M,\rho)}^{\mathfrak{F}}$,$\theta_{\phi}(g):=\sum_{\gamma\in P(F)\backslash G(F)}\phi(\gamma g)\wedge$.
converges absolutely and belongs to(ii) If we write
for the closure of the span of, then we have the orthogonal decomposition$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{X}$$\bigcup_{\mathfrak{F}}\bigcup_{(M,\rho)\in X}\{\theta_{\phi}|\phi\in$$\hat{P}_{(M,\rho)}^{\mathfrak{F}}\}$$\oplus$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})=$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{X}$.

$X\in X(G)$2.4

Cuspidal Eisenstein series

Next we set(1)$P_{(M,\rho)}^{\mathrm{i}Y}$

for the space of functions$(\alpha_{M}^{G})_{\mathbb{C}}^{*}\ni\lambda\mapsto\phi(\lambda;g)\in \mathbb{C}$(2)$\phi$:$(a_{M}^{G})_{\mathbb{C}}^{*}\cross \mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G}arrow \mathbb{C}$is of Paley-Wiener type for any$\phi(\lambda;mk)\in \mathbb{C}$$k\in \mathrm{K};m-,$belongs to$g\in \mathrm{M}^{1}\mathrm{K}$satisfying;$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$

for any$\lambda\in$$M(F)\mathfrak{U}_{M}\backslash \mathrm{M}\ni(a_{M}^{G})_{\mathbb{C}}^{*},$(3) $\mathrm{K}\ni k-\phi(\lambda;mk)\in \mathbb{C}$ belongs to the linear span of the matrix coefficients of.
-types in for any$\mathrm{K}$$\mathrm{i}Y$$\lambda\in(a_{M}^{G})_{\mathbb{C}}^{*},$$m\in \mathrm{M}^{1}$7

250
Obviously, any$\phi\wedge\in\hat{P}_{(M,\rho)}^{\mathfrak{F}}$is the Fourier transform(2.2)$\phi(uamk)\wedge:=\int_{i(\alpha_{M}^{G})^{*}}\phi(\lambda;mk)a^{\lambda}d\lambda$

of somesection”$\phi\in P_{(M,\rho)}^{\mathfrak{F}}$. On the other hand, associated to each$\phi\in P_{(M,\rho)}^{\mathfrak{F}}$is a$‘(\mathrm{P}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{y}$-Wiener$(\alpha_{M}^{G})_{\mathbb{C}}^{*}\ni\lambda\mapsto[g\mapsto\phi_{\lambda}(g):=e^{\langle\lambda+\rho_{P},H_{P}(g)\rangle}\phi(\lambda;g)]\in \mathcal{I}_{P}^{G}(\rho_{\lambda})$of the bundle of induced representations
in $P$ .
denotes the half of the sum of positive roots of, the associated cuspidal Eisenstein series is defined byFor$\mathcal{I}_{P}^{G}(\rho_{\lambda}):=\mathrm{i}\mathrm{n}\mathrm{d}_{\mathrm{P}}^{\mathrm{G}}[(e^{\lambda}\otimes\rho)\otimes 1_{\mathrm{U}}]arrow\lambda$

.

Here

$A_{M}$$\rho_{P}\in a_{M}^{*}$$\phi\in P_{(M,\rho)}^{\mathfrak{F}}$$E_{P}(x, \phi_{\lambda}):=\sum_{\delta\in P(F)\backslash G(F)}\phi_{\lambda}(\delta x)$

.

(2.3)

$P=MU,$ $P’=M’U’\in \mathcal{F}(P_{0})$ are said to be associated if the set $W(M, M’):=\{w\in$$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 $W(M, M’)$ is given by$W_{M,M’}:=\{w\in W(M, M’)|w(P_{0}^{M})\subset P_{0}\}$ . For $w\in W_{M,M’}$ we define the intertwining
operator by the integral$[M(w, \rho_{\lambda})\phi](x):=\int_{(\mathrm{U}’\cap w(\mathrm{U}))\backslash \mathrm{U}’}\phi_{\lambda}(w^{-1}ux)du\cdot e^{-\langle w(\lambda)+\rho_{P’},H_{P’}(x)\rangle}$.

Using the theory of resolvent, Langlands established the following properties [31], [33,
Chapt. II, IV].
andabsolutely converge for ${\rm Re}(\lambda)>>0$ . At
(1) Convergence.
, anddefines and automorphic form on
such. Moreover the following
defines an intertwining operator
holds.
$E_{P}(x, \phi_{\lambda})$$\lambda,$$M(w, \rho_{\lambda})\phi$$E_{P}(x, \phi_{\lambda})$$G(F)\mathfrak{U}_{G}\backslash \mathrm{G}$$\phi_{\lambda}\mapsto(M(w, \rho_{\lambda})\phi)_{w(\lambda)}$$\mathcal{I}_{P}^{G}(\rho_{\lambda})arrow \mathcal{I}_{P}^{G},(w(\rho_{\lambda}))$(i) Equivariance.

$E_{P}(x, \mathcal{I}_{P}^{G}(\rho_{\lambda}, f)\phi_{\lambda})=R(f)E_{P}(x, \phi_{\lambda})$(ii) Constant terms. The constant term of
given by

$E_{P}(x, \phi_{\lambda})$for any.
along $Q=LV\in \mathcal{F}(P_{0})$ is$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$$E_{P}(x, \phi_{\lambda})_{Q}=\sum_{w\in W_{M}(L)}E_{P_{w}^{L}}^{L}(x, (M(w, \rho_{\lambda})\phi)_{w(\lambda)})$

Here

and
denotes the unique element of
having $M_{w}:=w(M)$ as its Levi component.

$W_{M}(L):= \bigcup_{M\in \mathcal{L}^{L}(P_{0}^{L})},W_{M,M’}$$F^{L}(P_{0}^{L})$.

$P_{w}^{L}$(iii) Knctional equations.
for $w\in W_{M}(G)$ .
Also for $w\in W_{M,M’},$ $w’\in W_{M’,M’’}$ , we have $M(w’, w(\rho_{\lambda}))M(w, \rho_{\lambda})\phi=M(w’w, \rho_{\lambda})\phi$at where both operators converge absolutely.
$E_{P}(x, (M(w, \rho_{\lambda})\phi)_{w\lambda})=E_{P}(x, \phi_{\lambda})$$\lambda$(iv) Fourier transform. For$\lambda_{0}$whose real part is sufficiently positive, we have$\theta_{\phi}(uamk)=\int_{\lambda_{0}+i(\alpha_{M}^{G})^{*}}E_{P}(umk, \phi_{\lambda})a^{\lambda+\rho_{P}}d\lambda$

8

.

251
(2) Analytic continuation.
. The properties
wholefunctions.

$E_{P}(x, \phi_{\lambda})$$(a_{M}^{G})_{\mathbb{C}}^{*}$

andextend meromorphically to theof (1) still hold as equalities of meromorphic$M(w, \rho_{\lambda})\phi$$(\mathrm{i})-(\mathrm{i}\mathrm{v})$(hence those of $M(w,$(3) Singularities. The set of poles ofby (l-ii)) is alocallyoffinitecollectionofunion
affine hyperplanes whose vector parts are zeroesof some coroots.
$E_{P}(x, \phi_{\lambda})$$\rho_{\lambda})\phi$Residual Eisenstein series and the spectral decomposition

2.5

Essentially by the Perseval formula, we deduced from (ii), (iv) the
formula:$\sum$$\langle M(w, \rho_{\lambda})\phi, \phi’\rangle_{M}d\lambda$$\langle\theta_{\phi}, \theta_{\phi’}\rangle_{G}=\int_{\lambda_{0}+i(\alpha_{M}^{G})^{*}}w\in W_{M,M’}$$L^{2}$-inner product

.

(2.4)

denotes the hermitian inner product on. That is, the inner
Here ,
is the integral over the Pontrjagin dual of
product between twoof the Petersson
and . Certain residue analysis transforms (2.4) into
inner product of$\langle$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$\rangle_{G}$$\mathfrak{U}_{M}^{G}$$\theta_{\phi}’ \mathrm{s}$$M(w, \rho_{\lambda})\phi$$\phi’$$\langle\theta_{\phi}, \theta_{\phi’}\rangle=\tau\int_{i(a_{M}^{G})^{*}}\sum_{w\in W_{MM’}},\langle M(w, \rho_{\lambda})\phi, \phi’\rangle_{M}d\lambda$

(2.5)
.

$+ \sum_{6}\int_{o(\mathfrak{S})+i(\alpha_{M_{\mathrm{C}\backslash }}^{G})^{*}}\sum_{w\in W_{MM’}},{\rm Res}_{\mathfrak{S}}\langle M(w, \rho_{\lambda})\phi, \phi’\rangle_{M}d\lambda$

denotes a certain equivalence relation, the sum on the right runs over a finite setof singular hyperplanes ofandis a certain “origin” ofof intersections
equals the vector part of .
is such that.
means thebelongs to the discrete
iterated residue along . Noting that
and, at the same time, equals to the constant term ofspectrum ofcertain residual Eisenstein series, we arrive at the following theorem [31, Chapt. 7], [33,
Chapt. 6]$=\tau$$E_{P}(x, \phi_{\lambda})$$\mathfrak{S}$$\mathfrak{S}$$M_{\mathfrak{S}}\in \mathcal{L}(M_{0})$$o(\mathfrak{S})$$\mathfrak{S}$$i(\alpha_{M_{C\neg}}^{G})^{*}$$\mathfrak{S}$$\sum_{w\in W_{M,M}},$${\rm Res}_{\mathfrak{S}}$${\rm Res}_{\mathfrak{S}}M(w, \rho_{\lambda})\phi$$L^{2}(M_{6}(F)\mathfrak{U}_{M_{\mathrm{C}\backslash }}\backslash \mathrm{M}_{\mathfrak{S}})$and an irreducible subTheorem 2.4. We call a pair $(M, \pi)$ consisting of
discrete pair. A $G(F)$ -conjugacy class $[M, \pi]$ ofrepresentation of
discrete pairs is discrete datum. Write $[P]$ for the associated class of $P\in F(P_{0})$ .
in(1) For a discrete pair $(M, \pi)$ and a finite set of -types ff, we define the space
replaced bywiththethe same manner as
withintersection of the space of square-integrable automorphic forms. Then the residual Eisenstein series
the -isotypic subspacesatisfies the properties (1) (2) and (3) of 2.4 (But thisdefined by (2.3) withtime, the formula (l-ii) holds only for $Q\supset M.$ ).
(2) Letfor the Hilbert space of families of functions $F=\{F_{P’}\}_{P’\in[P]}$ such that
$M\in \mathcal{L}(P_{0})$$L^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$

$\pi$

$a$

$a$

$\mathrm{K}$$P_{(M,\pi_{J}^{\backslash }}^{\mathfrak{F}}$$P_{(M,\rho)}^{\mathrm{i}\mathrm{f}}$$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho}$$A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi f}$$A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$L^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi}$$\pi$$E_{P}(x, \phi_{\lambda})$$\phi\in P_{(M,\pi)}^{\mathfrak{F}}$$\hat{L}_{[M,\pi]}$(i)$F_{P’}$:$i(a_{M}^{G},)^{*}arrow L^{2}(\mathrm{U}’M’(F)\mathfrak{U}_{M’}\backslash \mathrm{M}’)_{\pi’}$$[M, \pi]$.

9is a measurable function. Here $(M’, \pi’)\in$252(ii) $F_{P’}(w(\lambda))=M(w, \pi_{\lambda})F_{P}(\lambda),$ $w\in W_{M,M’}$ .
(iii) The norm$||F||^{2}$$:=, \sum_{(M’,\pi)\in[M,\pi]}\frac{1}{|W_{M}(G)|}\int_{i(\alpha_{M’}^{G})^{*}}||F_{P’}(\lambda’)||_{M’}^{2}d\lambda’$is finite.
, whoseThen there exists a -equivariant unitary injection
restriction to the automorphic spectrum valued Paley-Wiener sections is given by$\hat{L}_{[M,\pi]}arrow L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$\mathrm{G}$$F-, \sum_{(M’,\pi)\in[M,\pi]}\frac{1}{|W_{M}(G)|}\int_{i(\alpha_{M’}^{G})^{*}}E_{P’}(x, F_{P’}(\lambda’))d\lambda’$

If we write$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]}$

position.

for the image of this map, we have the orthogonal decom-$L^{2}(G(F) \mathfrak{U}_{G}\backslash \mathrm{G})=\bigoplus_{[M,\pi]}L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]}$.

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 constructionof the trace formula much harder in the case when the $F$-rank is more than one.

2.6

Spectral kernel

Th. 2.4 allows us to deduce the spectral expression of the kernel $K(x, y)(2.1)$ .
Choose a discrete datum $[M, \pi]$ , a cuspidal datum $X\in X(G)$ and a finite set of K-types$S$ . Set,

$A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi,X}$$:=A_{2}(M(F) \mathfrak{U}_{M}\backslash \mathrm{M})_{\pi}\cap\bigoplus_{X^{M}\in X(M)\cap X}L^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{X^{M}}$for the subspace of functions which transform under
and write,
the set of irreducible components of the restrictions toof elements in { , un. This is a finite dimensional subspace of, so that we can choose ader. Similarly we have an orthoortho-normal basisof it satisfying
ifnormal basisof the induced space.
Associated to eachis the ‘constant section”$S^{M}$$A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi,X}^{\mathrm{i}\zeta^{M}}$$\mathrm{K}^{M}:=\mathrm{K}\cap \mathrm{M}$$\mathrm{K}^{M}$$\zeta$$L^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$\mathfrak{B}_{\pi,X}^{\mathfrak{F}^{M}}\subset \mathfrak{B}_{\pi,X}^{\{\zeta^{\prime M}}$$\mathfrak{B}_{\pi,X}^{S^{M}}$$\mathrm{i}Y\subset S’$$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}^{S}=\mathrm{I}\mathrm{n}\mathrm{d}_{\mathrm{K}^{M}}^{\mathrm{K}}A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi,X}^{\mathfrak{F}^{M}}$$\mathfrak{B}_{\pi,X}^{\mathfrak{F}}$$\zeta$$\varphi\in A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}^{\mathfrak{F}}$$\varphi_{\lambda}(uamk):=a^{\lambda+\rho_{P}}\varphi(mk)$of the bundle
Now the spacein$\mathcal{I}_{P}^{G}(\pi_{\lambda})arrow\lambda\in(\alpha_{M}^{G})_{\mathbb{C}}^{*}$.

$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]_{)}X}^{\mathfrak{F}}$for functions transforming according to { under$\zeta$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]}\cap L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{X}$10$\mathrm{K}$253
is spanned by$\Phi_{F}(x):=\int_{i(\alpha_{M}^{G})^{*}}E(x, F(\lambda)_{\lambda})d\lambda$

where $F$ :sion,

is a Paley-Wiener section. Using the expan, we have$i(a_{M}^{G})^{*}arrow A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,\mathrm{X}}^{\mathfrak{F}}$$F( \lambda)=\sum_{\varphi\in \mathfrak{B}_{\pi}^{i?_{x}}},\langle F(\lambda)_{\lambda}, \varphi_{\lambda}\rangle\varphi_{\lambda}$$\sum$$\langle F(\lambda)_{\lambda}, \varphi_{\lambda}\rangle E_{P}(x, \varphi_{\lambda})d\lambda$$[R(f) \Phi_{F}](x)=R(f)\int_{i(a_{M}^{G})^{*}}\varphi\in \mathfrak{B}_{\pi}^{S},x$$\sum$$R(f)E_{P}(x, \varphi_{\lambda})\langle F(\lambda)_{\lambda}, \varphi_{\lambda}\rangle d\lambda$$= \int_{i(\alpha_{M}^{G})^{*}}\varphi\in \mathfrak{B}_{\pi,x}^{\mathrm{i}?}$for$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})^{\mathfrak{F}}$. Now recall the$\mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$,-equivariance$R(f)E_{P}(x, \varphi_{\lambda})=E_{P}(x,\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})$and the following formula obtained by the Fourier inversion formula applied to$\mathfrak{U}_{M}/\mathfrak{U}_{G}\cross$$(a_{M}^{G})^{*}:$$\langle F(\lambda)_{\lambda}, \varphi_{\lambda}\rangle=\langle\int_{i(\alpha_{M}^{G})^{*}}E_{P}(F(\mu)_{\mu})d\mu, E_{P}(\varphi_{\lambda})\rangle_{G}$.

We see that $[R(f)\Phi_{F}](x)$ equals to$\int_{i(a_{M}^{G})^{*}}\sum_{\varphi\in \mathfrak{B}_{\pi}^{\mathrm{i}\sigma_{x}}},E_{P}(x,\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})\langle\int_{i(\alpha_{M}^{G})^{*}}E_{P}(F(\mu)_{\mu})d\mu, E_{P}(\varphi_{\lambda})\rangle_{G}d\lambda$.

$= \int_{G(F)\mathfrak{U}_{G}\backslash \mathrm{G}}\int_{i(\mathfrak{a}_{M}^{G})^{*}}\sum_{\varphi\in \mathfrak{B}_{\pi x}\mathrm{f}\mathrm{f}},E_{P}(x,\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})\overline{E_{P}(y,\varphi_{\lambda})}d\lambda\Phi_{F}(y)dy$

In other words,$R(f)(f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})^{\mathfrak{F}})$

restricted to$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi],X}^{\mathfrak{F}}$has the kernel$E_{P}(x,\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})\overline{E_{P}(y,\varphi_{\lambda})}d\lambda$$\sum$$K_{[\pi],X}^{\mathfrak{F}}(x, y):= \int_{i(a_{M}^{G})^{*}}\varphi\in \mathfrak{B}_{\pi,x}^{S}$.

It was shown in [2] that
$K_{X}(x, y)$$:= \sum_{P\in i^{\tau_{(P_{0})}}}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\frac{1}{|W_{M}(G)|}$

(2.6)
,

$\int_{i(\{\iota_{M}^{G})^{*}}\sum_{\varphi\in \mathfrak{B}_{\pi x}},E_{P}(x,\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})\overline{E_{P}(y)\varphi_{\lambda})}d\lambda$

is empty if does not appear
defined as a certain limit, exists. Here
. Also the factor $1/|W_{M}(G)|$ is necessary
in the discrete spectrum of
because we take the sum over all the standard parabolics, not over their associated classes.
We obtain the spectral kernel:$\mathfrak{B}_{\pi,X}:=\bigcup_{\mathfrak{F}}\mathfrak{B}_{\pi,X}^{\mathfrak{F}}$$\pi$$L^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$K(x, y)=$$\sum$$X\in X(G)$11$K_{X}(x, y)$

.

(2.7)

254

3

Convergence3.1. buncated kernel
One can easily see that the operator $R(f)$ is no longer of trace class and$\int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}K(x, x)dx$

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.
inand the set of corresponding
for the set of simple roots of
We writeand$P=MU\in\mathcal{F}(P_{0})$,setcoroots,For
andrespectively.
simplecomponent of $X=X^{M}\oplus X_{M}\in$, where $X_{M}$ denotes the$W$. Using this, we define the positive
under the -invariant decompositionchamber$:=\{X\in\alpha_{0}|\alpha(X_{M})>0, \forall\alpha\in\triangle_{P}\}$ ,$\triangle_{0}$$A_{0}$$\triangle_{0}^{\mathrm{v}}$$P_{0}$$\triangle_{P}:=\{\alpha|_{\mathfrak{U}_{M}}|\alpha\in\triangle_{0}\backslash \triangle_{0}^{M}\}$$\triangle_{P}^{\vee}:=\{(\alpha^{\vee})_{M}|\alpha\in\triangle_{0}\backslash \triangle_{0}^{M}\}$$a_{M}$$\alpha_{0}=a_{0}^{M}\oplus\alpha_{M}$$a_{0}$$a_{P}^{\dashv-}$

whose characteristic function is denoted by . Next let. Then the convex open cone
the basis$\triangle_{P}\wedge\subset(a_{M}^{G})^{*}$$\tau_{P}$be the dual basis of$\triangle_{P}^{\mathrm{v}}\subset a_{M}^{G}$$+_{\alpha_{P}:=\{X\in a_{0}|\varpi(X)}=\varpi(X_{M})>0,$$\forall\varpi\in\triangle_{P}\}\wedge$

with an open cone.).
(Strictly speaking, this is the direct sum ofis spanned byfor the characteristic function of this cone.
Write
be the set of semisimple classes in $G(F)$ . For each $0\in \mathrm{O}(G)$ , we writeLet
under the Jordan decomposition
for the set of $\gamma\in G(F)$ whose semisimple part
belongs to . We have$a_{G}$$\triangle_{P}^{\mathrm{v}}$$\tau_{P}\wedge$$\mathfrak{O}(G)$$\overline{\mathit{0}}$$\gamma_{s}$$\gamma=\gamma_{s}\gamma_{u}=\gamma_{u}\gamma_{s}$$\mathit{0}$$K(x, y)= \sum_{0\in \mathrm{J}\supset(G)}K_{\mathit{0}}(x, y)$,$K_{0}(x, y):= \sum_{\gamma\in\overline{\mathit{0}}}f(x^{-1}\gamma y)$.

