The trace fortmila

THE TRACE FORMULA FOR REDUCTIVE 'GROUPS*

James ARTHUR

This paper is a report on the present state of the trace formula for a general reductive group. The trace formula is not so much an end in itself as it is a key to deep results on automorphic representations. However, such applications have only been carried out for groups of low dimension
( 51,

171,

[8(c)],

[ 3 ] ) . We will not try to discuss them here.

For reports on progress towards applying the trace formula for general groups, see the papers of Langlands [8(d) I and Shelstad [ll] in these proceedings Our discussion will be brief and largely confined to a description of the main results. On occasion we will try to give some idea of the proofs, but more often we shall simply refer the reader to papers in the bibliography. Section 1 will be especially sparse, for it contains a review of results which were summarized in more detail in [l(e)]. This report contains no mention of the twisted trace formula. Such a formula is not available at the present, although I do not think that its proof will require any essentially new ideas. Twisted

~ e c t u r e sfor Journges Automprphes, Dijon, Feb. 16-19, 1981. I Wish to thank the University of Dijon for its hospitality.
1

Janes ARTHUR

trace formulas for special groups were proved in r8(c) I and [ 3 1 . As for the untwisted trace formula, we draw attention to Selberg's original papers [10(a)l, [10(b)], and also point out, in addition to the papers cited in the text, the articles [ 5 1 ,
C9 (b)I .

[l(a) ], [ 4 ] ,

[12], [9 (a)l and

1, THE TRACE FORMULA - FIRST VERSION,

Le G be a reductive algebraic group defined over split component of the center of G, and set
2).

Let AG be the

where X(G)^ is the group of characters of G defined over 9. Then a r e vector space whose dimension equals that of AG. Let G ( A )

is be the

kernel of the map H

:

G (A) *

^,

which is defined by

Then G (Q) embeds diagonally as a discrete subgroup of G (A)

and the

coset space G (Q)\G (A) has finite invariant volume. We are interested in the regular representation R of G (A) on (G(Q)\G (A) 1)

. If
1)

f e c (G(A) 1) , R(f) is an integral operator on L2 (G(Q)\G

(A)

. The

sour-

ce of the trace formula is the circumstance that there are two ways to express the integral kernel of R(f).

We shall state the trace formula in its roughest form, recalling

briefly how each side is obtained from an expression for the integral kernel. This version of the trace formula depends on a fixed minimal

The trace f o m ' i a

parabolic subgroup Po, with Levi component Mo and unipotent radical N o , and also on an appropriate maximal compact subgroup K = II K v In addition, it depends on a point T in of G ( A ) 1

which is suitably regular with respect to that

a(T)

is large for each root

of

Po, Po

in the sense
A0 =

On

\.

The trace formula is then an identity

We describe the left hand side first.

0 denotes the set

of equivalence classes in

G($),

in which two elements in

G ( Q ) are deemed equivalent if their semisimple components are

G(Q)

conjugate.

This relation is just G ( Q )

conjugacy if G

is anisotropic, but it is weaker than conjugacy for general G If

P

is a parabolic subgroup of Po, and

which is standard with

respect to

o-

0,

set

where

is the unipotent radical of

P

and

Mp

is the unique

Levi component of

which contains M

Then

James ARTHUR

is the integral kernel of the operator convolving f on

Rp(f)

obtained by

L ( N @)M (Q) (A) \G I). If P = G

it is

just the integral kernel of T k(x,f) =

R(f).

Define

P3P0 The function

dim ( A / A 1 (-1)

' 6eP (Q)\