Next, for $P=MU\in \mathcal{F}(P_{0})$ write$K_{P}(x, y)= \sum_{\mathit{0}\in \mathfrak{O}(G)}K_{P,0}(x, y)$,$K_{P,0}(x, y):= \sum_{\gamma\in P(F)\cap\overline{\mathit{0}}}\int_{U(F)\backslash \mathrm{U}}f(x^{-1}\gamma uy)du$for the kernel of the right translation operator $R_{P}(f)$ on
similarly as $R(f)$ . Choosing $T\in a_{0}$ , define the truncated kernel$L^{2}(\mathrm{U}M(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$k^{T}(x, f):= \sum_{\mathit{0}\in \mathfrak{O}(G)}k_{\mathit{0}}^{T}(x, f)$,$k_{\mathit{0}}^{T}(x, f)$$:= \sum_{P=MU\in F(P_{0})}(-1)^{a_{M}^{G}}\sum_{\delta\in P(F)\backslash G(F)}K_{P,0}(\delta x, \delta x)_{\mathcal{T}_{P}}^{\wedge}(H_{0}(\delta x)-T)$

12.

defined255
Example 3.1. We present the explicit formulae in the case of $G=GL(2)[\mathit{2}\mathit{2}]$ . The
minimal parabolic subgroup will be denoted by $B=TU$ . The
is given as follows.
(i) If is $elliptic_{f}$ then$k_{\mathit{0}}^{T}(x, f)$$\mathit{0}$$\sum_{\gamma\in \mathit{0}}f(x^{-1}\gamma x)$(ii) If$\mathit{0}\ni(^{\alpha}\beta)$

.

(3.1)

is hyperbolic regular, then$\sum_{\gamma\in \mathit{0}}f(x^{-1}\gamma x)-\sum_{\delta\in B(F)\backslash G(F)}\int_{\mathrm{U}}f(x^{-1}\delta^{-1}u\delta x)_{\mathcal{T}_{B}}^{\wedge}(H_{0}(\delta x)-T)du$(3.2)$- \sum_{\delta\in B(F)\backslash G(F)}\int_{\mathrm{U}}f(x^{-1}\delta^{-1}u\delta x)_{\mathcal{T}_{B}}^{\wedge}(H_{0}(\delta x)-T)du$(iii) If$\mathit{0}=\{\zeta\cdot 1\}f\zeta\in F^{\cross}f$

.

then

.

$\sum_{\gamma\in\overline{\mathit{0}}}f(\zeta x^{-1}\gamma x)-\sum_{\delta\in B(F)\backslash G(F)}\int_{\mathrm{U}}f(\zeta x^{-1}\delta^{-1}u\delta x)_{\mathcal{T}_{B}}^{\wedge}(H_{0}(\delta x)-T)du$(3.3)

Note that this last term unifies the unipotent and identity terms.

Theorem 3.2 ([2] Th. 7.1). If we choose, then$T\in\alpha_{0}$sufficiently positive with respect to$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f$$J^{T}(f):= \sum_{\mathit{0}\in \mathfrak{O}(G)}J_{\mathit{0}}^{T}(f)$,$J_{\mathit{0}}^{T}(f):= \int_{G(F)\mathfrak{U}_{G}\backslash \mathrm{G}}k_{\mathit{0}}^{T}(x, f)dx$

converges absolutely.
Let us sketch the proof. In the higher rank case, the following combinatorial argument
is fundamental.
, where\S 2.1 combined with the Iwasawa decomposition yields. If we write
for the characteristic function of$\mathrm{G}=P(F)\mathfrak{S}_{P}(T_{0})$$\mathfrak{S}_{P}(T_{0})=\omega_{2}\omega_{1}\mathfrak{U}_{0}^{M}(T_{0})\mathfrak{U}_{M}\mathrm{K}$$F_{Q}^{P}(\bullet, T)$$Q(F)\{x\in \mathfrak{S}_{P}(T_{0})|\varpi(H_{0}(x)-T)\leq 0, \forall\varpi\in\triangle_{0}^{L}\}\wedge$

and$\mathfrak{J}_{Q}^{P}(\delta x, T):=F_{Q}^{P}(x, T)\tau_{Q}^{P}(H_{0}(x)-T)$, then we have [2, Lem. 6.4]:

(3.4)

.
$Q \subset P\sum_{Q=LV\in F(P_{0})}\sum_{\delta\in Q(F)\backslash P(F)}3_{Q}^{P}(\delta x, T)=1$

Next we put $\mathrm{R}_{Q}^{P}(x, T):=F_{Q}^{P}(x, T)\sigma_{Q}^{P}(H_{0}(x)-T)$ , whereof$\{X\in a_{0}|(2)(1)(3)$$\sigma_{Q}^{P}$

is the characteristic function$\alpha(X_{L})>0,\forall\alpha\in\triangle_{Q^{M}}\alpha(X_{L})\leq 0,\forall\alpha\in\triangle_{Q.\backslash \triangle_{Q^{M}}}\varpi(X)>0,\forall\varpi\in\triangle_{Q}\wedge’,$

13

$\}$

256
Example 3.3. If $G=SL(3)$ , the chambers inand$\sigma_{Q}^{P}$

are illustrated as

are the characteristic functions of the regions:$\sigma_{0}^{G}$$\sigma_{0}^{P_{2}}$$\sigma_{0}^{P_{1}}.\cdot$:$\wedge\tau_{P_{1}}=\sigma_{P_{1}}^{G}$

where$a_{0}$$\wedge\tau_{0}$:$P_{i}=M_{i}U_{i}\in \mathcal{F}(P_{0})$:$\wedge\tau_{P_{2}}=\sigma_{P_{2}}^{G}.\cdot$

are such thatAs these illustrate, we have$\triangle_{P_{0}^{M_{i}}}=\{\alpha_{i}\},$$\sum_{R\supset P}\sigma_{Q}^{R}=\tau_{Q^{M^{\wedge}}}\tau_{P}$$(i=1,2)$ .

and hence.

$R \in F(P_{0})\sum_{R\supset P}\mathrm{R}_{Q}^{R}(x, T)=\mathfrak{J}_{Q}^{P}(x, T)$

14

(3.5)

257
Applying these combinatorial formulae to our integrand, we have$k_{\mathit{0}}^{T}(x, f)(34)=$$\sum$$Q \subset P\in \mathcal{F}(P_{0})\sum_{\delta\in Q(F)\backslash G(F)}(-1)^{a_{M}^{G}}K_{P,\mathit{0}}(\delta x, \delta x)\mathfrak{J}_{Q}^{P}(\delta x, T)_{\mathcal{T}_{P}}^{\wedge}(H_{0}(\delta x)-T)$$(3.5)=$$\sum$$Q \subset R\in \mathcal{F}(P_{0})\sum_{\delta\in Q(F)\backslash G(F)}\mathrm{R}_{Q}^{R}(\delta x, T)\sum_{P\in \mathcal{F}(P_{0})}(-1)^{a_{M}^{G}}K_{P,\mathit{0}}(\delta x, \delta x)$.

Thus it suffices to prove the convergence of$\int_{Q(F)\mathfrak{U}_{G\backslash \mathrm{G}}}|\mathrm{R}_{Q}^{R}(x, T)\sum_{P\in F(P_{0})}(-1)^{a_{M}^{G}}K_{P}(x, x)|dx$.

cut off the domain of integration to a product of a compact set and a transported. But on such cone the alternating sum in the integrand is rapidly
positive cone indecreasing by Lem. 2.2.
$\mathrm{R}_{Q}^{R}(x, T)$$\alpha_{L}^{M_{R}}$3.2

Tkuncation operator and the basic identity

Next we turn to the convergence of the spectral side [3]. Already it follows from Th. 3.2that the integral of the total kernel$\int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\sum_{x\in X(G)}k_{X}^{T}(x, f)dx$

converges absolutely. Here we have written$k_{X}^{T}(x, f)$$:= \sum_{P=MU\in F(P_{0})}(-1)^{a_{M}^{G}}\sum_{\delta\in P(F)\backslash G(F)}K_{P,X}(\delta x, \delta x)_{\mathcal{T}_{P}}^{\wedge}(H_{0}(\delta x)-T)$

,

where
$K_{P,X}(x, y)= \sum_{Q\subset P}\sum_{\tau Q=LV\in F(P_{0})\in\Pi(\mathrm{L}^{1})}\frac{1}{|W_{L}^{M}(M)|}$$\int_{i(\alpha_{L}^{G})^{*}}\sum_{\varphi\in \mathfrak{B}_{T}x},E_{Q}^{P}(x,\mathcal{I}_{Q^{M}}^{M}(\tau_{\lambda}, f)\varphi_{\lambda})\overline{E_{Q}^{P}(y,\varphi_{\lambda})}d\lambda$

is the kernel of $R_{P}(f)$ restricted tomutativity of the integration and the summation.
, define the truncation $operator\wedge^{T}$ on
For$L^{2}(\mathrm{U}M(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{X}$$T\in\alpha_{0}$. Thus the problem is the com-$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$by$(\wedge^{T}\phi)(x)$$:= \sum_{P=MU\in \mathcal{F}(P_{0})}(-1)^{a_{M}^{G}}\sum_{\delta\in P(F)\backslash G(F)}\phi_{P}(\delta x)_{\mathcal{T}_{P}}^{\wedge}(H_{0}(\delta x)-T)$.

As in the proof of Th. 3.2, we can rewrite this as$( \wedge^{T}\phi)(x)=\sum_{Q\subset P_{1\in i}r_{(P_{0})}}\sum_{\delta\in Q(F)\backslash G(F)}\mathrm{R}_{Q}^{P_{1}}(\delta x, T)\sum_{P,Q\subset P\subset P_{1}}(-1)^{a_{M}^{G}}\phi_{P}(\delta x)$

and obtain the following properties:

15

,

258
is an orthogonal projection.

(1)$\wedge^{T}$(2)$\wedge^{T}\emptyset=\phi$for$\phi\in L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$.

is slowly increasing and sufficiently smooth relative to $\dim U_{0}$ , then(3) Ifrapidly decreasing.
$\phi$We have similar operators$\wedge^{T,P}$

on$L^{2}(\mathrm{U}M(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$\wedge^{T}\emptyset$

is.

Example 3.4. Consider the simplest example, the $GL(2)$ case. Note that if $T$ is sufficiently positive the sum on contains only one term. (Recall the classical situation where
which preserves the region
any element of$\delta$$SL_{2}(\mathbb{Z})$with sufficiently large is upper triangular.) $Then\wedge^{T}$ is the operator which cuts off theconstant term in the neighborhood $\{x|\tau_{P}(\wedge H_{0}(x)-T)>0\}$ of cusps, the complement of acompact set in the Siegel domain. The properties stated above are rather obvious in thisexample.
$t$

A similar combinatorics as in the geometric side yield$k_{X}^{T}(x, f)=$$\sum$$Q \subset R\in \mathcal{F}(P_{0})\sum_{\delta\in Q(F)\backslash G(F)}\sigma_{Q}^{R}(H_{0}(\delta x)-T)$$\cross\sum_{P\in F(P_{0})}(-1)^{a_{M}^{G}}\bigwedge_{2}^{T,Q}K_{P,X}(\delta x, \delta x)$.

means the truncation operator is applied in the second variable. This combinedHerewith (3) above shows that the integration and the summation commute. Moreover thealternating sum on the right gives$\bigwedge_{2}^{T}$$J_{X}^{T}(f):= \int_{G(F)\mathfrak{U}_{G}\backslash \mathrm{G}}k_{X}^{T}(x, f)dx$$= \sum_{Q\subset R\in \mathcal{F}(P_{0})}\int_{Q(F)\mathfrak{U}_{G}\backslash \mathrm{G}}\sigma_{Q}^{R}(H_{0}(x)-T)\sum_{P\in \mathcal{F}(P_{0})}(-1)^{a_{M}^{G}}\wedge^{T,Q}K_{P,X}(x, x)dx$$= \int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\wedge^{T}K_{X}(x, x)dx$.

That is, the term associated to $Q,$ $R$ vanishes unless $Q=R=G$ .

16

(3.6)

259
We now come to the coarse Arthur-Selberg trace formula$\sum_{\mathit{0}\in \mathfrak{O}(G)}J_{\mathit{0}}^{T}(f)=\sum_{X\in X(G)}J_{X}^{T}(f)$for sufficiently positive$T\in\alpha_{0}$

(3.7)

,

.

The fine O-expansion4

in (3.7). Recall that, in the $GL(2)$and
Next we need to calculate each terms
case [24], [22], certain part of the hyperbolic terms and Eisenstein terms cancel each other.
The resulting equality has a meaning when $Tarrow\infty$ . Since such an explicit cancellation is
not available in the present case, taking the limit under $Tarrow\infty$ is not allowed. Instead
we calculate the special value at “ $T=0$ ”.
$J_{X}^{T}(f)$$J_{\mathit{0}}^{T}(f)$

4.1
For

$J^{T}(f)$as a polynomial$Q\subset P\in \mathcal{F}(P_{0})$, consider the function$\Gamma_{Q}^{P}(X, Y):=\sum_{P_{1;}Q\subset P_{1}\subset P}(-1)^{a_{M_{1}}^{M}}\tau_{Q^{M_{1}}}(X)_{\mathcal{T}_{P_{1}^{M}}}^{\wedge}(X-Y)$

and. Noting that the matricesoneach other, one sees that this function expresses the variation of$((-1)^{a_{L}}\tau_{Q^{M}})_{Q,P}$$a_{0}\cross\alpha_{0}$$((-1)^{a_{L^{\wedge}}}\tau_{Q^{M}})_{Q,P}$$\wedge$$\tau_{Q^{M}}$are inverse to:$\wedge\tau_{Q^{M}}(X-Y)=.\sum_{P_{1)}Q\subset P_{1}\subset P}(-1)^{a_{M_{1^{\wedge}}}^{M}}\tau_{Q^{M_{1}}}(X)\Gamma_{P_{1}}^{P}(X, Y)$

(4.1)

.

Some calculation shows that
$J^{T+X}.(f)= \sum_{P\in \mathcal{F}(P_{0})}J_{M}^{M,T}.(\overline{f}_{P})\int_{a_{M}^{G}}\Gamma_{P}^{G}(H, X)dH$

where$\overline{f}_{P}$,is the descent$\overline{f}_{P}(m):=m^{\rho_{P}}\int_{\mathrm{K}}\int_{\mathrm{U}}f(k^{-1}muk)dudk$

in $H$ is a polynomial function in $X$ , the same
to $M$ . Since the integral ofis true for $J^{T}.(f)$ [ , Prop. 2.3].
Recall that the distributions in (3.7) depend on the choice of the set of representatives
if $G$ is split. In general, which can be chosen in
attached toof the reflection, the representative
and, forby. We define the (analytic) origin
for somesatisfiesof$\Gamma_{P}^{G}’(H, X)$

$f$

$4$

$\mathrm{K}\mathrm{M}_{0}\cap G(F)$$\alpha\in\triangle_{0}$$T_{1}:= \sum_{\alpha\in\triangle_{0}}h_{\alpha}\varpi_{\alpha}^{\vee}$is the basis of$J_{X}(f):=J_{X}^{T_{1}}(f)[4, \S 1, 2]$ .
$\{\varpi_{\alpha}^{\vee}\}_{\alpha\in\triangle_{0}}$$\mathfrak{a}_{0}^{G}$

dual to$T_{1}\in a_{0}$.

$\triangle_{0}\in(\alpha_{0}^{G})^{*}$

17$\alpha$$r_{\alpha}$$w_{r_{\alpha}}$$h_{\alpha}\in \mathbb{R}$$H_{0}((w_{r_{\alpha}})^{-1})=h_{\alpha}\alpha^{\vee}$Here$W\subset$$G(F)\cap \mathrm{K}$$W\in \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(A_{0)}G)$.

We define$J_{\mathit{0}}(f):=J_{\mathit{0}}^{T_{1}}(f)$,260
Example 4.1. Let us compare this to the classical $GL(2)$ case. The distributions.
are calculated as follows(1) If is elliptic, we have

$J_{\mathit{0}}^{T}(f)$$[\mathit{2}\mathit{2}J$$\mathit{0}$(4.2)$J_{\mathit{0}}^{T}(f)= \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(G_{\gamma}(F)\mathfrak{U}_{G}\backslash \mathrm{G}_{\gamma})\int_{\mathrm{G}_{\gamma}\backslash \mathrm{G}}f(x^{-1}\gamma x)dx$

as in the anisotropic case. Note that we have(2) If is the hyperbolic regular class ofand $w\in W$ of$\mathit{0}$in the special case of $GL(2)$ .
equals the sum over $\delta\in B(F)\backslash G(F)$$G^{\gamma}=G_{\gamma}$$(^{\alpha}\beta),$$(3.2)$.

$\frac{1}{2}\sum_{\iota/\in U(F)}f(\mathrm{A}\mathrm{d}(wu\delta x)^{-1})-\int_{\mathrm{U}}f(\mathrm{A}\mathrm{d}(wu\delta x)^{-1})du\tau_{B}\wedge(H_{0}(\delta x)-T)$

One sees that

$J_{\mathit{0}}^{T}(f)$equals$\sum_{w\in W}\int_{B(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\frac{1}{2}\sum_{\nu\in U(F)}f(\mathrm{A}\mathrm{d}(wux)^{-1})$$- \int_{\mathrm{U}}f(\mathrm{A}\mathrm{d}(wux)^{-1})du^{\wedge}\tau_{B}(H_{0}(x)-T)dx$$= \sum_{w\in W}\int_{\mathrm{U}T(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\int_{\mathrm{U}}f(\mathrm{A}\mathrm{d}(wux)^{-1})(\frac{1}{2}-\tau_{B}\wedge(H_{0}(x)-T))dudx$$= \sum_{w\in W}\int_{T(F)\mathfrak{U}_{G\backslash \mathrm{G}}}f(w^{-1}x^{-1}xw)(\frac{1}{2}-\tau_{B}\wedge(H_{0}(x)-T))dx$$= \int_{T(F)\mathfrak{U}_{G\backslash \mathrm{G}}}f(x^{-1}x)(1-\tau_{B}\wedge(H_{0}(x)-T)-\tau_{B}(\wedge H_{0}(wx)-T))dx$

using the Iwasawa decomposition,$= \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(T(F)\mathfrak{U}_{G}\backslash \mathrm{T}^{1})\int_{\mathrm{K}}\int_{\mathrm{U}}f(k^{-1}u^{-1}uk)$.

$\int_{\mathfrak{U}_{T}^{G}}1-\tau_{B}(\wedge H_{0}(a)-T)-\tau_{B}(\wedge H_{0}(wau)-T)dadudk$

Noting $H_{0}(wau)=w(H_{0}(a))+H_{0}(wu)f$ the inner integral becomes$\int_{\alpha(H_{0}(wu)-T)\alpha^{\vee}/2}^{\alpha(T)\alpha^{\vee}/2}dH=\frac{\alpha(2T-H_{0}(wu))}{\sqrt{2}}$Here$\alpha$denotes the unique positive root of$T$.

in G. We obtain$J_{\mathit{0}}^{T}(f)=- \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(T(F)\mathfrak{U}_{G}\backslash \mathrm{T}^{1})\int_{\mathrm{K}}\int_{\mathrm{U}}f(k^{-1}u^{-1}uk)\frac{\alpha(H_{0}(wuk))}{\sqrt{2}}dudk$$+\sqrt{2}\alpha(T)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(T(F)\mathfrak{U}_{G}\backslash \mathrm{T}^{1})\overline{f}_{P}()$

18

.

(4.3)
(4.4)

261
(3) The case of central$\coprod_{\delta\in B(F)\backslash G(F)}\mathrm{A}\mathrm{d}(\delta)^{-1}U(F)$is easily reduced to the caseallows us to write (3.3) as$\mathit{0}$$\mathit{0}=\{1\}$. If$\mathit{0}=\{1\},$$\overline{\mathit{0}}=$$f(1)+ \sum_{\delta\in B(F)\backslash G(F)}(\sum_{\nu\in U(F)\backslash \{1\}}f(x^{-1}\delta^{-1}I\nearrow\delta x)-\int_{\mathrm{U}}f(x^{-1}\delta^{-1}u\delta x)du^{\wedge}\tau_{B}(H_{0}(\delta x)-T))$.

Thus, by the Iwasawa $decomposition_{f}J_{\{1\}}^{T}(f)$ equals the sum of

meas

(4.5)

$(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})f(1)$and$\int_{T(F)\mathfrak{U}_{G\backslash \mathrm{T}}}(\sum_{I/\in U(F)\backslash \{1\}}\overline{f}(t^{-1}\nu t)-\int_{\mathrm{U}}\overline{f}(t^{-1}ut)du\tau_{B}\wedge(H_{0}(t)-T))t^{-2\rho_{B}}dt$

(4.6)

,

where
$\overline{f}(x):=\int_{\mathrm{K}}f(k^{-1}xk)dk$.

Fix a non-trivial character of. Write for the Lie algebra of andThen the Poisson summation formula for the Fourier transformation$\psi$$F\backslash \mathrm{A}$$U$$\mathrm{u}$$\mathrm{u}^{\vee}$

be its dual.

$( \mathrm{A}\mathrm{d}\overline{(t)\overline{f}\circ}\exp)(X^{\vee}):=\int_{\mathrm{u}(\mathrm{A})}(\mathrm{A}\mathrm{d}(t)\overline{f})(\exp X)\psi(\langle X^{\vee}, X\rangle)dX=t^{2\rho_{B}}(\overline{\overline{f}\circ \mathrm{e}\mathrm{x}}\mathrm{p})(\alpha(t)X^{\vee})$

implies that (4.6) equals–$\sum$$(\overline{f}\circ\exp)(x^{-1}\xi)|x|_{\mathrm{A}}^{-1}-(\overline{f}\circ\exp)(0)\tau_{>\alpha(T)}(\log|x|_{\mathrm{A}})dx^{\cross}$$\sqrt{2}\cdot \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(F^{\cross}\backslash \mathrm{A}^{1})\int_{F^{\cross}\backslash \mathrm{A}^{\cross}}\xi\in \mathrm{u}(F)\backslash \{0\}$$\sum$$(\overline{f}\circ\exp)(x^{-1}\xi)|x|_{\mathrm{A}}^{-1}dx^{\cross}$$= \sqrt{2}\cdot \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(F^{\cross}\backslash \mathrm{A}^{1})(\int_{F^{\cross}\backslash \mathrm{A}_{<1}}\xi\in \mathrm{u}(F)\backslash \{0\}$$+ \int_{F^{\cross}\backslash \mathrm{A}}\geq 1\sum_{\xi^{\vee}\in \mathrm{u}^{\vee}(F)}(\overline{\overline{f}\circ \mathrm{e}\mathrm{x}}\mathrm{p})(x\xi^{\vee})-\overline{f}(1)|x|_{\mathrm{A}}^{-1}-(\overline{\overline{f}\circ \mathrm{e}\mathrm{x}}\mathrm{p})(0)\tau_{>\alpha(T)}(\log|x|_{A})dx^{\cross})$$\sum$$(\overline{f}\circ\exp)(x\xi)|x|_{\mathrm{A}}dx^{\cross}$$= \sqrt{2}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(F^{\cross}\backslash \mathrm{A}^{1})(\int_{F^{\cross}\backslash \mathrm{A}_{>1}}\xi\in \mathrm{u}(F)\backslash \{0\}$$+ \int_{F^{\cross}\backslash \mathrm{A}_{\underline{>}1}}\sum_{\xi^{\vee}\in \mathrm{u}^{\vee}(F)\backslash \{0\}}(\overline{\overline{f}\circ \mathrm{e}\mathrm{x}}\mathrm{p})(x\xi^{\vee})-\overline{f}(1)|x|_{\mathrm{A}}^{-1}dx^{\cross}$$+( \overline{\overline{f}\circ \mathrm{e}\mathrm{x}}\mathrm{p})(0)\int_{F^{\cross}\backslash \mathrm{A}}\geq 11-\tau_{>\alpha(T)}(\log|x|_{\mathrm{A}})dx^{\cross})$Here$\tau_{>A}$is the characteristic function of$\mathbb{R}_{>A}$. Consider the integral$Z( \overline{f}, s):=\int_{\mathrm{A}^{\cross}}(\overline{f}\circ\exp)(x)|x|_{\mathrm{A}}^{s}dx^{\cross}$

19

.

262
This converges absolutely for ${\rm Re}(s)>1$ and extends to a meromorphic function in .
It turns out that the sum of the first two integrals in the above equals the regular part.
while the last equals meas
this ‘zeta function” at
$s$

$t$

Noting that$(F^{\cross}\backslash \mathrm{A}^{1})\alpha(T)$$s=1_{\rangle}$$\mathrm{r}\mathrm{e}\mathrm{g}_{s=1}Z(\overline{f}, s)\underline{of}$$(\overline{f}\circ\exp)(0)=\overline{f}_{P}(1)$, we conclude that

$J_{\{1\}}^{T}(f)=\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})f(1)+\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Z(F)\mathfrak{U}_{G}\backslash \mathrm{Z})\mathrm{r}\mathrm{e}\mathrm{g}_{s=1}Z(\overline{f}, s)$$+\sqrt{2}\alpha(T)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(T(F)\mathfrak{U}_{G}\backslash \mathrm{T}^{1})\overline{f}_{P}(1)$We choose $T_{1}=0$ . Then taking the special value atsum of (4.4) and (4.8):$T_{1}$

(4.7)
(4.8)

.

is equivalent to throw away the$\sum_{\alpha,\beta\in F^{\cross}}\sqrt{2}\alpha(T)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(T(F)\mathfrak{U}_{G}\backslash \mathrm{T}^{1})\overline{f}_{P}()$.

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. Notealso that this term can be viewed as the geometric side of the “trace formula” for the Levi.
subgroup $T$ applied to$\overline{f}_{P}$4.2

Reduction by the Jordan decomposition

as a sum of terms, each of
Calculate the geometric terms. Our goal is to express
which is a product of a global constant and an Euler product of local distributions.
withFirst we replace$J_{\mathit{0}}(f)$$k_{0}^{T}(x, f)$$j_{\mathit{0}}^{T}(x, f):= \sum_{P\in F(P_{0})}(-1)^{a_{M}^{G}}\sum_{\delta\in P(F)\backslash G(F)}j_{P,0}(\delta x)_{\mathcal{T}_{P}}^{\wedge}(H_{0}(\delta x)-T)$$j_{P,0}(x)$$:=$$\sum$$\sum_{\gamma\in M(F)\cap\overline{\mathit{0}}\nu\in U^{\gamma s}(F)\backslash U(F)}\int_{\mathrm{U}^{\gamma s}}\phi(X-1I/-1\gamma u\iota/x)du$,.

We have$K_{P,\mathit{0}}(x, x)= \int_{U(F)\backslash \mathrm{U}}j_{P,0}(ux)du$and hence$J_{\mathit{0}}^{T}(f)= \int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}j_{\mathit{0}}^{T}(x, f)dx$.

which is elliptic in some $P_{1}=M_{1}U_{1}\in \mathcal{F}(P_{0})$ , that is,
Take a representative. Thenis a$\sigma\in M_{1}(F)$ but it is not contained in any proper parabolic subgroup of,
is conjugate to some element of. Eachminimal parabolic subgroup of, the connected centralizer of . Thus writingis the unipotent variety of
where, we have$\sigma\in 0$$M_{1}$$G_{\sigma}$$\gamma\in\overline{\mathit{0}}$$G_{\sigma}$$\mathcal{U}_{G_{\sigma}}$$P_{1,\sigma}$$\sigma \mathcal{U}_{G_{\sigma}}(F)$$\sigma$$\iota^{G}(\sigma):=[G^{\sigma}(F) : G_{\sigma}(F)]$$j_{Q,0}(x)= \iota^{G}(\sigma)^{-1}\sum_{w\in W_{M_{1}}(L,G_{\sigma})}$.

$\sum_{\pi\in(^{w}G_{\sigma}\cap Q)(F)\backslash Q(F)}\sum_{\nu\in w^{-1}(L)(F)\cap \mathcal{U}_{G_{\sigma}}(F)}\int_{w^{-1}(\mathrm{V})\cap \mathrm{G}_{\sigma}}f(x^{-1}\pi^{-1w}(\sigma uv)\pi x)dv$

20

263
Here
$W_{M_{1}}(L, G_{\sigma})$$\Sigma_{P}$$:=\{w\in W_{M_{1}}(L)|(ii)(i)$$w(\beta)>0,\forall\beta\in\Sigma_{P_{1,\sigma}}w^{-1}(\alpha)>0,\forall\alpha\in\triangle_{w}P_{1}^{L},$$\})$being the set of positive roots of $P$ . If we write $R$ for the standard parabolic subgroupofand $\mathcal{F}(M_{1})_{R}:=\{P\in F(M_{1})|P_{\sigma}=R\}$ , the above gives$w^{-1}Q\cap G_{\sigma}$$G_{\sigma}$$J_{\mathit{0}}^{T}(f)= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma}\backslash \mathrm{G}}\int_{G_{\sigma}(F)\mathfrak{U}_{G\backslash \mathrm{G}_{\sigma}}}\sum_{R\in \mathcal{F}(P_{1,\sigma})}\sum_{\delta\in R(F)\backslash G_{\sigma}(F)}$(4.9)$\sum_{\nu\in \mathcal{U}_{M_{R}}(F)}\int_{\mathrm{U}_{R}}f(y^{-1}x^{-1}\delta^{-1}\sigma\iota/v\delta xy)dv$$\sum_{P\in i^{F(M_{1})_{R}}}(-1)^{a_{M}}\tau_{P}(H_{0}(\delta xy)-w_{P}^{-1}(T-T_{1})-T_{1})dxdyG_{\wedge}$.

. $R=M_{R}U_{R}$ denotes the standard Levi decompositionis such thatof . We shall write this in terms of the unipotent termsof the trace formula for.
the Levi subgroups of
As in the case of $G$ , we have the origin. We writeforfor the image
and setof $T-T_{1}+T_{1,\sigma}$ in$w_{P}\in W$$w_{PP}\in \mathcal{F}(P_{0})$$R$$J_{\{1\}}^{M_{S},T}$$G_{\sigma}$$T_{1,\sigma}\in\alpha_{M_{1}}$$G_{\sigma}$$T_{\sigma}$$\alpha_{M_{1}}$$Y_{P}^{T}(x, y)$$:=-H_{P}(k_{P_{\sigma}}(x)y)+w_{P}^{-1}(T-T_{1})-T_{\sigma}+T_{1}$,$P\in \mathcal{F}(M_{1})_{R}$.

Then the family $\mathcal{Y}_{P}^{T}(x, y):=\{Y_{P}^{T}(x, y)|P\in \mathcal{F}(M_{1})_{R}\}$ is compatible in the sense of [9,\S 4]. Generalizing (4.1), we see that the last line in (4.9) can be written as,$S \in \mathcal{F}^{G_{\sigma}}(M_{1,\sigma})\sum_{S\supset R}(-1)^{a_{M_{R\tau_{R^{M_{S}}}(H_{R}(\delta x)-T_{\sigma})\Gamma_{S}^{G}(H_{S}(\delta x)-T_{\sigma},\mathcal{Y}_{S}^{T}(\delta x,y))}}^{M_{S_{\wedge}}}}$

where$\Gamma_{R}^{G}(x, y_{R})$$:= \sum_{S\supset R}\tau_{R^{M_{S}}}(X)S\in F^{G_{\sigma}}(M_{1,\sigma})(_{Q\in i}\sum_{\tau_{(M_{1})_{S}}}(-1)^{a_{L}^{G_{\wedge}}}\tau_{Q}(X-Y_{Q}))$.

This combined with the Iwasawa decomposition yields$J_{\mathit{0}}^{T}(f)= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma}\backslash \mathrm{G}}\sum_{S\in F^{G_{\sigma}}(P_{1,\sigma})}(\int_{\mathrm{K}_{\sigma}}\int_{\mathfrak{U}_{S}^{G}}J_{\{1\}}^{M_{S},T_{\sigma}}(\Phi_{S,a,k,y}^{T})dadk)dy$(4.10),

with($\Phi_{S,a,k,y}^{T}(m):=m^{\rho_{S}}\int_{\mathrm{U}_{S}}f$ $y^{-1}\sigma k^{-1}$muky)$du\Gamma_{S}^{G}(H_{S}(a)-T_{\sigma}, \mathcal{Y}_{S}^{T}(k, y))$.

Finally if we specialize this to $T=T_{1},$reduces to $\Gamma_{Q}^{G}(X, -H_{Q}(ky)+$. Applying this to the integral over in (4.10), we obtainfor any$\Gamma_{S}^{G}(X, \mathcal{Y}_{S}^{T_{1}}(k, y))$$T_{1}-T_{1,\sigma})$$Q\in \mathcal{F}(M_{1})_{S}$$a$$J_{\mathit{0}}(f)= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma}\backslash \mathrm{G}}\sum_{S\in F^{G_{\sigma}}(P_{1,\sigma})}J_{\{1\}}^{M_{S}}(\Phi_{S,y,T_{1}-T_{1,\sigma}})dy$(4.11)$= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma}\backslash \mathrm{G}}(\sum_{R\in F^{G_{\sigma}}(M_{1,\sigma})}\frac{|W^{M_{R}}|}{|W^{G_{\sigma}}|}J_{\{1\}}^{M_{R}}(\Phi_{R,y,T_{1}-T_{1,\sigma}}))dy$

21

,

264
where
,

$\Phi_{S,y,T}(m):=m^{\rho s}\int_{\mathrm{K}_{\sigma}}\int_{\mathrm{U}_{S}}f(y^{-1}\sigma k^{-1}muky)v_{S}’(ky, T)dudk$.

$v_{S’}(ky, T):= \sum_{Q\in \mathcal{F}(M_{1})_{S}}\int_{a_{L}^{G}}\Gamma_{Q}^{G}(X, -H_{Q}(ky)+T)dX$

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.
, we define two denominator functions
For$Q\subset P\in \mathcal{F}(M_{0})$$\theta_{Q}^{P}(\lambda):=\frac{1}{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(\alpha_{L}^{M}/\mathbb{Z}[\triangle_{Q^{M}}^{\mathrm{v}}])}\prod_{\alpha\in\triangle_{Q^{M}}}\alpha^{\vee}(\lambda)$,.

$\theta_{Q}^{P}(\lambda)\wedge:=\frac{1}{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(a_{L}^{M}/\mathbb{Z}[\triangle_{Q^{M}}^{\mathrm{v}}])\wedge}$$\prod_{\wedge,\varpi^{\vee}\in\triangle_{Q^{M}}^{\vee}}\varpi^{\vee}(\lambda)$

Heredenotes the basis of
the Fourier transform of$\triangle_{P}^{\mathrm{v}}\wedge$$a_{M}^{G}$$\Gamma_{Q}^{G}(X, Y)$. These functions first appeared indual toin $X$ [ , Lem. 2.2]:$\triangle_{P}\subseteq(a_{M}^{G})^{*}$$4$,$\int_{\mathfrak{a}_{L}^{G}}\Gamma_{Q}^{G}(X, Y)e^{\lambda(X)}dX=\sum_{P\supset Q}(-1)^{a_{L}^{M}}\frac{e^{\langle\lambda_{M},Y\rangle}}{\wedge,\theta_{Q}^{P}(\lambda)\theta_{P}^{G}(\lambda)}$but its real nature is to produce certain “difference”(4.12)$c_{Q}’( \lambda):=\sum_{P_{1}\supset Q}(-1)^{a_{L}^{M_{1}}}\frac{c_{P_{1}}(\lambda)}{\wedge,\theta_{Q}^{P_{1}}(\lambda)\theta_{P_{1}}^{G}(\lambda)}$of a smooth function
component of under$\lambda$$c_{P}(\lambda)$

on$i\alpha_{M}^{*},$$Q\supset P$. Here.

$c_{P_{1}}(\lambda):=c_{P}(\lambda_{M_{1}}),$$\lambda_{M_{1}}$being the$ia_{M_{1}}^{*}$$i\alpha_{M}^{*}=i(\alpha_{M}^{M_{1}})^{*}\oplus i\alpha_{M_{1}}^{*}$

Example 4.2. In the case of $GL(2)$ , we have$c_{B}’( \lambda)=\frac{\sqrt{2}}{\alpha^{\vee}(\lambda)}(c_{B}(\lambda)-c_{G}’(\lambda))=\frac{\sqrt{2}}{\alpha^{\vee}(\lambda)}(c_{B}(\lambda)-c_{B}(\lambda_{G}))$

Since$\sqrt{2}$is the length of$\alpha^{\mathrm{v}_{\mathrm{Z}}}$

this is literally the difference of$c_{B}(\lambda)$.

around$\lambda_{G}$.

A familyof smooth functions on
is a $(G, M)$ -family if
for any $P,$ $P’\in P(M)$ whose associated chambers share a wall andin that, the function (4.12) is well-defined because it iswall. For a $(G, M)$ -family$P\inP^{P_{1}}(M)$.
independent of the choice of$\{c_{P}(\lambda)\}_{P\in P(M)}$$c_{P}(\lambda)=$$ia_{M}^{*}$$c_{P’}(\lambda)$$\lambda$$\{c_{P}(\lambda)\}_{P\in P(M)}$

22

265
Lemma 4.3 ([4] Lem. 6.1, 6.2, 6.3). Let(1)extends to a smooth function on.
(2) The function

be a $(G, M)$ -family.

$\{c_{P}(\lambda)\}_{P\in P(M)}$$c_{Q}’(\lambda)$$ia_{L}^{*}$$c_{M}^{Q}( \lambda):=\sum_{P\subset Q}P\in P(M)\frac{c_{P}(\lambda)}{\theta_{P}^{Q}(\lambda)}$.
extends to a smooth function on$(G,M)$(3) Ifis another-family,family and we have$i\alpha_{M}^{*}$$\{d_{P}(\lambda)\}_{P(M)}$$(cd)_{P}(\lambda):=c_{P}(\lambda)d_{P}(\lambda)$$(cd)_{M}( \lambda)=\sum_{Q\in F(M)}c_{M}^{Q}(\lambda)d_{Q}’(\lambda)$

An example of $(G, L)$ -family is the family$P=MU\in \mathcal{F}(L)$ , then we have the bijection$/\mathcal{P}^{P}(L)\ni Q\mapsto Q^{M}$

Thus, from a $(G, L)$ -family. Applying this to$\{c_{Q}(\lambda)\}_{Q\in P(L)}$form a $(G, M)-$.

$\{v_{Q}(\lambda, x):=e^{-\lambda(H_{Q}(x))}\}_{Q\in P(L)}$for$x\in \mathrm{G}$. If$:=Q\cap M\in P^{M}(L)$ .

, one deduces a $(M, L)$ -family, we have the smooth function$\{c_{Q^{M}}(\lambda)$$:=$$\{v_{Q}(\lambda, x)\}_{Q\in P(L)}$$c_{Q}(\lambda)\}_{Q^{M}\in P^{M}(L)}$$v_{L}^{P}( \lambda, x):=\sum_{Q^{M}\in P^{M}(L)}\frac{v_{Q}(\lambda,x)}{\theta_{Q}^{P}(\lambda)}$by Lem. 4.3 (2). This depends on $P\in P(M)$ (as the notation suggests) because thedepends not only
but also . Set
function.
$F$
placesof . We now define the weighted orbital integral
Letbe a finite set of. Writeandforfor the Jordan decomposition$v\inS$, we define
at anyof . If$Q^{M}$$H_{Q}$$Q$$v_{L}^{P}(x):= \lim_{\lambdaarrow 0}v_{L}^{P}(\lambda, x)$$S$$J_{M}(\gamma, f)$$\gamma_{v}$$M\in \mathcal{L}(M_{0})$$\gamma\in \mathrm{M}_{S}$$\gamma_{v}=\gamma_{v,s}\gamma_{v,u}$$G_{\gamma_{v,s}}\subset M_{\gamma_{v,s}}$$J_{M}( \gamma, f)=|D^{G}(\gamma)|_{\mathrm{A}_{S}}^{1/2}\int_{\mathrm{M}_{S}\backslash \mathrm{G}_{S}}\int_{0^{\mathrm{M}_{S}}(\gamma)}f(x^{-1}\mu x)v_{M}(x)d\mu dx$$=|D^{G}( \gamma)|_{\mathrm{A}_{S}}^{1/2}\int_{\mathrm{G}_{S,\gamma}\backslash \mathrm{G}_{S}}f(x^{-1}\gamma x)v_{M}(x)dx$.

is given byHerein , the Lie algebra of. Alsobeing the fixed part ofis theis assured by [10, Lem. 2.1] and the Deligne-Rao
orbit of . The convergence oftheorem [36]. In the general case, we define$D^{G}(\gamma)=(D^{G}(\gamma_{v}))_{v\in S}$$D^{G}(\gamma):=\det(1-\mathrm{A}\mathrm{d}(\gamma_{v,s})|\mathrm{g}(F_{v})/\mathrm{g}_{\gamma_{vs}}(F_{v})),$$\mathrm{g}_{\gamma_{v,s}}$’$\mathrm{A}\mathrm{d}(\gamma_{v,s})$$\gamma$$G\otimes_{F}F_{v}$$\mathrm{g}_{v}$$\mathrm{o}^{\mathrm{M}_{S}}(\gamma)$$\mathrm{M}_{S^{-}}$$J_{M}(\gamma, f)$$J_{M}(\gamma, f)$$:= \lim_{aarrow 1}\sum_{L\in \mathcal{L}(M)}r_{M}^{L}(\gamma, a)J_{L}(a\gamma, f)$.

Here $r_{M}^{L}(\gamma, a)=r_{M}^{Q}(\gamma, a)(Q\in P(L))$ is constructed from the $(G, M)$ -family,$r_{P}( \lambda, \gamma, a)=\prod_{v\in S}\prod_{\beta\in\Sigma_{P_{\gamma v,s}}^{r}}|a^{\beta}-a^{-\beta}|_{v}^{\rho(\beta,\gamma_{v,u})\lambda(\beta^{\vee})/2}$

23$P\in P(M)$(4.13)

266
are defined in [10, \S 3],
by the process explained above. The non-negative integers
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 $Q\in P(L)$ [ $10$ , Lem. 5.1]. Note that, if $M=G$ this reduces tothe ordinary orbital integral$\rho(\beta, u)$$J_{G}( \gamma, f)=|D^{G}(\gamma)|_{\mathrm{A}_{S}}^{1/2}\int_{\mathrm{G}_{S,\gamma}\backslash \mathrm{G}_{\mathrm{S}}}f(x^{-1}\gamma x)dx$.

We need the following descent formula for weighted orbital integrals.
Lemma 4.4 ([10] Cor. 8.7). Suppose that(i)$\gamma\in \mathrm{G}_{S}$satisfies$(\gamma_{v,s})_{v\in S}\in G(F),\cdot$(ii)$a_{M_{\gamma s}}=\alpha_{M}$($\gamma_{s}$is elliptic in $M(F)$ ).

Then we have
$J_{M}( \gamma, f)=|D^{G}(\gamma_{s})|_{\mathrm{A}_{S}}^{1/2}\int_{\mathrm{G}_{\gamma_{S},S}\backslash \mathrm{G}_{S}}(\sum_{R\in F^{G_{\gamma_{S}}}(M_{\gamma s})}J_{M_{\gamma_{S}}}^{M_{R}}(\gamma_{u}, \Phi_{R,y,T}))dy$

,

where
$\Phi_{R,y,T}(m):=m^{\beta R}\int_{\mathrm{K}_{\gamma_{S},S}}\int_{\mathrm{U}_{R,S}}f(y^{-1}\gamma_{s}k^{-1}muky)v_{R}’(ky, T)dudk$is the local analogue of

on

$T$

$\Phi_{R,y,T}$in (4.11). In particular the right hand side does not depend.

We now go back to the calculation of (4.11). Take a finite set
and so thatsufficiently large with respect to
$G$

$S$

of places of $F$$f$$J_{\mathit{0}}(f)= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma,S}\backslash \mathrm{G}_{S}}(\sum_{R\in \mathcal{F}^{G_{\sigma}}(M_{1,\sigma})}\frac{|W^{M_{R}}|}{|W^{G_{\sigma}}|}J_{\{1\}}^{M_{R}}(\Phi_{R,y,T_{1}-T_{1,\sigma}}))dy$

We have following formulae on the variation of$J_{\{1\}}(f)$and$J_{M}(\gamma, f)$$J_{\{1\}}( \mathrm{A}\mathrm{d}(g^{-1})f)=\sum_{Q\in F(M_{0})}\frac{|W^{L}|}{|W|}J_{\{1\}}^{L}(\overline{f}_{Q,g})$$J_{M}( \gamma, \mathrm{A}\mathrm{d}(g^{-1})f)=\sum_{Q\in \mathcal{F}(M_{0})}J_{M}^{L}(\gamma, \overline{f}_{Q,g})$

,

,

where
$\overline{f}_{Q,g}(\ell)=\ell^{\rho_{Q}}\int_{\mathrm{K}}\int_{\mathrm{V}}f(k^{-1}\ell vk)u_{Q}’(k, y)dvdk$$u_{Q}’(k, y)= \int_{\alpha_{L}^{G}}\Gamma_{Q}^{G}(X, -H_{0}(ky))dX$

24.

(4.14)

under conjugations:267
From this Arthur deduced [8, Cor. 8.3] that there exists a family of complex numbers
such that$\{a^{L}(S, u)|L\in \mathcal{L}(M_{0}), u\in \mathcal{U}_{L}(F)\}$$J_{\{1\}}(f)=$$L \in$\sum$c(M_{0})\sum_{u\in u_{L}(F)/\mathrm{A}\mathrm{d}(\mathrm{L}_{S})}\frac{|W^{L}|}{|W|}a^{L}(S, u)J_{L}(u, f_{S})$.

Using this (4.14) becomes$J_{\mathit{0}}(f)= \iota^{G}(\sigma)^{-1}\int_{\mathrm{G}_{\sigma,\mathrm{S}}\backslash \mathrm{G}_{S}}\sum_{L\in \mathcal{L}^{G_{\sigma}}(M_{1,\sigma})}\sum_{R\in F^{G_{\sigma}}(L)}\frac{|W^{L}|}{|W^{G_{\sigma}}|}$$\cross\sum_{u\in \mathcal{U}_{L}(F)/\mathrm{A}\mathrm{d}(\mathrm{L}_{S})}a^{L}(S, u)J_{L}^{M_{R}}(u, \Phi_{R,y,T_{1}-T_{1,\sigma}})dy$

writing$\mathcal{L}_{\sigma}^{0}(M_{1}):=\{M\in \mathcal{L}(M_{1})|a_{M}=a_{M_{\sigma}}\}$,$= \iota^{G}(\sigma)^{-1}\sum_{M\in \mathcal{L}_{\sigma}^{0}(M_{1})}\frac{|W^{M_{\sigma}}|}{|W^{G_{\sigma}}|}\sum_{u\in \mathcal{U}_{M_{\sigma}}(F)/\mathrm{A}\mathrm{d}(\mathrm{M}_{\sigma,S})}$$a^{M_{\sigma}}(S, u)|D^{G}( \sigma)|_{\mathrm{A}_{S}}^{1/2}\int_{\mathrm{G}_{\sigma,S}\backslash \mathrm{G}_{S}}\sum_{R\in F^{G_{\sigma}}(M_{\sigma})}J_{M_{\sigma}}^{M_{R}}(u, \Phi_{R,y,T_{1}-T_{1,\sigma}})dy$

Going back from.

to $G$ by Lem. 4.4, we obtain$G_{\sigma}$$J_{\mathit{0}}(f)= \iota^{G}(\sigma)^{-1}\sum_{M\in \mathcal{L}_{\sigma}^{0}(M_{1})}\frac{|W^{M_{\sigma}}|}{|W^{G_{\sigma}}|}\sum_{u\in \mathcal{U}_{M_{\sigma}}(F)/\mathrm{A}\mathrm{d}(\mathrm{M}_{\sigma,S})}a^{M_{\sigma}}(S, u)J_{M}(\sigma u, f)$.

(4.15)

The fine O-expansion

4.4

Our final step is to get rid of from (4.15). We say that $\gamma’\in M(F)$ is $(M, S)$ -equivalent) if there exists $\delta\in M(F)$ such that
(notated as$\sigma$$\gamma,$$\gamma\sim\gamma’$$(M,S)$$\bullet$$\gamma_{s}’=\delta\gamma_{s}\delta^{-1}$$\bullet$$\gamma_{u}’$

and

and

$\delta\gamma_{u}\delta^{-1}$

are conjugate in$\mathrm{M}_{S}$.

for the (finite) set of $(M, S)$ -equivalence classes in
We write, we set
Noting that $\sigma\in M(F)$ is elliptic if and only if$(M(F)\cap\overline{\mathit{0}})_{M,S}$$M\in \mathcal{L}_{\sigma}^{0}(M_{1})$,$a^{M}(S, \gamma):=\frac{\epsilon^{M}(\gamma_{s})}{\iota^{M}(\gamma_{s})}\sum_{(M,S)}u\in \mathcal{U}_{M_{\gamma}}(F)/\mathrm{A}\mathrm{d}(\mathrm{M}_{\gamma s})\gamma_{s}^{s}u\sim\gamma a^{M_{\gamma s}}(S, u)$

where

.

1
$\epsilon^{M}(\sigma):=\{$$0$

if is elliptic in $M(F)$ ,otherwise.
$\sigma$25$M(F)\cap\overline{\mathit{0}}$.

268This allows us to write$J_{0}(f)= \sum_{M\in \mathcal{L}(M_{1})}\frac{|W^{M}|}{|W|}\frac{|W|\cdot|.W^{M_{\sigma}}|\cdot\iota^{M}(\sigma)}{|W^{M}||W^{G_{\sigma}}|\cdot\iota^{G}(\sigma)}\sum_{\gamma\in(M(F)\cap\overline{\mathit{0}})_{M,S}}a^{M}(S, \gamma)J_{M}(\gamma, f)$$= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}$$\sum_{\mathit{0}_{M\in \mathfrak{O}(M)},\mathrm{A}\mathrm{d}(G(F))0_{M}=0}\sum_{\gamma\in(M(F)\cap\overline{\mathit{0}})_{M,S}}a^{M}(S, \gamma)J_{M}(\gamma, f)$$= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\sum_{\gamma\in(M(F)\cap\overline{0})_{M,S}}a^{M}(S, \gamma)J_{M}(\gamma, f)$.

Theorem 4.5 ([9] Th.9.2). For each compact neighborhood
such that we haveists a finite set of places$\Omega\subset \mathfrak{U}_{G}\backslash \mathrm{G}$of 1, there ex-$S_{\Omega}$$J(f)= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\sum_{\gamma\in M(F)_{M,S}}a^{M}(S, \gamma)J_{M}(\gamma, f)$for$S\supset S_{\Omega}$and$f\in C_{c}^{\infty}(\mathrm{G}/\mathfrak{U}_{G})$supported on$\Omega$.

are constants which are not well-understood. The only
We must remark that
information for these is the following theorem.
$a^{M}(S, \gamma)$

Theorem 4.6 ([9] Th. 8.2). For a semisimple $\gamma\in G(F)$ and sufficiently large
have
if is liptic in $G(F)$ ,$\frac{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(G_{\gamma}(F)\mathfrak{U}_{G}\backslash \mathrm{G}_{\gamma})}{\iota^{G}(\gamma)}$$\gamma$

$S$

, we

$el$

$a^{G}(S, \gamma)=\{$

otherwise.

$0$

5

Fine

$\chi$-expansion

Next we calculate the spectral terms. Let us recall the definition of. We have theinduced space\S 2.6. Define the linear transformation
spacebyon this$J_{X}^{T}(f)$$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}$$\Omega_{\pi,X}^{T}(P, \lambda)$$\int_{\mathrm{K}}\int_{M(F)\mathfrak{U}_{M\backslash \mathrm{M}}}[\Omega_{\pi,X}^{T}(P, \lambda)\phi](mk)\overline{\phi’(mk)}dmdk=\int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\wedge^{T}E_{P}(x, \phi_{\lambda})\overline{\wedge^{T}E_{P}(x,\phi_{\lambda}’)}dx$

for any$\phi,$$\phi’\in A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,\mathrm{X}}$. From (2.6), we know that$\bigwedge_{2}^{T}K_{X}(x, y)=\wedge^{T}K_{X}(x, y)=\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\frac{1}{|P(M)|}$$\int_{i(a_{M}^{G})^{*}}\sum_{\varphi\in \mathfrak{B}_{\pi}x},\wedge^{T}E_{P}(x, \mathcal{I}_{P}^{G}(\pi_{\lambda}, f)\varphi_{\lambda})\wedge^{T}E_{P}(y, \varphi_{\lambda})d\lambda$

26

.

,

269
Replacing this in (3.6), we have
$J_{X}^{T}(f)=$

(5.1)

,

$\sum$$P \in F(P_{0})\sum_{\pi\in\Pi(\mathrm{M}^{1})}\int_{i(\alpha_{M}^{G})^{*}}\frac{1}{|P(M)|}\mathrm{t}\mathrm{r}(\Omega_{\pi,X}^{T}(P, \lambda)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))d\lambda$

denotes the representation ofwhere
to obtain a more explicit expression for this.
$\mathcal{I}_{P,X}^{G}(\pi_{\lambda})$$\mathrm{G}$on$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}$. We would likeAn application of the Paley-Wiener theorem

5.1
We put$\omega^{T}(\lambda, \lambda’, \phi, \phi’)$$:= \sum_{Q=LV\in F(P_{0})}\sum_{w\in W_{M,L}},\sum_{w\in W_{ML}},$meas$(\alpha_{L}^{G}/\mathbb{Z}[\triangle_{Q}^{\mathrm{v}}])$,$\langle M(w, \pi_{\lambda})\phi, M(w’, \pi_{\lambda}’,)\phi’\rangle e^{(w(\lambda)-w’(\lambda’))(T)}$$\cross\overline{\prod_{\alpha\in\Delta_{Q}}(w(\lambda)-w’(\lambda’))(\alpha^{\vee})}$’,. Ex,
for$GL(2)$
case, Langlands proved that if and belong to the space of (induced
tending thefrom) cusp forms, the -inner product of truncated Eisenstein series. For general
we have the following weaker result.
is given by$\phi\in A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X},$$\phi’\in A_{2}(\mathrm{U}’M’(F)\mathfrak{U}_{M’}\backslash \mathrm{G})_{\pi’,X},$$\lambda\in ia_{M}^{*}$$\lambda’\in ia_{M}^{*}$$\phi’$$\phi$$L^{2}$$\langle\wedge^{T}E_{P}(\phi_{\lambda}), \wedge^{T}E_{P’}(\phi_{\lambda}’,)\rangle$$\omega^{T}(\lambda, \lambda’, \phi, \phi’)[30, \S 9]$$\phi,$

Proposition 5.1 ([5] Th. 9.1). There exist, such thaton$\phi’$$\epsilon>0$

and a locally bounded function$\rho(\lambda, \lambda’)$$i\alpha_{M}^{*}\cross ia_{M}^{*}$.

$|\langle\wedge^{T}E_{P}(\phi_{\lambda}), \wedge^{T}E_{P’}(\phi_{\lambda’}’)\ranglearrow\omega^{T}(\lambda, \lambda’, \phi, \phi’)|\leq\rho(\lambda, \lambda’)||\phi||\cdot||\phi’||e^{-\epsilon||T||}$

Here the norms on the right are given by$|| \phi||^{2}:=\int_{\mathrm{K}}\int_{M(F)\mathfrak{U}_{M}\backslash \mathrm{M}}|\phi(mk)|^{2}dmdk$

Example 5.2. In the $GL(2)$ case, we write$\chi$$\chi’$

)

for

$\pi$

and

.

$\pi’\in\Pi(\mathrm{T}^{1})$. Then we have$\langle\wedge^{T}E_{B}(\phi_{\lambda}), \wedge^{T}E_{B}(\phi_{\lambda’}’)\rangle=\langle E_{B}(\phi_{\lambda}), \wedge^{T}E_{B}(\phi_{\lambda’}’)\rangle$$= \int_{B(F)\mathfrak{U}_{G\backslash \mathrm{G}}}E_{B}(x, \phi_{\lambda})(\overline{\phi_{\lambda’}’(x)}-\overline{E_{B}(x,\phi_{\lambda’}’)}_{B}\tau_{B}(\wedge H_{0}(x)-T))dx$$= \int_{\mathrm{U}T(F)\mathfrak{U}_{G\backslash \mathrm{G}}}E_{B}(x, \phi_{\lambda})_{B}(\overline{\phi_{\lambda’}’(x)}-\overline{E_{B}(x,\phi_{\lambda’}’)}_{B}\tau_{B}(\wedge H_{0}(x)-T))dx$

using \S 2.4 (l-ii),$= \int_{\mathrm{U}T(F)\mathfrak{U}_{G\backslash \mathrm{G}}}(\phi_{\lambda}(x)+M(w, \chi_{\lambda})\phi_{\lambda}(x))$$\cross(\overline{\phi_{\lambda}’,(x)}(1-\tau_{B}(\wedge H_{0}(x)-T))-\overline{M(w,\chi_{\lambda}’,)\phi_{\lambda}’,(x)}_{\mathcal{T}_{B}}^{\wedge}(H_{0}(x)-T))dx$

27

.

270
Using the Iwasawa decomposition this becomes, for${\rm Re}\alpha^{\vee}(\lambda’)>\pm{\rm Re}\alpha^{\vee}(\lambda)$,$\int_{-\infty}^{\alpha(T)}\langle\phi, \phi’\rangle e^{t\alpha^{\vee}(\lambda+\overline{\lambda}’)/2}+\langle M(w, \chi_{\lambda})\phi, \phi’\rangle e^{t\alpha^{\vee}(w(\lambda)+\overline{\lambda}’)/2_{\frac{dt}{\sqrt{2}}}}$$- \int_{\alpha(T)}^{\infty}\langle\phi, M(w, \chi_{\lambda’}’)\phi’\rangle e^{t\alpha^{\vee}(\lambda+w(\overline{\lambda}^{J}))/2}+\langle M(w, \chi_{\lambda})\phi, M(w, \chi_{\lambda’}’)\phi’\rangle e^{t\alpha^{\vee}(w(\lambda+\overline{\lambda’}))/2_{\frac{dt}{\sqrt{2}}}}$$= \frac{\sqrt{2}e^{(\lambda+\overline{\lambda}^{J})(T)}}{\alpha^{\vee}(\lambda+\overline{\lambda}’)}\langle\phi, \phi’\rangle+\frac{\sqrt{2}e^{(w(\lambda)+\overline{\lambda}^{J})(T)}}{\alpha^{\vee}(w(\lambda)+\overline{\lambda}’)}\langle M(w, \chi_{\lambda})\phi, \phi’\rangle$$+ \frac{\sqrt{2}e^{(\lambda+w(\overline{\lambda}^{J}))(T)}}{\alpha^{\vee}(\lambda+w(\overline{\lambda}’))}\langle\phi, M(w, \chi_{\lambda’}’)\phi’\rangle+\frac{\sqrt{2}e^{w(\lambda+\overline{\lambda}^{J})(T)}}{\alpha^{\vee}(w(\lambda+\overline{\lambda}’))}\langle M(w, \chi_{\lambda})\phi, M(w, \chi_{\lambda’}’)\phi’\rangle$

If we restrict the last formula to$(\lambda, \lambda’)\in i\alpha_{T}^{*}\cross i\alpha_{T\rangle}^{*}$

we obtain$\omega^{T}(\lambda, \lambda’, \phi, \phi’)$.

. In the higher

an irreducible quotient of an induced fromcan be in the residual
rank cases,
cuspidal representation. Since it throw away some submodules which might contribute to
the inner product $formula_{f}$ we can control the inner product only asymptotically.
$spectrum_{f}$$\pi$for the inner product of truncated Eisenstein
We would like to replaceseries in (5.1). But this is not allowed because the domain of the integral is not compact.
We bypass this difficulty in the following way.
which is contained inofFix an -minimal parabolic subgroupfor the Lie algebra of the. Writethe global minimal parabolic subgroupin the Lie algebra ofsubalgebra.
aChoose Cartanofsplit component. Writefor the, a Cartan subalgebra ofand set-invariant compactly
be the space ofand letin
Weyl group of
Paley-Wiener
theoremcorollaryof thesupported distributions on . The following is afor real reductive groups.
$\omega^{T}(\lambda, \lambda’, \phi, \phi’)$$P_{\infty}=M_{\infty}U_{\infty}$$\mathbb{R}$$\mathbb{R}-$$\mathrm{P}_{0,\infty}$$\mathrm{A}_{\infty}$$\mathrm{K}_{\infty}\cap M_{\infty}$$\mathrm{G}_{\infty}$$a_{\infty}$$Z(M_{\infty})$$\mathfrak{h}_{\mathrm{K}_{\infty}}$$\mathfrak{h}:=i\mathfrak{h}_{\mathrm{K}_{\infty}}\oplus\alpha_{\infty}$$\mathfrak{h}_{\mathbb{C}}$$W(\mathfrak{h}_{\mathbb{C}})$$\mathcal{E}(\mathfrak{h})^{W(\mathfrak{h}_{\mathbb{C}})}$$\mathrm{g}_{\infty}(\mathbb{C})$$W(\mathfrak{h}_{\mathbb{C}})$$\emptyset\infty$$\mathfrak{h}$Proposition 5.3 (Multiplier theorem, [7] Th. 4.2). For
such that, we can find$\gamma\in \mathcal{E}(\mathfrak{h})^{W(\mathfrak{h}_{\mathbb{C}})}$

and$f_{\infty}\in$$f_{\infty,\gamma}\in \mathcal{H}(\mathrm{G}_{\infty})$$\mathcal{H}(\mathrm{G}_{\infty})$$\pi_{\infty}(f_{\infty,\gamma})=\gamma(\wedge\chi_{\pi_{\infty}})\pi_{\infty}(f_{\infty})$denotes the Fourier transform offinitesimal character of .
$Here\wedge\gamma$$\gamma$and,$\forall\pi_{\infty}\in\Pi(\mathrm{G}_{\infty})$$\chi_{\pi_{\infty}}\in \mathfrak{h}_{\mathbb{C}}^{*}$.

is a representative of the in-$\pi_{\infty}$

for the kernel ofWrite, we have:component of$\mathfrak{h}\exparrow \mathrm{G}_{\infty}H_{G}arrow\alpha_{G}$$\mathfrak{h}^{G}$. Applying this proposition to the infinite$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$Corollary 5.4. For
that$\gamma\in \mathcal{E}(\mathfrak{h}^{G})^{W(\mathfrak{h}_{\mathrm{C}})}$, we have a linear map$\pi(f_{\gamma})=\gamma(\wedge\chi_{\pi_{\infty}})\pi(f))$$f\mapsto f_{\gamma}$$\forall\pi\in\square (\mathrm{G}^{1})$.

We use the abbreviation$\Psi_{X,\pi}^{T}(\lambda, f):=\frac{1}{|P(\mathcal{M})|}\mathrm{t}\mathrm{r}(\Omega_{\pi,X}^{T}(P, \lambda)\mathcal{I}_{P_{)}X}^{G}(\pi_{\lambda}, f))$

28

.

on

$\mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$

such

271
There exists $C_{0}>0$ such that if $\alpha(T)>C_{0},$$\forall\alpha\in\triangle_{0}$we have$J_{\chi}^{T}(f_{\gamma})= \sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\int_{i(\alpha_{M}^{G})^{*}}\Psi_{\mathrm{X},\pi}^{T}(\lambda, f_{\gamma})d\lambda$$= \sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\square (\mathrm{M}^{1})}\int_{i(\alpha_{M}^{G})^{*}}\wedge\gamma(\chi_{\pi_{\infty}}+\lambda)\Psi_{X,\pi}^{T}(\lambda, f)d\lambda$$= \sum_{P\in F(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\int_{i(\alpha_{M}^{G})^{*}}\Psi_{X,\pi}^{T}(\lambda, f)\int_{\mathfrak{h}^{G}}\gamma(X)e^{(\chi_{\pi}+\lambda)(X)}\infty dXd\lambda$,$= \int_{\mathfrak{h}^{G}}\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\psi_{X,\pi}^{T}(X, f)\gamma(X)e^{\chi_{\pi}(X)}\infty dX$

where$\psi_{X,\pi}^{T}(X, f):=\int_{i(a_{M}^{G})^{*}}\Psi_{X,\pi}^{T}(\lambda, f)e^{\lambda(X)}d\lambda$

In particular, if we write
tions), then for $T$ with $\alpha(T)>C_{0},$$\gamma_{H}:=|W(\mathfrak{h}\mathrm{c})|^{-1}\sum_{w\in W(\mathfrak{h}_{\mathrm{C}})}\delta_{w^{-1}(H)}$$\forall\alpha\in\triangle_{0}$.

(average of Dirac distribu-

we have$J_{X}^{T}(f_{\gamma_{H}})= \frac{1}{|W(\mathfrak{h}_{\mathbb{C}})|}\sum_{w\in W(\mathfrak{h}_{\mathbb{C}})}\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\psi_{X,\pi}^{T}(w^{-1}(H), f)e^{\chi_{\pi}(w^{-1}(H))}\infty$

(5.2)

.

Let $p^{T}(H)$ be the right hand side. This is a polynomial in and is smooth in $H$ . We canas $p^{T}(0)=‘(\delta_{0}(p^{T})$ ”.
recover
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$T$$J_{X}^{T}(f)$$\psi_{\lambda}^{T}(H):=\frac{1}{|W(\mathfrak{h}_{\mathbb{C}})|}$$\sum$$P \in \mathcal{F}(P_{0})(w,\pi)\in w\sum_{W\mathfrak{y}_{\mathrm{c}})\cross\Pi(\mathrm{M}^{1})}\psi_{X,\pi}^{T}(w^{-1}(H), f)e\infty{\rm Im}\chi_{\pi}(w^{-1}(H))$, the sum is finite. We canof each real exponent ofasymptotically approximate these by polynomial functions:of polyLemma 5.5 ([6] , Prop. 5.1). There exists a unique (finite) family
$C,$
any
. For
which satisfies the following conditions for somediffernomials in$D$$c_{D}>0$
that
such
we can findonential operator$\lambda$$p^{T}(H):p^{T}(H)= \sum_{\lambda}\psi_{\lambda}^{T}(H)e^{\lambda(H)}$$\{p_{\lambda}^{T}(H)\}$$\mathrm{I}$$\epsilon>0$$T$$\mathfrak{y}^{c_{;}}$(1)$|D( \psi_{\lambda}^{T}(H)-p_{\lambda}^{T}(H))|\leq c_{D}\exp(-\epsilon\inf_{\alpha\in\triangle_{0}}\alpha(T))(1+||T||)^{d0_{J}}$ciently positive $T$ ,(2) $|Dp_{\lambda}^{T}(H)|\leq c_{D}(1+||H||)^{d_{0}}(1+||T||)^{d_{0}}$ , forHere$d_{0}$

is the maximum of the degrees of

By construction, these$p_{\lambda}^{T}(H)$$p_{\lambda}^{T}$$H\in \mathfrak{h}^{G}$

are tempered and we can consider,$T$and $T\in a_{0}$ .

.

$p_{\lambda}^{T}( \beta):=\int_{\mathfrak{h}^{G}}p_{\lambda}^{T}(H)\beta(H)dH$

a polynomial infor. Moreover we have29$\beta\in S(\mathfrak{h}^{G})$,$H\in \mathfrak{h}^{G}$and suffi-

272(1) Forsatisfying$\beta\in S(\mathfrak{h}^{G})$$\int_{\mathfrak{h}^{G}}\beta(H)dH=1$

we have(2) For$\lim_{\epsilonarrow 0}p_{\lambda}^{T}(\beta_{\epsilon})=p_{\lambda}^{T}(0)$

where

,

$\beta_{\epsilon}(H):=\epsilon^{-\dim \mathfrak{h}^{G}}\beta(\epsilon^{-1}H)$.

, we have$\beta\in S(\mathfrak{h}^{G})$$\int_{\mathfrak{h}^{G}}\psi_{\lambda}^{T}(H)\beta(H)dH-p_{\lambda}^{T}(\beta)arrow 0$

as$\alpha(T)arrow\infty$

for any$\alpha\in\triangle_{0}$.

Now we are ready to calculate $F_{X}(f)$ . Take
. There is a
restriction of this to$i(\alpha_{M}^{G})^{*}\subset i(\mathfrak{h}^{G})^{*}$$\beta\in S(\mathfrak{h}^{G})$$B( \lambda)=\int_{\mathfrak{h}^{G}}\beta(H)e^{\lambda(H)}dH$

If we put(1), (2) imply:$P^{T}(B):= \sum_{\lambda}p_{\lambda}^{T}(\beta)$(1) If $B(\mathrm{O})=1$ we have

and write
such that$B\in C_{c}^{\infty}(i(\mathfrak{h}^{G})^{*})$$B_{M}$

for the.

, then the theory of Fourier transformation and the above$\lim_{\epsilonarrow 0}P^{T}(B_{\epsilon})=J_{X}^{T}(f)$, where$B_{\epsilon}(\lambda):=B(\epsilon\lambda)$.

(2) $P^{T}(B)$ is the unique polynomial in $T$ such that$\sum_{P\in F(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\int_{i(\alpha_{M}^{G})^{*}}\Psi_{X,\pi}^{T}(\lambda, f)B_{M}(\lambda)d\lambda-P^{T}(B)$goes to zero as$\alpha(T)$tends to infinity for any$\alpha\in\triangle_{0}$.

Consider the linear transformation$A_{P,\pi,X}( \lambda, \lambda’):=\sum_{Q\in F(P_{0})},\sum_{w,w\in W_{M,L}}\frac{e^{(w’(\lambda’)-w(\lambda))(T)}}{\theta_{Q}^{G}(w’(\lambda’)-w(\lambda))}M(w, \pi_{\lambda})^{-1}M(w’, \pi_{\lambda}’)$on
imaginary axis, we have$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}$. Since the global intertwining operators are unitary on the$\langle A_{P,\pi,X(\lambda,\lambda’)\phi’,\phi\rangle=\sum_{Q\in \mathcal{F}(P_{0})}},\sum_{w,w\in W_{M,L}}\frac{\langle M(w’,\lambda’)\phi’,M(w,\lambda)\phi\rangle e^{(w’(\lambda’)-w(\lambda))(T)}}{\theta_{Q}^{G}(w’(\lambda’)-w(\lambda))}$$=\omega^{T}(\lambda’, \lambda, \phi’, \phi)$.

It is shown thatis holomorphic on[ , Cor.9.2], and we.
can define
for
Since
are compactly supported,
we can apply Prop. 5.1 to see that $P^{T}(B)$ is the unique polynomial in $T$ such that$i(a_{M}^{G})^{*}\cross i(a_{M}^{G})^{*}$$\omega^{T}(\lambda’, \lambda, \phi’, \phi)$$\omega_{X,\pi}^{T}(P, \lambda):=A_{P,\pi,X}(\lambda, \lambda)$$5$$B_{M}$$\lambda\in i(\alpha_{M}^{G})^{*}$$\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\frac{1}{|P(M)|}\int_{i(a_{M}^{G})^{*}}\mathrm{t}\mathrm{r}[\omega_{X,\pi}^{T}(P, \lambda)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)]B_{M}(\lambda)d\lambda-P^{T}(B)$goes to zero as$\alpha(T)$tends to infinity for any

30$\alpha\in\triangle_{0}$.

273
Logarithmic derivatives

5.2

Yet we need to give an explicit expression for$\omega_{X,\pi}^{T}(P, \lambda)$. We first look at the simplest

case.
Example 5.6. Suppose $G=GL(2)$ . In the notation of Ex. 5.2, we have$A_{B,\chi,X}( \lambda, \lambda’)=\frac{\sqrt{2}e^{(\lambda’-\lambda)(T)}}{\alpha^{\vee}(\lambda’-\lambda)}+\frac{\sqrt{2}e^{(w(\lambda’)-\lambda)(T)}}{\alpha^{\vee}(w(\lambda’)-\lambda)}M(w, \chi_{\lambda’})$$+ \frac{\sqrt{2}e^{(\lambda’-w(\lambda))(T)}}{\alpha^{\vee}(\lambda’-w(\lambda))}M(w, \chi_{\lambda})^{-1}+\frac{\sqrt{2}e^{w(\lambda’-\lambda)(T)}}{\alpha^{\vee}(w(\lambda’-\lambda))}M(w, \chi_{\lambda})^{-1}M(w, \chi_{\lambda’})$$= \frac{\sqrt{2}}{\alpha^{\vee}(\lambda’-\lambda)}(e^{(\lambda’-\lambda)(T)}-e^{(\lambda’-\lambda)(w(T))}M_{\overline{B}|B}(1, \chi_{\lambda})^{-1}M_{\overline{B}|B}(1, \chi_{\lambda’}))$$+ \frac{\sqrt{2}}{\alpha^{\vee}(w(\lambda’)-\lambda)}(e^{(w(\lambda’)-\lambda)(T)}M(w, \chi_{\lambda’})-e^{(w(\lambda’)-\lambda)(w(T))}M_{\overline{B}|B}(1, \chi_{\lambda})^{-1}M_{\overline{B}|B}(w, \chi_{\lambda’}))$

where we have written
becomes holomorphic ontrates how
the first row in the right hand side restricts to$M_{\overline{B}|B}(1, \chi_{\lambda}):=w\circ M(w, \chi_{\lambda})$$A_{B,\chi,X}(\lambda, \lambda’)$and. Moreover, writing$M_{\overline{B}|B}(w, \chi_{\lambda}):=w$$\lambda=\lambda’$$\lambda=\lambda’$. This illus,$\lambda’=\lambda+t\alpha/2$

as$\lim_{tarrow 0}\frac{\sqrt{2}}{t}[e^{\alpha(T)t/2}-e^{-\alpha(T)t/2}M_{\overline{B}|B}(1, \chi_{\lambda})^{-1}M_{\overline{B}|B}(1, \chi_{\lambda+t\alpha/2})]$$= \lim_{tarrow 0}\frac{\sqrt{2}}{t}[1-e^{-\alpha(T)t}M_{\overline{B}|B}(1, \chi_{\lambda})^{-1}M_{\overline{B}|B}(1, \chi_{\lambda+t\alpha/2})]$,

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 twokinds of $(G, M)$ -families. We need the following recapitulation of intertwining operators.
. For $P\in \mathcal{P}(M),$ $P’\in P(M’)$ and $w\in W_{M,M’}$ , we define
Take $M,$$M’\in \mathcal{L}(M_{0})$,$[M_{P’|P}(w, \pi_{\lambda})\phi](x)=\int_{(\mathrm{U}’\cap^{w}\mathrm{U})\backslash \mathrm{U}’}\phi(w^{-1}ux)e^{\langle\lambda+\rho_{P},H_{P}(w^{-1}ux)\rangle}du\cdot e^{\langle-(w(\lambda)+\rho_{P’}),H_{P’}(x)\rangle}$a linear operator
show that$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}arrow A_{2}(\mathrm{U}’M’(F)\mathfrak{U}_{M’}\backslash \mathrm{G})_{w(\pi),X}$. One can easily$v\circ M_{P’|P}(w, \pi_{\lambda})=e^{-\langle(w(\lambda)+\rho_{P};),T_{1}-v^{-1}(T_{1})\rangle}M_{v(P’)|P}(vw, \pi_{\lambda})$.

$M_{P’|P}(w, \pi_{\lambda})\circ v^{-1}=e^{\langle\lambda+\rho_{P},T_{1}-v^{-1}(T_{1})\rangle}M_{P^{l}|v(P)}(wv^{-1}, v(\pi_{\lambda}))$If we choose$w=v_{1}’w_{1}v_{1}^{-1}$$P_{1},$$P_{1}’\in \mathcal{F}(P_{0})$for someand$v_{1},$$w_{1}\in W_{M_{1},M_{1}’}$, we may write
such that $P=^{v_{1}}P_{1},$. Then the above formulae imply that$v_{1}’\in W$$P’=^{v_{1P_{1}’}’}$$M_{P’|P}(w, \pi_{\lambda})$$=e^{\langle w_{1}v_{1}^{-1}(\lambda)+\rho_{P_{1}’},T_{1}-v_{1}^{l-1}(T_{1})\rangle}e^{-\langle(\lambda)(T_{1})\rangle}-v_{1}^{-1}v_{1}’\circ M_{P_{1}’|P_{1}}v_{1}^{-1}+\rho_{P_{1}},T_{1}(w_{1}, v_{1}^{-1}(\lambda))\circ v_{1}^{-1}$

31

.

(5.3)

,

274
.
to those ofThis relation allows us to translate the basic properties of
In particular the latter operator absolutely converges if the real part of belongs to acone, and extends meromorphically to. This defines an intertwining operator atwhere it is holomorphic. Finally this fits into the functional equations:$M(w, \pi_{\lambda})$$M_{P’|P}(w, \pi_{\lambda})$$\lambda$$\lambda$$a_{M,\mathbb{C}}^{*}$$E_{P’}(x, M_{P’|P}(w, \pi_{\lambda})\phi_{\lambda})=E_{P}(x, \phi_{\lambda})$$M_{P’’|P’}(w’, w(\pi_{\lambda}))M_{P’|P}(w, \pi_{\lambda})=M_{P’’|P}(w’w, \pi_{\lambda})$

These results allow us to write$A_{P,\pi,X}(\lambda, \lambda’)$

,

(5.4)
(5.5)

,

$P”\in P(M’’),$ $w’\in W_{M’,M’’}$ .

as:$A_{P,\pi,X}( \lambda, \lambda’)=\sum_{P_{1}\supset P_{0}}\sum_{w,w’\in W_{M,M_{1}}}\frac{e^{(w’(\lambda’)-w(\lambda))(T)}}{\theta_{P_{1}}^{G}(w’(\lambda’)-w(\lambda))}M_{P_{1}|P}(w, \pi_{\lambda})^{-1}M_{P_{1}|P}(w’, \pi_{\lambda}’)$$= \sum_{P_{1}\supset P_{0}}v\in W_{M,M_{1}}\sum_{w\in W_{M,M}}\frac{e^{v(w(\lambda’)-\lambda)(T)}}{\theta_{P_{1}}^{G}(v(w(\lambda’)-\lambda))}M_{P_{1}|P}(v, \pi_{\lambda})^{-1}M_{P_{1}|P}(vw, \pi_{\lambda})$using (5.5)$= \sum_{P_{1}\supset P_{0}}v\in W_{M,M_{1}}\sum_{w\in W_{M,M}}\frac{e^{v(w(\lambda’)-\lambda)(T)}e^{(w(\lambda’)-\lambda)(T_{1}-v^{-1}(T_{1}))}}{\theta_{v^{-1}(P_{1})}^{G}(w(\lambda’)-\lambda)}$$\cross M_{v^{-1}(P_{1})|P}(1, \pi_{\lambda})^{-1}M_{v^{-1}(P_{1})|P}(w, \pi_{\lambda}’)$putting $Q:=v^{-1}(P_{1})\in P(M)$$= \sum_{Q\in P(M)}\sum_{w\in W_{MM}},\frac{e^{\langle w(\lambda’)-\lambda,Y_{Q}(T)\rangle}}{\theta_{Q}^{G}(w(\lambda’)-\lambda)}M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(w, \pi_{\lambda}’)$.

Here we have written $Y_{Q}(T):=T_{1}+v^{-1}(T-T_{1})$ for $Q=v^{-1}(P_{1}),$ $v\in W_{M,M_{1}}$ . To
, which equals the restriction of
calculate$\mathrm{t}\mathrm{r}[\omega_{\pi,X}^{T}(P, \lambda)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)]$

(5.6)$\sum_{w\in W_{MM}},\sum_{Q\in P(M)}\frac{e^{(w(\lambda’)-\lambda)(Y_{Q}(T))}}{\theta_{Q}^{G}(w(\lambda^{l})-\lambda)}\mathrm{t}\mathrm{r}(M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(w, \pi_{\lambda’})\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))$to$\lambda=\lambda’$, we note that$c_{Q}(T, \Lambda):=e^{\Lambda(Y_{Q}(T))}$,$d_{Q}^{w}(\Lambda):=$tr$(M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(w, \pi_{\lambda+\Lambda})\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))$form $(G, M)$ -families. Then we know from Lem. 4.3 that (5.6) is smooth at any
andtr$(\lambda, \lambda’)$.

$[ \omega_{\pi,X}^{T}(P, \lambda)\mathcal{I}_{P,\mathrm{X}}^{G}(\pi_{\lambda}, f)]=\sum_{w\in W_{MM}},\sum_{P_{1\in}\mathcal{F}(M)}c_{M}^{P_{1}}(T, w(\lambda)-\lambda)d_{P_{1}}^{w/}(\lambda_{L_{w}}, w(\lambda)-\lambda)$

32

275
Here

$L_{w}\in \mathcal{L}(M)$is such that $\alpha_{L_{w}}=\{H\in\alpha_{M}|w(H)=H\}$ and,$d_{Q}^{w}(\lambda_{L_{w}}, \Lambda):=\mathrm{t}\mathrm{r}(M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(w, \pi_{\lambda+(})\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))$. (Notice that and can be recovered from
if $\Lambda=w(\lambda)-\lambda+\zeta,$with the result of \S 5.1 to see that
the
above
now
combineWe$\lambda$$\zeta\in i\alpha_{M}^{*}$$\zeta$$\Lambda$

and$\lambda_{L_{w}}.$)$\sum_{P\in F(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\frac{1}{|P(M)|}\sum_{w\in W_{MM}}$

,

(5.7)

$\int_{i(\alpha_{M}^{G})^{*}}\sum_{P_{1}\in \mathcal{F}(M)}c_{M}^{P_{1}}(T, w(\lambda)-\lambda)d_{P_{1}}^{w\prime}(\lambda_{L_{w}}, w(\lambda)-\lambda)B_{M}(\lambda)d\lambda$,
is asymptotic to $P^{T}(B)$ . Write
of $\{Y_{Q}(T)|Q\in \mathcal{P}(M), Q\subset P_{1}\}$ , then$\chi_{M}^{P_{1}}(T$$\bullet$$)$for the characteristic function of the convex hullis its Fourier transform [4, \S 6]:$c_{M}^{P_{1}}(T)$$c_{M}^{P_{1}}(T, \mu)=\int_{(Y_{Q}(T))_{M_{1}}+a_{M}^{M_{1}}}\chi_{M}^{P_{1}}(T, X)e^{\mu(X)}dX$

Then the integral over$i(a_{M}^{G})^{*}$.

can be calculated as$\frac{1}{|\det(w-1|\alpha_{M}^{L_{w}})|}\sum_{P_{1\in}\mathcal{F}(\cdot M)}\int_{(Y_{Q}(T))_{M_{1}}+\alpha_{M}^{M_{1}}}\chi_{M}^{P_{1}}(T, H)$$\int_{\mathrm{i}(a_{M}^{L_{w}})^{*}}\int_{i(\alpha_{L_{w}}^{G})^{*}}e^{\mu(H)}d_{P_{1}}^{w\prime}(\lambda, \mu)B_{M}((w-1)^{-1}(\mu)+\lambda)d\lambda d\mu dH$

In this, the terms associated to
becomes
associated to$P_{1}\not\supset L_{w}$goes to$0$

as $\alpha(T)arrow\infty,$$\forall\alpha\in\triangle_{0}$. Those$P_{1}\supset L_{w}$.

$\frac{1}{|\det(w-1|\alpha_{M}^{L_{w}})|}c_{L_{w}}^{P_{1}}(T, 0)\int_{i(a_{L_{w}}^{G})^{*}}d_{P_{1}}^{w\prime}(\lambda, 0)B_{M}(\lambda)d\lambda$

We conclude that (5.7) equals$\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\frac{1}{|P(M)|}\sum_{w\in W_{MM}}$

,

(5.8)

$\sum$$\frac{1}{|\det(w-1|\alpha_{M}^{L_{w}})|}\int_{i(\alpha_{L_{w}}^{G})^{*}}P_{1}\in \mathcal{F}(L_{w})$Final step towards the finederivat ”$X$$c_{L_{w}}^{P_{1}}(T, 0)d_{P_{1}}^{w\prime}(\lambda, 0)B_{M}(\lambda)d\lambda$.

-expansion is to look at the nature of the “logarithmic$\mathrm{i}\mathrm{v}\mathrm{e}$$\sum_{P_{1}\in \mathcal{F}(L_{w})}c_{L_{w}}^{P_{1}}(T, 0)d_{P_{1}}^{w\prime}(\lambda, 0)=\sum_{Q\in P(L_{w})}\frac{c_{Q}(T,\Lambda)d_{Q}^{w}(\lambda,\Lambda)}{\theta_{Q}^{G}(\Lambda)}|_{\Lambda=0}$

As in Ex. 5.6, we divide$d_{Q}^{w}(\lambda, \Lambda)$as$d_{Q}^{w}(\lambda, \Lambda)=\mathrm{t}\mathrm{r}[(M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(1, \pi_{\lambda+\Lambda}))\mathrm{o}(M_{P|P}(w, \pi_{\lambda+\Lambda})\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))]$

33

(5.9)

276
The latter simply restricts to $\Lambda=0$ . But in the former,

A4$Q(P, \pi, \lambda, \Lambda):=M_{Q|P}(1, \pi_{\lambda})^{-1}M_{Q|P}(1, \pi_{\lambda+\Lambda})$is a $(G, M)$ -family in$\Lambda$,$Q\in P(M)$and so is$\mathcal{M}_{Q}^{T}(P, \pi, \lambda, \Lambda):=c_{Q}(T, \lambda)\mathcal{M}_{Q}(P, \pi, \lambda, \Lambda)$.

This combined with Lem. 4.3 justifies to write the specialization of (5.9) to $\Lambda=0$ as.

$\mathrm{t}\mathrm{r}[(\sum_{Q\in P(L_{w})}\frac{1}{\theta_{Q}^{G}(\Lambda)}\mathcal{M}_{Q}^{T}(P, \pi, \lambda, \Lambda)|_{\Lambda=0})M_{P|P}(w, \pi)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)]$

Note that $M_{P|P}(w, \pi_{\Lambda+\lambda})=M_{P|P}(w, \pi)$ for. Since this is a linear combination$\Lambda=0$
of certain derivatives atof the exponential functions $c_{Q}(T, \Lambda)=e^{\Lambda(Y_{Q}(T))}$ , theexponents 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:$\Lambda+\lambda\in ia_{L}^{*}$$P^{T}(B)= \sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\sum_{L\in \mathcal{L}(M)}\sum_{w\in W_{MM}^{L,\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{1}{|P(M)|}\frac{1}{|\det(w-1|\alpha_{M}^{L})|}$(5.10).

$\cross\int_{i(\alpha_{L}^{G})^{*}}\mathrm{t}\mathrm{r}(\mathcal{A}4_{L}^{T}(P, \pi, \lambda, 0)M_{P|P}(w, \pi)\mathcal{I}_{P,X}(\pi_{\lambda}, f))B_{M}(\lambda)d\lambda$

Here. Once we have an equality, we canspecialize it to $T=T_{1}$ . Since $Y_{Q}(T_{1})=T_{1}$ , we have
$W_{M,M}^{L,\mathrm{r}\mathrm{e}\mathrm{g}}:=\{w\in W_{M,M}^{L}|\det(w-1|a_{M}^{L})\neq 0\}$$\mathrm{A}4_{L}^{T_{1}}(P, \pi, \lambda, 0)=e^{\Lambda(T_{1})}\mathcal{M}_{L}(P, \pi, \lambda, \Lambda)|_{\Lambda=0}=\mathcal{M}_{L}(P, \pi, \lambda, 0)$.

Using p. 30 (1), we conclude from (5.10) that
$J_{X}(f)= \lim_{\epsilonarrow 0}\sum_{P\in \mathcal{F}(P_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\sum_{L\in \mathcal{L}(M)}\sum_{w\in W_{MM}^{L,\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{1}{|P(M)|}\frac{1}{|\det(w-1|\alpha_{M}^{L})|}$$\cross\int_{i(a_{L}^{G})^{*}}\mathrm{t}\mathrm{r}(\mathcal{M}_{L}(P, \pi, \lambda, 0)M_{P|P}(w, \pi)\mathcal{I}_{P,X}(\pi_{\lambda}, f))(B_{\epsilon})_{M}(\lambda)d\lambda$$= \lim_{\epsilonarrow 0}\sum_{M\in \mathcal{L}(M_{0})}\sum_{L\in \mathcal{L}(M)}\sum_{\pi\in\square (\mathrm{M}^{1})}\sum_{w\in W_{MM}^{L,\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{|W^{M}|}{|W|}\frac{1}{|\det(w-1|a_{M}^{L})|}$(5.11)$\cross\int_{i(\alpha_{L}^{G})^{*}}\frac{1}{|P(M)|}\sum_{P\in P(M)}\mathrm{t}\mathrm{r}(\mathcal{M}_{L}(P, \pi, \lambda, 0)M_{P|P}(w, \pi)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))(B_{\epsilon})_{M}(\lambda)d\lambda$

Here$B\in C_{c}^{\infty}(i(\mathfrak{h}^{G})^{*})$

is such that $B(\mathrm{O})=1$ .

34

.

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5.3

Normalization and estimation of intertwining operators

We hope to get rid of the limit
and the factor
from (5.11). For this we haveto estimate the integrand and show that it converges without the factor. For such an
estimation, we adopt the usual approach of normalizing intertwining operators.
The normalization is constructed locally. Consider a connected reductive group overa local field $F$ of characteristic zero. We adopt the local analogue of various notation presented above. In particular we write $\mathrm{G}=G(F)$ and fix its maximal compact subgroupso that the Iwasawa decomposition
holds for any $P\in F(M_{0})$ . For,
write
for the set of isomorphic classes of irreducible admissible representations(irreducible-modules if $F$ is archimedean) of M. For
and $P\in P(M)$ ,write
for the space of smooth right -finite functions on
satisfying$\epsilonarrow 0$$B_{\epsilon}$$B_{\epsilon}$$G$$\mathrm{K}$$\mathrm{G}=\mathrm{P}\mathrm{K}$$M\in \mathcal{L}(M_{0})$$\Pi_{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M})$$(\mathfrak{m}_{\mathbb{C}},$$\mathrm{K}^{M})$$\pi\in\square (\mathrm{M})$$\mathrm{K}$$\mathcal{V}_{P}(\pi)$$\phi(umg)=\pi(m)\phi(g)$ ,The parabolically induced representation$\mathrm{G}$$u\in \mathrm{U},$$\mathcal{I}_{P}^{G}(\pi_{\lambda}),$$m\in \mathrm{M},$$\lambda\in a_{M,\mathbb{C}}^{*}$$[\mathcal{I}_{P}^{G}(\pi_{\lambda}, g)\phi](x):=\phi(xg)e^{\langle\lambda+\rho \mathrm{p}_{\rangle}H_{P}(xg)\rangle}e^{-\langle\lambda+\rho_{P},H_{P}(x)\rangle}$,$g\in \mathrm{G}$.

is defined by$g\in \mathrm{G},$$\phi(x)\in \mathcal{V}_{P}(\pi)$.

This is isomorphic to the usual parabolically induced representation by$\mathcal{I}_{P}^{G}(\pi_{\lambda})\ni\phi(x)-\phi(x)e^{\langle\lambda+\rho_{P)}H_{P}(x)\rangle}\in \mathrm{i}\mathrm{n}\mathrm{d}_{\mathrm{P}}^{\mathrm{G}}[\pi_{\lambda}\otimes 1_{\mathrm{U}}]$

As in the global case, we define the intertwining integral$M’U’\in \mathcal{F}(M_{0}),$ $w\in W_{M,M’})$ by.

$M_{P’|P}(w, \pi_{\lambda}),$$(P=MU,$ $P’=$.

$[M_{P’|P}(w, \pi_{\lambda})\phi](x):=\int_{(\mathrm{U}’\cap^{w}\mathrm{U})\backslash \mathrm{U}’}\phi(w^{-1}ux)e^{\langle\lambda+\rho_{P},H_{P}(w^{-1}ux)\rangle}du\cdot e^{\langle-w(\lambda)+\rho_{P};),H_{P’}(x)\rangle}$

Proposition 5.7 ([27], [39] \S 2.2). (i) $[M_{P’|P}(w, \pi_{\lambda})\phi](x)$ converges absolutely if
belongs to some open cone in.
; and meromorphically continued to the whole$W_{M}(G)$(ii) If we write
lengththeonrespectwith
to$P\in P(M)$forfunction[ , I. 1. , then for $w\in W_{M,M’}$ and $w’\in W_{M’,M’’}$ with $P_{P}(w’w)=p_{P}(w)+p_{P’}(w’)_{;}$ thefunctional equation${\rm Re}(\lambda)$$\alpha_{M}^{*}$$\alpha_{M,\mathbb{C}}^{*}$$\ell_{P}(w)$$7J$$\mathit{3}\mathit{3}$$M_{P’’|P’}(w’, w(\pi_{\lambda}))M_{P’|P}(w, \pi_{\lambda})=M_{P’’|P}(w’w, \pi_{\lambda})$holds.

Consider first the following two special cases.
(1)

$F$

is archimedean.

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

35

278
(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
for the simple roots of $T$ in B. $G$ is quasisplit if it
admits a splitting $\mathrm{s}\mathrm{p}1_{G}=(B, T, \{X\})$ which is stable under. Then a characterof the unipotent radicalof is non-degenerate if its stabilizer in equals the centerof G. For such , a -Whittaker functional on
is a linear functional:on a realization of such that$\{X\}\subset \mathrm{g}$$\mathrm{G}\mathrm{a}1(\overline{F}/F)$$\theta$$\mathrm{N}$$\theta$$\mathrm{B}$$\mathrm{B}$$\pi\in\Pi(\mathrm{G})$$\theta$$\Lambda_{\pi}^{\theta}$$V_{\pi}arrow \mathbb{C}$$\pi$$\Lambda_{\pi}^{\theta}(\pi(n)\xi)=\theta(n)\Lambda_{\pi}^{\theta}(\xi)$,$n\in \mathrm{N},$$\xi\in V_{\pi}$.

is -generic if it admits a non-trivial -Whittaker functional.) In theseWe say thatcases, we have the automorphic and -factors $L(s, \pi, r)$ andof attached tocertain finite dimensional continuous representation of the -group $LG$ of $G[17],$ beinga non-trivial character of F. (N.B. The -group should be the Weil form $LG=\hat{G}x_{\rho_{G}}W_{F}$instead of the Galois form
adopted in [17], since some important cocycles
ondoes not split while its inflation to
does.) In the case (1), these aredefined in terms of the local Langlands correspondence established in [32]. In the case(2), the definition is given in [40, \S 7]. Now let $P,$ $P’,$
be as
, set
above. Writing$\pi$$\theta$$\theta$$L$$\epsilon(s, \pi, r, \psi)$$\epsilon$

$\pi$

$L$

$r$

$\psi$$L$$\hat{G}\cross_{\rho_{G}}\mathrm{G}\mathrm{a}1(\overline{F}/F)$$W_{F}$$\mathrm{G}\mathrm{a}1(\overline{F}/F)$$\pi\in\Pi_{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M}^{1}),$$\lambda\in i\alpha_{M}^{*}$$\wedge \mathrm{u}_{w}:=\mathrm{u}/w^{-1}\mathfrak{U}’\cap n$$r_{w}$: $LM\ni m\lambda w-\mathrm{A}\mathrm{d}(m)\circ\rho c(w)|_{u_{w}}\wedge\in GL(u_{w})\wedge$ .

Define the normalization factor for$M_{P’|P}(w, \pi_{\lambda})$by.

$r_{P’|P}(w, \pi_{\lambda}, \psi):=\frac{L(0,\pi_{\lambda},r_{w})}{\epsilon(0,\pi_{\lambda},r_{w},\psi)L(1,\pi_{\lambda},r_{w})}$

The normalized operator
ing properties 13, \S \S 2,3 , [40, Th. 7.9]:$N_{P’|P}(w, \pi_{\lambda}):=r_{P’|P}(w, \pi_{\lambda}, \psi)^{-1}M_{P’|P}(w, \pi_{\lambda})$

$[$

(N1)

$\mathrm{I},$enjoys the follow-$]$$N_{P’|P}(w, \pi_{\lambda})\mathcal{I}_{P}^{G}(\pi_{\lambda}, f)=\mathcal{I}_{P}^{G},(w(\pi_{\lambda}), f)N_{P’|P}(w, \pi_{\lambda}),$$f\in \mathcal{H}(\mathrm{G})$.

(N2) Without any length condition, the functional equation$N_{P’’|P’}(w’, w(\pi_{\lambda}))N_{P’|P}(w, \pi_{\lambda})=N_{P’’|P}(w’w, \pi_{\lambda})$holds.
(N3) For$\lambda\in i\alpha_{M}^{*},$$N_{P’|P}(w, \pi_{\lambda})$(N4) In the case (1),
it is a rational function infield of $F$ .

is unitary.

$N_{P’|P}(w, \pi_{\lambda})$(N5) In the case (1), and if${\rm Re}(\alpha^{\vee}(\lambda))>0,$$\pi$$\forall\alpha\in\triangle_{P}$is a rational function in. In the case (2), whereis the cardinality of the residue$(\alpha^{\vee}(\lambda))_{\alpha\in\triangle_{P}}$$(q_{F}^{-\alpha^{\vee}(\lambda)})_{\alpha\in\triangle_{P}}$$q_{F}$is tempered, then$r_{P’|P}(w, \pi_{\lambda}, \psi)$

has no poles in the region.

(N6) If $G$ is unramified in case (2) andhyperspecial maximal compact subgroup

admits a fixed vector
under the$N_{P’|P}(w,\pi_{\lambda})\phi^{0}(k)=\phi^{0}(k),$, then.

$\mathcal{I}_{P}^{G}(\pi_{\lambda})$36$\mathrm{K}$$\phi^{0}$$k\in \mathrm{K}$279
The property (N5) should also hold in the case (2) [40, Conj. 7.1].

Example 5.8. Consider the case of $GL(2)$ . Iftion factor equals$\chi_{\lambda}=\chi_{1}||_{F}^{\lambda_{1}}\otimes\chi_{2}||_{F}^{\lambda_{2}}$, then the normaliza-$r_{B|B}(w, \pi_{\lambda}, \psi)=r_{\overline{B}|B}(1, \pi_{\lambda}, \psi)=\frac{L(\lambda_{1}-\lambda_{2},\chi_{1}\chi_{2}^{-1})}{\in(\lambda_{1}-\lambda_{2},\chi_{1}\chi_{2}^{-1},\psi)L(1+\lambda_{1}-\lambda_{2},\chi_{1}\chi_{2}^{-1})}$.

is clearly valid in this case.

$(N\mathit{5})$In the general case, a normalization factor satisfying (N1) to (N6) was constructedby Langlands using the Plancherel measure [20, Lect.15]. We still use the notation
for this normalization factor, since these two essentially coincide in thein terms of and -factors
above special cases. To obtain an expression ofas illustrated above is also important in the arithmetic applications.
Going back to the global setting, we define$r_{P’|P}(w, \pi_{\lambda}, \psi)$$L$$r_{P’|P}(w, \pi_{\lambda}, \psi)$$r_{P’|P}(w, \pi_{\lambda}):=\prod_{v}r_{P’|P}(w, \pi_{v,\lambda}, \psi_{v})$Hereestimate$\psi=\otimes_{v}\psi_{v}$,$N_{P’|P}(w, \pi_{\lambda}):=\bigotimes_{v}N_{P’|P}(w, \pi_{v,\lambda})$is a non-trivial character of$\mathrm{A}/F$

and$\pi=\otimes_{v}\pi_{v}$$\sum_{\pi\in\Pi(\mathrm{M}^{1})}\int_{i(a_{L}^{G})^{*}}||\lambda 4_{L}(P, \pi, \lambda, 0)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)||d\lambda$

where$||||$

We apply Lem. 4.3 to A4

Since

.

. We use these to,is the trace class norm. Define two $(G, M)$ -families$N_{Q}(P, \pi, \lambda, \Lambda):=N_{Q|P}(1, \pi_{\lambda})^{-1}N_{Q|P}(1, \pi_{\lambda+\Lambda})$1$\epsilon$,.

$r_{Q}(P, \pi, \lambda, \Lambda):=\frac{r_{Q|P}(1,\pi_{\lambda+\Lambda})}{r_{Q|P}(1,\pi_{\lambda})}$$Q(P, \pi, \lambda, \Lambda)=r_{Q}(P, \pi, \lambda, \Lambda)N_{Q}(P, \pi, \lambda, \Lambda)$

to have$L(P, \pi, \lambda, 0)\mathcal{I}_{P,\mathrm{X}}^{G}(\pi_{\lambda}, f)=\sum_{P_{1}\in \mathcal{F}(L)}r_{L}^{P_{1}}(P, \pi, \lambda, 0)N_{P_{1}}’(P, \pi, \lambda, 0)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)$$N_{P_{1}}’(P, \pi, \lambda, 0)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f)$.

is rapidly decreasing in , it suffices to show that$\lambda$$\int_{i(a_{L}^{G})^{*}}|r_{L}^{P_{1}}(P, \pi, \lambda, 0)|(1+||\lambda||)^{-N}d\lambda$

converges absolutely for sufficiently large $N$ . Once we are reduced to the estimation ofsuch a scalar valued function, we can deduce it from that of the inner product of two. Finally we havetruncated Eisenstein series (using Prop. 5.1). This is done in$[6, \mathrm{I}\mathrm{I}, \S 9]$

the following.

Theorem 5.9 (The fine $X$-expansion). For$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$, we have$J_{X}(f)= \sum_{M\in \mathcal{L}(M_{0})}\sum_{L\in C(M)}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\sum_{w\in W_{M,M}^{L\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{|W^{M}|}{|W|}\frac{1}{|\det(w-1|a_{M}^{L})|}$.

$\cross\int_{i(\alpha_{L}^{G})^{*}}\frac{1}{|P(M)|}\sum_{P\in P(M)}\mathrm{t}\mathrm{r}(\mathcal{M}_{L}(P, \pi, \lambda, 0)M_{P|P}(w, \pi)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))d\lambda$

37

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5.4

Weighted characters

To obtain an expression of $J_{X}(f)$ analogous to Th. 4.5, we need weighted characters, thespectral counter part of weighted orbital integrals [13].
Let be a finite set of places of $F$ . Consider the local analogue$S$$N_{Q}(P, \pi, \lambda, \Lambda)=N_{Q|P}(1, \pi_{\lambda})^{-1}N_{Q|P}(1, \pi_{\lambda+\Lambda})$,$\pi\in\Pi_{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M}_{S}),$of the $(G, M)$ -family defined above. Since the singularity ofis isolated, its “logarithmic derivative” at$\Lambda\in i\alpha_{M}^{*}$$N_{Q|P}(1, \pi_{\lambda})$as a function in$\lambda$$\lambda$$N_{M}(P, \pi, \lambda):=\lim_{\Lambdaarrow 0}\sum_{Q\in P(M)}\frac{1}{\theta_{Q}^{G}(\Lambda)}N_{Q}(P, \pi, \lambda, \Lambda)$

whose singularity set is a locally finite unionof affine hyperplanes whose vector parts are the zeros of coroots. Define the weighted
character to be

can be defined. This is meromorphic in$\lambda$$J_{M}(\pi_{\lambda}, f):=\mathrm{t}\mathrm{r}(N_{M}(P, \pi, \lambda)\mathcal{I}_{P}^{G}(\pi_{\lambda}, f))$.

, we defineMore generally, forwith
any $Q\in P^{M}(L)$ . This again is a meromorphic function having the same type singularities
as $N_{M}(P, \pi, \lambda)$ has. Note that, by taking trace, this is independent of $P\in P(M)$ as thenotation suggests. We also need the distribution$\tau\in\Pi_{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{L})(L\subset M)$$J_{M}(\pi, X, f)$$J_{M}(\tau_{\lambda}, f):=J_{M}(\mathcal{I}_{Q}^{M}(\tau)_{\lambda}, f)$$:= \int_{ia_{M,S}^{G*}}J_{M}(\pi_{\lambda}, f)e^{-\lambda(X)}d\lambda$,$f\in \mathcal{H}(\mathrm{G}_{S})$.

(according to either contains anisitself or a lattice in
Here
for
or
archimedean place or not), and we have written
accordingly.
in the latter.
denotes the dual lattice of
We look at the discrete part (i.e. the term associated to $L=G$ ) of the fine X-expansion
Th. 5.9:$\alpha_{M,S}:=H_{M}(\mathrm{M}_{S})$$S$$a_{M}$$a_{M}$$\alpha_{M}^{*}/(a_{M,S}^{\vee})$$a_{M}^{*}$$\alpha_{M,S}^{*}$$a_{M,S}$$a_{M,S}^{\vee}$.

$\sum_{M\in \mathcal{L}(M_{0})}\sum_{\pi\in\Pi(\mathrm{M}^{1})}\sum_{w\in W_{MM}^{\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{|W^{M}|}{|W|}\frac{1}{|\det(w-1|\alpha_{M}^{G})|}\mathrm{t}\mathrm{r}(M_{P|P}(w, 0)\mathcal{I}_{P_{)}X}^{G}(\pi_{\lambda}, f))$is an invariant distribution and
Sincefind that this is a finite linear combination of characters:$\mathrm{t}\mathrm{r}(M_{P|P}(w, 0)\mathcal{I}_{P,X}^{G}(\pi_{\lambda}, f))$$\sum_{\pi\in\Pi(\mathrm{G}^{1})}a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{G}(\pi, X)\mathrm{t}\mathrm{r}\pi(f)$$\mathcal{I}_{P,X}^{G}(\pi_{\lambda})$is admissible, we.

are merely some scalars and are not theHere we note thatmultiplicity of in the discrete spectrum (cf. Th. 2.4)
$a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{G}( \pi):=\sum_{X\in X(G)}a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{G}(\pi, X)$$\pi$$L_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{2}(G(F) \mathfrak{U}_{G}\backslash \mathrm{G})=\bigoplus_{[G,\pi]}L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]}$

38

.

281
Set

We now give an expression analogous to Th. 4.5 for the sum $J(f)= \sum_{X\in X(G)}J_{X}(f)$ .
$|(2)(1)$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M):=\{\pi\in JH(\mathcal{I}_{Q}^{M}(\sigma_{\mu}))Q\in P^{M}(L)$$\square (M_{1}):=\prod_{M\in \mathcal{L}^{M_{1}}(M_{0})}\{\pi_{\lambda}|\pi\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M), \lambda\in i(a_{M}^{M_{1}})^{*}\}$

Note again thatFor$\pi_{\lambda}\in\Pi(M_{1}),$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M)$.

is not the set of discrete automorphic representations of $M$ .
being as above, define$\pi\in JH(\mathcal{I}_{Q}^{M}(\sigma_{\mu}))$,$a^{M_{1}}(\pi_{\lambda}):=a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{L}(\sigma)r_{L}^{M_{1}}(\pi_{\lambda})$$r_{L}^{M_{1}}( \pi_{\lambda}):=\lim_{\Lambdaarrow 0}\sum_{Q\subset P_{1}}Q\in \mathcal{F}(L)\frac{r_{Q}(Q_{1},\pi,\lambda,\Lambda)}{\theta_{Q}^{P_{1}}(\Lambda)}$Here$r_{L}^{M_{1}}(\pi_{\lambda})$,$w(\sigma_{\mu})=\sigma_{\mu},\exists w\in W_{L,L}^{M,\mathrm{r}\mathrm{e}\mathrm{g}}a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{L}(\sigma)\neq 0,\}$is independent ofCorollary 5.10 ([12]). For$Q_{1}$

and$P_{1}$.

.

$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$, we have$J(f)= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\int_{\Pi(M)}a^{M}(\pi)J_{M}(\pi, f)d\pi$Here we have written $J_{M}(\pi, f):=J_{M}(\pi, 0, f)$ and the measure$d\pi$.

on$\square (M)$is such that$\int_{\square (M)}\phi(\pi)d\pi=\sum_{L\in \mathcal{L}^{M}(M_{0})}\frac{|W^{L}|}{|W^{M}|}\sum_{\pi\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(L)}\int_{i((\iota_{L}^{M})^{*}}\phi(\pi_{\lambda})d\lambda$

holds.

Proof. Writefor the right regular representation of
on the discrete spectrumof $M$ . We may consider the induced representation, whichfrom Th. 2.4 is isomorphic to. Th. 5.9 asserts that$\mathrm{M}$$R_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{M}$$L_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})$$\mathcal{I}_{P}^{G}(R_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c},\lambda}^{M})$$\oplus_{X\in X(G)}\oplus_{\pi\in\square (\mathrm{M}^{1})}\mathcal{I}_{P,X}^{G}(\pi_{\lambda})$$J(f)= \sum_{L\in \mathcal{L}(M_{0})}\sum_{M\in \mathcal{L}^{L}(M_{0})}\sum_{w\in W_{M,M}^{L\mathrm{r}\mathrm{e}\mathrm{g}}},\frac{|W^{M}|}{|W|}\frac{1}{|\det(w-1|a_{M}^{L})|}$.

$\cross\frac{1}{|P(M)|}\sum_{P\in P(M)}\int_{i(a_{L}^{G})^{*}}\mathrm{t}\mathrm{r}(\Lambda t_{L}(P, \lambda, 0)M_{P|P}(w, 0)\mathcal{I}_{P}^{G}(R_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c},\lambda}^{M}, f))d\lambda$

Here A4 $L(P, \lambda, 0),$ $M_{P|P}(w, 0)$ are defined similarly as A4 $L(P, \pi, \lambda, 0),$ $M_{P|P}(w, \pi)$ with.
replaced with
One can easily see that the operator
vanishes on$(Q\in P(L)$ ,the orthogonal complement of a subspace which is a direct sum of-component,. On theequals$\pi$$R_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{M}$$\lambda 4_{L}(P, \lambda, 0)M_{P|P}(w, 0)\mathcal{I}_{P}^{G}(R_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c},\lambda}^{M}, f)$$\mathcal{I}_{Q}^{G}(\pi_{\lambda}),$$\pi\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(L))$$\mathcal{I}_{Q}^{G}(\pi_{\lambda})$$\mathcal{M}_{L}(P, \lambda, 0)$$\mathcal{M}_{L}(P, \pi, \lambda, 0)=\sum_{Q\in P(L)}\frac{1}{\theta_{Q}^{G}(\Lambda)}r_{Q}(P, \pi, \lambda, \Lambda)N_{Q}(P, \pi, \lambda, \Lambda)|_{\Lambda=0}$

39

.

282
Noting that
we can write this as$r_{L}^{P_{1}}(P, \pi, \lambda, \Lambda)$(cf. Lem. 4.3) depends only on$M_{1}$and not on $P_{1}\in P(M_{1})$ ,.

$\sum_{M_{1}\in \mathcal{L}(L)}r_{L}^{M_{1}}(P, \pi, \lambda, \Lambda)N_{M_{1}}(P, \pi, \lambda, \Lambda)|_{\Lambda=0}=r_{L}^{M_{1}}(\pi_{\lambda})N_{M_{1}}(P, \pi, \lambda)$

This combined with the definition of$a_{\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{C}}}^{L}(\sigma)$

yields$J(f)= \sum_{L\in \mathcal{L}(M_{0})}\sum_{M_{1}\in \mathcal{L}(L)}\frac{|W^{L}|}{|W|}\sum_{\pi\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(L)}\int_{i(a_{L}^{G})^{*}}a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{L}(\pi)r_{L}^{M_{1}}(\pi_{\lambda})\mathrm{t}\mathrm{r}(N_{M_{1}}(Q, \pi, \lambda)\mathcal{I}_{Q}^{G}(\pi_{\lambda}, f))d\lambda$

decomposing$i(a_{L}^{G})^{*}=i(a_{L}^{M_{1}})^{*}\oplus i(a_{M_{1}}^{G})^{*}$,$= \sum_{M_{1}\in \mathcal{L}(M_{0})}\sum_{L\in \mathcal{L}^{M_{1}}(M_{0})}\frac{|W^{L}|}{|W|}\sum_{\pi\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(L)}\int_{i(a_{L}^{M_{1}})^{*}}a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{L}(\pi)r_{L}^{M_{1}}(\pi_{\lambda})J_{M_{1}}(\pi_{\lambda}, f)d\lambda$

writing $M$ for$M_{1}$

and using the definition of the measure$= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\int_{\Pi(M)}a^{M}(\pi)J_{M}(\pi, f)d\pi$$d\pi$,.
$\square$6

The invariant trace formula

Recall, in many application of the trace formula, we need to compare the trace formulaeof 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], [i1], [12].

6.1

Non-invarianceFirst we measure the non-invariance of the terms in the trace formula. Recall the geometof the induced operator $R_{Q}(f)$ on
ric kernel, we have\S 3.1. Since$L^{2}(\mathrm{V}L(F)\mathfrak{U}_{G}\backslash \mathrm{G})$$K_{Q}^{f}(x, y)= \sum_{\mathit{0}\in \mathfrak{O}(G)}K_{Q_{\mathit{0}}}^{f},(x, y)$$K_{Q,\mathit{0}}^{\mathrm{A}\mathrm{d}(y^{-1})f}(x, x)=K_{Q_{\mathit{0}}}^{f},(xy^{-1}, xy^{-1})$$\mathrm{A}\mathrm{d}(y)J_{\mathit{0}}^{T}(f)=\int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\sum_{Q\in \mathcal{F}(P_{0})}(-1)^{a_{L}^{G}}\sum_{\delta\in Q(F)\backslash G(F)}K_{Q_{\mathit{0}}},(\delta x, \delta x)_{\mathcal{T}_{Q}}^{\wedge}(H_{Q}(\delta xy)-T)dx$

40

283
using (4.1),$= \int_{G(F)\mathfrak{U}_{G\backslash \mathrm{G}}}\sum_{Q\subset P\in F(P_{0})}(-1)^{a_{L}^{M}}\sum_{\delta\in Q(F)\backslash G(F)}K_{Q_{\mathit{0}}},(\delta x, \delta x)_{\mathcal{T}_{Q^{M}}}^{\wedge}(H_{Q}(\delta x)-T)$$\cross\Gamma_{P}^{G}(H_{Q}(\delta x)-T, -H_{Q}(k_{Q}(\delta x)y))dx$$= \sum_{P\in F(P_{0})}\int_{P(F)\mathfrak{U}_{G\backslash \mathrm{G}}}Q\subset P\sum_{Q\in \mathcal{F}(P_{0})}(-1)^{a_{L}^{M}}\sum_{\delta\in Q(F)\backslash P(F)}K_{Q_{\mathit{0}}},(\delta x, \delta x)_{\mathcal{T}_{Q^{M}}}^{\wedge}(H_{Q}(\delta x)-T)$$\cross\Gamma_{P}^{G}(H_{P}(\delta x)-T, -H_{P}(k_{Q}(\delta x)y))dx$

Here we have written. If we writetothen$\delta x=q(\delta x)k_{Q}(\delta x)$$\mathrm{G}=\mathrm{Q}\mathrm{K}$for the Iwasawa decomposition of$x\in P(F)\mathfrak{U}_{G}\backslash \mathrm{G}$as $x=umak,$$K_{Q_{\mathit{0}}}^{f},(\delta x, \delta x)=K_{Q^{M_{\mathit{0}}M}}^{f_{P}^{k}},(\delta ma, \delta ma)$,.
$\delta x$

with respect,$(u\in \mathrm{U}, m\in \mathrm{M}, a\in \mathfrak{U}_{M}^{G}, k\in \mathrm{K})$$H_{Q^{M}}(\delta x)=H_{Q^{M}}(\delta m)$,,$\Gamma_{P}^{G}(H_{P}(\delta x)-T, -H_{P}(k_{Q}(\delta x)y))=\Gamma_{P}^{G}(H_{P}(a)-T, -H_{P}(ky))$where$f_{P}^{k}(m):=m^{\rho_{P}} \int_{\mathrm{U}}f(k^{-1}muk)du$Using the function
formula is similar.)$\overline{f}_{Q,g}$,$m\in \mathrm{M}$.

(see p. 24), we obtain the following. (The proof for the spectral

Lemma 6.1 ([4] Th. 3.2). For$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$, we have,$\mathrm{A}\mathrm{d}(y)J_{\mathit{0}}(f)=\sum_{Q\in F(M_{0})}\frac{|W^{L}|}{|W|}J_{\mathit{0}}^{L}(\overline{f}_{Q,y})$.

$\mathrm{A}\mathrm{d}(y)J_{\mathrm{X}}(f)=\sum_{Q\in F(M_{0})}\frac{|W^{L}|}{|W|}J_{X}^{L}(\overline{f}_{Q,y})$We now explain the rough idea of the combinatorial part of the construction. Suppose
satisfying:
we are given a family of continuous linear maps$\phi_{L}^{M}$(1)$\phi_{L}^{M}(\mathrm{A}\mathrm{d}(y^{-1})f)=\sum_{P_{1}\in F^{M}(L)}\phi_{L}^{M_{1}}(\overline{f}_{P_{1,y}})$(2)$\phi_{M}^{M}$:(3) Any.

$\mathcal{H}(\mathrm{M})arrow \mathcal{I}(\mathrm{M})$$(\mathrm{A}\mathrm{d}(\mathrm{M})-)$$\mathcal{H}(\mathrm{M})arrow \mathcal{I}(\mathrm{L})$,is surjective,

invariant distribution$I^{M}$$\phi_{M}^{M}$

41

on$\mathcal{H}(\mathrm{M})$passes through$\phi_{M}^{M}:I^{M}=I^{M}\circ\wedge$

284
Then we define a family of distributions$\{I_{\mathit{0}}^{M}, I_{X}^{M}\}_{M\in \mathcal{L}(M_{0})}$by$I_{\mathit{0}}^{M}(f^{M})$$:=J_{\mathit{0}}^{M}(f^{M})- \sum_{L\in \mathcal{L}^{M}(M_{0}),\neq M}\frac{|W^{L}|}{|W^{M}|}I_{\mathit{0}}^{L}(\phi_{L}^{M}(f^{M}))\wedge$$I_{X}^{M}(f^{M})$,$:=J_{X}^{M}(f^{M})- \sum_{L\in \mathcal{L}^{M}(M_{0}),\neq M}\frac{|W^{L}|}{|W^{M}|}\hat{I}_{X}^{L}(\phi_{L}^{M}(f^{M}))$

Then it follows from Lem. 6.1 and (1) that ( is$\bullet$$\mathit{0}$

or

$X$

.

)

.

$I.( \mathrm{A}\mathrm{d}(y^{-1})f-f)=J.(\mathrm{A}\mathrm{d}(y^{-1})f-f)-\sum_{L\in \mathcal{L}(M_{0}),\neq G}\frac{|W^{L}|}{|W|}\acute{I}^{\mathrm{Y}}$$(\phi_{L}^{G}(\mathrm{A}\mathrm{d}(y^{-1})f-f))$$= \sum_{P\in \mathcal{F}(M_{0}),\neq G}\frac{|W^{M}|}{|W|}J^{M}.(\overline{f}_{P,y})-\sum_{L\in \mathcal{L}(M_{0}),\neq G}\frac{|W^{L}|}{|W|}\sum_{P\in F(L)}I^{L}.(\phi_{L}^{M}(\overline{f}_{P,y}))\wedge$$= \sum_{P\in F(M_{0}),\neq G}\frac{|W^{M}|}{|W|}(J^{M}.(\overline{f}_{P,y})-\sum_{L\in \mathcal{L}^{M}(M_{0})}\frac{|W^{L}|}{|W^{M}|}I^{L}.(\phi_{L}^{M}(\overline{f}_{P,y})))\wedge$

$=0$

.

.

That is,are all invariant distributions. On the other hand we deduce from (3.7) theinvariant trace formula$I^{M}$$\sum_{\sigma\in \mathfrak{O}(G)}I_{\mathit{0}}(f)=\sum_{X\in X(G)}I_{X}(f)$6.2.

Application of the trace Paley-Wiener theorem

Of course the most difficult point is to construct the maps
. Arthur used the distribution $J_{M}(\pi, X, f)$ (\S 5.4) for this. In fact, he defined the “Fourier transform”
ofby$\phi_{L}^{M}$$\phi_{M}^{G}(f)$$f\in \mathcal{H}(\mathrm{G}_{S})$$\phi_{M}^{G}(f)$:$\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(\mathrm{M}_{S})\cross a_{M,s\ni}(\pi, X)-J_{M}(\pi, X, f)\in$

C.

Here
denotes the subset of tempered elements in. Of course thiscan be extended toby analytic continuation. It was shown in [13, I.
Lem. 6.2] that this satisfies (1) above:$\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(\mathrm{M}_{S})$$\square _{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M}_{S})$$\square _{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M}_{S})\cross a_{M,S}$$J_{M}( \pi, X, \mathrm{A}\mathrm{d}(y^{-1})f)=\sum_{Q\in F(M)}J_{M}^{L}(\pi, X, \overline{f}_{Q,y})$.

Here we overlook the technical imprecision thatis not stable under
. The
image
of
under
is described by the trace Paley-Wiener theorem [19],[16].
The next problem is that the image
does not contain
if $G\neq M$ .
Then Arthur enlarged the range a little by relaxing the support condition in the directionof the center to obtain. To assure the surjectivity, he also enlarged the domain$\mathcal{H}(\mathrm{G}_{S})$$\mathcal{I}(\mathrm{G}_{S})$$\mathcal{H}(\mathrm{G}_{S})$$\mathrm{A}\mathrm{d}(\mathrm{G}_{S})$$\phi_{G}^{G}$$\mathcal{I}(\mathrm{M}_{S})$$\mathcal{I}_{\mathrm{a}\mathrm{c}}(\mathrm{M}_{S})$42$\phi_{M}^{G}(\mathcal{H}(\mathrm{G}_{S}))$

285. Finally we have the surjective maps
to a little larger space:.
for any
pass throughThe final problem is to show that the distributions
in order. More precisely, we need to establish the following inductive statement.
to define$\phi_{L}^{M}$$\mathcal{H}_{\mathrm{a}\mathrm{c}}(\mathrm{M}_{S})$$\mathcal{H}_{\mathrm{a}\mathrm{c}}(\mathrm{M}_{S})arrow$$L\subset M\in \mathcal{L}(M_{0})$$\mathcal{I}_{\mathrm{a}\mathrm{c}}(\mathrm{L}_{S})$$I_{\mathit{0}}^{G},$$I_{\mathit{0}}^{G}\wedge,$$\phi_{G}^{G}$$I_{X}^{G}$$I_{X}^{G}\wedge$

Problem 6.2. Suppose for any
which pass throughon

we are given the distributions,, so that we can define, and satisfy$M\in \mathcal{L}(M_{0})\rangle\neq G_{2}$$I_{L}^{M}(\pi)$$\mathcal{H}(\mathrm{M})$$\phi_{M}^{M}$$I_{L}^{M}(\gamma)$$I_{L}^{M}(\gamma))\wedge I_{L}^{M}(\pi)\wedge$$J_{L}^{M}( \pi, f^{M})=I_{L}^{M}(\pi, f^{M})+\sum_{M_{1}\in \mathcal{L}^{M}(M_{0}),\neq M}I_{L}^{M_{1}}(\gamma, \phi_{M_{1}}^{M}(f^{M}))\wedge$$J_{L}^{M}( \gamma, f^{M})=I_{L}^{M}(\gamma, f^{M}).+\sum_{M_{1}\in C^{M}(M_{0}),\neq M}I_{L}^{M_{1}}(\pi, \phi_{M_{1}}^{M}(f^{M}))\wedge$,.

Then the distributions$I_{L}^{G}(\gamma, f)$$:=J_{L}^{G}( \gamma, f)-\sum_{M\in \mathcal{L}(M_{0}),\neq G}I_{L}^{M}(\gamma, \phi_{M}^{G}(f))\wedge$,$I_{L}^{G}( \pi, f):=J_{L}^{G}(\pi, f)-\sum_{M\in \mathcal{L}(M_{0}),\neq G}I_{L}^{M}(\pi, \phi_{M}^{G}(f))\wedge$

pass through$\phi_{G}^{G}$.

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, then we haveficiently large for

$S$

suf-

$f\in \mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$$\sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\sum_{\gamma\in(M(F))_{M,S}}a^{M}(S, \gamma)I_{M}(\gamma, f)$$= \sum_{M\in \mathcal{L}(M_{0})}\frac{|W^{M}|}{|W|}\int_{\Pi(M)}a^{M}(\pi)I_{M}(\pi, f)d\pi$

References[1] J. Arthur, Eisenstein series and the trace formula, in “Automorphic Forms, Representations, and -functions”, Proc. Sympos. in Pure Math. 33, part 1, AMS, Providence, 1979, pp. 253-274.
$L$[2] –, A trace formula for reductive groups I. Terms associated to classes inDuke Math. J. 45 (1978), pp. 911-952.

$G(\mathbb{Q})$,[3] –, A trace formula for reductive groups II.. Applications of a truncation operator, Comp. Math. 40 (1980), pp. 87-121.

[4] –, The trace formula in invariant form, Ann. of Math. 114 (1981), pp. 1-74.

43

286
[5] –, On the inner product of truncated Eisenstein series, Duke Math. J. 49(1982), pp. 35-70.
[6] –, On a family of distributions obtained from Eisenstein series I.. Applicationof the Paley-Wiener theorem, Amer. Jour. of Math. 104 (1982), pp. 1243-1288, II..
Explicit formulas, Amer. Jour. of Math. 104 (1982), pp. 1289-1336.

[7] –, A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983),
pp. 1-89.
[8] –, A measure on the unipotent variety, Canad. J. Math. 37 (1985), no. 6, pp.

1237-1274.
[9] –, On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986),
no. 1, pp. 179-214.

[10] –, The local behavior of weighted orbital integrals, Duke Math. J. 56, No. 2(1988), pp. 223-293.
[11] –, The invariant trace formula I. Local theory, Jour. of the Amer. Math. Soc.
1 (1988), pp. 323-383.

[12] –, The invariant trace formula II. Global theory, Jour. of the Amer. Math. Soc.
1 (1988), pp. 501-554.
[13] –, Intertwining operators and residues I. Weighted characters, Jour. of Funct.
Anal. 84 (1989), pp. 19-84, II. Invariant distributions, Composit. Math. 70 (1989),
pp. 51-99.
[14] –, Unipotent automorphic representations: global motivation, in
forms, Shimura varieties, and -functions”, Vol. I (Ann Arbor, MI, 1988), pp. 1-75,
Perspect. Math., 10, Academic Press, Boston, MA, 1990.

$‘(\mathrm{A}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{c}$$L$[15] –and L. Clozel, Simple algebras, base change, and the advanced theory of
the trace formula, Annals of Mathematics Studies, 120, Princeton University Press,
Princeton, NJ, 1989.
[16] J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive-adic groups, J. Analyse Math. 47 (1986), pp. 180-192.
$p$[17] A. Borel, Automorphic -functions, in “Automorphic Forms, Representations, and-functions”, Proc. Sympos. in Pure Math. 33, part 2, AMS, Providence, 1979, pp.
27-62.
$L$$L$[18] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of
Math. 75 (1962), pp. 485-535.

[19] L. Clozel and P. Delorme, Le th\’eor\‘eme de Paley-Wiener invariant pour les groupesde Lie r\’eductifs, Invent. Math. 77 (1984), no. 3, pp. 427-453, II, Ann. Sci. \’Ecole
Norm. Sup. (4) 23 (1990), no. 2, pp. 193-228.

44

287
[20] L. Clozel, J-P. Labesse, R.P. Langlands, Morning Seminar on the Trace Formula,mimeographed notes, IAS, Princeton Univ., 1984.

[21] M. Duflo and J.-P. Labesse, Sur la formule des traces de Selberg, Ann. Sci. \’Ec. Norm.
Sup., 4e. s\’erie, 4 (1971), pp. 193-284.
[22] S. Gelbart and H. Jacquet, Forms offrom the analytic point of view, informs, representations and -functions”, (Proc. Sympos. Pure Math.,
Corvallis, Ore., 1977), Part 1, pp. 213-251, Proc. Sympos. Pure
Univ.,
Oregon State
Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
$\mathrm{G}\mathrm{L}(2)$$L$$‘(\mathrm{A}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{c}$[23] R. Godement, Domaines fondamentaux des groupes arithm\’etiques, S\’eminaire Bourbaki, 1962/63, n. 257.
[24] H. Jacquet and R.P. Langlands, Automorphic forms on $GL(2)$ , Lect. Notes in Math.
114, Springer Verlag, 1970.
[25] H. Jacquet, I.I. Piatetskii-Shapiro and J.A. Shalika, Rankin-Selberg convolutions,Amer. J. Math. 105 (1983), no. 2, pp. 367-464.

[26] H. Jacquet and J.A. Shalika, On Euler products and the classification of automorphic
representations II, Amer. Jour. of Math. 103 (1981), pp. 777-815.

[27] A.W. Knapp and E.M. Stein, Intertwining operators for semisimple Lie groups II,
Invent. Math. 60 (1980), pp. 9-84.
[28] J.P. Labesse, La formule des traces d’Arthur-Selberg, S\’eminaire Bourbaki, (1984-85),
n. 636.
[29] –, The present state of the trace formula, in “Automorphic Forms, Shimura
Varieties, and -functions”, L. Clozel and J.S. Milne eds., Perspect. in Math. Vol.
10, Volume I, 211-226, Academic Press 1990.
$L$[30] R.P. Langlands, Eisenstein series in “Algebraic Groups and Discontinuous Subgroups”, Proc. Sympos. in Pure Math. 9, Amer. Math. Soc., Providence, 1966, pp.

235-252.
[31] R.P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lect.
Notes in Math. 544, Springer, Berlin, Heidelberg, 1976.
[32] –, On the classification of irreducible representations of real algebraic groups,
theory and harmonic analysis on semisimple Lie groups”, Math.
in
31, Amer. Math. Soc., Providence, RI, 1989, pp. 101-170.
Monographs,
Surveys$‘\zeta \mathrm{R}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$[33] C. Moeglin and J.L. Waldspurger, D\’ecomposition Spectrale et S\’eries d’Eisenstein,
Une Paraphrase de l’\’Ecriture, Progress in Math. 113, Birkh\"auser, 1994.

45

288
[34] M.S. Osborne and G. Warner, The Selberg trace formula. I. -rank one lattices, J.
Reine Angew. Math. 324 (1981), pp. 1-113, II. Partition, reduction, truncation,
Pacific J. Math. 106, (1983), no. 2, pp. 307-496, III. Inner product formulae (initial considerations), Mem. Amer. Math. Soc. 44 (1983), no. 283, IV. Inner productgroup representations I”, (College Park, Md.
formulae (final considerations) in
1982/1983), Lecture Notes in Math. 1024, Springer, Berlin-New York, 1983, pp.
112-263, V. Questions of trace class, Trans. Amer. Math. Soc. 286 (1984), no. 1,
pp. 351-376, VI. Implications of estimability, Amer. J. Math. 107 (1985), no. 6, pp.
1369-1437, VII. Application of the truncation process to the continuous spectrum, Pacific J. Math. 140 (1989), no. 2, pp. 263-352, VIII. Contribution from the continuous
spectrum, Trans. Amer. Math. Soc. 324 (1991), no. 2, pp. 623-653.
$\Gamma$$‘(\mathrm{L}\mathrm{i}\mathrm{e}$[35] I.I. Piatetskii-Shapiro and J.W. Cogdell, Converse theorem for $GL(n)$ , Inst. Hautes\’Etudes Sci. Publ. Math. No. 79, (1994), pp. 157-214.
[36] R. Rao, Orbital integrals on reductive groups, Ann. of Math. 96 (1972), pp. 505-510.
[37] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric RieSelberg Collected Pamannian spaces with applications to Dirichlet series, inpers” Vol. 1, Springer Verlag, Berlin Heidelberg, 1989, pp. 423-463.
$‘(\mathrm{A}\mathrm{t}\mathrm{l}\mathrm{e}$[38] –, Discontinuous groups and harmonic analysis in Proc. Int. Cong. Math.,
1962, pp. 177-189.
[39] F. Shahidi, On certain$L$-functions, Amer. Jour. of Math. 103 (1981), pp. 297-355.

[40] –, A proof of Langlands’ conjecture on Plancherel measures; Complementary
series for -adic groups, Ann. of Math. 132 (1990), pp. 273-330.
$p$

46

289

Index
$(M(F)\cap\overline{\mathit{0}})_{M,S},$25$A_{G},$$\mathfrak{U}_{G},$$a_{G},$$A_{P,\pi,X}(\lambda, \lambda’))$25$a_{M}^{G},$W$=W^{G}$$W_{M,M’},$

$8$

y),
$Y_{Q}(T),$ 32
A,$Y_{P}^{T}(x,$39$a^{M}(\pi),$T), 22the relative Weyl group, 5$v_{S}’(ky,$

4
30

26

$a^{M}(S, \gamma),$23$v_{L}^{P}(x),$(M,$S)$ -equivalence,6$\mathrm{A}_{S},$38$a_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}^{G}(\pi),$

5
of

”

$c_{P}(\lambda),$

22

K

32
23

$\mathrm{G}s:=G(\mathrm{A}_{S}),$$5$$= \prod_{v}\mathrm{K}_{v},$$\triangle_{0},$32$\triangle_{P},$$a_{M,S},$$\triangle_{P}^{\mathrm{v}},$

12

$6$

$\mathfrak{U}_{0}(T),$8$\triangle_{P}\wedge,$$\triangle_{0}^{\mathrm{v}},$

38$a_{M,S}^{\vee},$3$H_{G},5G^{\gamma},G_{\gamma},$$4$$\chi_{M}^{P_{1}}(T,$$d_{Q}^{w}(\lambda_{L_{w}}, \Lambda),$$E_{P}(x, \phi_{\lambda}),$$4$$\mathrm{A}_{f},$X), 33$c_{Q}(T, \Lambda),$$d_{Q}^{w}(\Lambda),$y), 21$\mathrm{G}^{1},4$$‘(\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$$D^{G}(\gamma),$$\mathrm{A}_{\infty},$$\mathrm{G}:=G(\mathrm{A}),$$C_{c}^{\infty}(\mathrm{G}/\mathfrak{U}_{G}),$$c_{Q}’(\lambda),$$\mathcal{Y}_{P}^{T}(x,$$\mathrm{J}^{P}(\delta x,T),13\mathrm{R}_{Q}(x,T),13@$$H_{P},$

5

12
a Siegel domain, 6$\mathrm{O}(G),$f),f), 23f), 38$I_{L}^{G}(\pi,$$I_{L}^{G}(\gamma,$f), 43$\mathfrak{S}(T_{0})$$X(G),$ $7$$J_{M}(\gamma,$$J_{M}(\pi_{\lambda},$$\Gamma_{Q}^{P}(X,$1316f), 20y), 12$K_{P,X}(x, y))$ 15y), 12Y), 1728$J_{\mathit{0}}^{T}(f),$$\mathcal{E}(\mathfrak{h})^{W(\mathfrak{h}_{\mathbb{C}})},$$J_{X}^{T}(f),$$F(M),$$5$$\mathcal{F}(P_{0}),$43$j_{\mathit{0}}^{T}(x,$$\mathcal{H}_{\mathrm{a}\mathrm{c}}(\mathrm{M}_{S}),$

43$K_{P,\mathit{0}}(x,$$\mathcal{I}(\mathrm{M}_{S}),$$\mathcal{I}_{\mathrm{a}\mathrm{c}}(\mathrm{M}_{S}),$$\mathcal{I}_{P,X}^{G}(\pi_{\lambda}),$$K_{\mathit{0}}(x,$y), 11f), 12f), 15-cusp form,f), 39X, f), 38$K_{X}(x,$$J_{M}(\pi,$$k^{T}(x,$$J_{M}(\pi,$$k_{X}^{T}(x,$$L^{2}$$\mathcal{L}(M),$$L_{0}^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G}),$$5$$N_{M}(P, \pi, \lambda),$10$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{[\pi]},$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})_{X},$$P_{(M,\rho)}^{\mathfrak{F}},$$5$$P(M),$

7

26

25$\overline{f}_{Qg}(l))’$30$\omega^{T}(\lambda, \lambda’, \phi, \phi’),$

$8$

37

$5$

$\Omega_{\pi,X}^{T}(P, \lambda),$intertwining operator, 8$\mathcal{M}_{Q}^{T}(P, \pi, \lambda, \Lambda),$38$N_{Q}(P, \pi, \lambda, \Lambda),$$7$$L_{0}^{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho},$$P^{T}(B),$$\mathcal{L}(P_{0}),$$6$$\mathcal{A}4_{Q}(P, \pi, \lambda, \Lambda),$$L^{2}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G}),$$M(w, \rho_{\lambda})$

27

$8$

$\mathcal{I}_{P}^{G}(\rho_{\lambda}),$27

21$\Phi_{S,y,T}(m),$ 22$\Phi_{S,a,k,y}^{T}(m),$$p_{\lambda}^{T}(H),$$p_{\lambda}^{T}(\beta),$29

R $=R_{G)}5$$r_{Q}(P, \pi, \lambda, \Lambda),$$\Pi(\mathrm{G})=\Pi(G(\mathrm{A})),$

37$\square _{\mathrm{a}\mathrm{d}\mathrm{m}}(\mathrm{M}),$

analytic origin, 17$u_{Q}’(k, y))$ 25$T_{1}$$\square _{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M),$35$\Pi(M),$f), 29$\Psi_{X,\pi}^{T}(\lambda,$

47

$2$

39

34

290
$\phi_{M}^{G},$$\phi_{P}$$\phi_{\lambda}$$42$

the constant term along $P,$for$\phi\in P_{(M,\rho)}^{\mathfrak{F}},$$\psi_{X,\pi}^{T}(X, f),$

$29$

$13$

$12$

$\tau_{P},\tau_{P}\wedge,$$\theta_{Q}^{P}(\lambda)\wedge,$$\theta_{Q}^{P}(\lambda),$$\theta_{\phi}(g),$$\wedge^{T}$

$8$

$29$

$\psi_{\lambda}^{T}(H),$$\sigma_{Q}^{P},$

$6$

$22$

$7$

, truncation operator, 15$\hat{P}_{(M,\rho)}^{\mathfrak{F}},$$\triangle_{P}^{\mathrm{v}}\wedge,$

$7$

$22$
$23$

$\{v_{Q}(\lambda, x)\}_{Q\in P(L)},$$A_{0}(G(F)\mathfrak{U}_{G}\backslash \mathrm{G})$the space fo cusp forms,

6$A_{0}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\rho},$$7$$A_{2}(M(F)\mathfrak{U}_{M}\backslash \mathrm{M})_{\pi},$$9$$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X},$$10$$A_{2}(\mathrm{U}M(F)\mathfrak{U}_{M}\backslash \mathrm{G})_{\pi,X}^{\mathfrak{F}},$$10$$\mathcal{H}(\mathrm{G}/\mathfrak{U}_{G})$the Hecke algebra, 5cuspidal datum, 7
cuspidal pair, 7disrete pair, datum, 9Eisenstein series (cuspidal), 8
geometric side, 3Hilbert-Schmidt class, 2
spectral side, 3
trace class, 3weighted character, 38

48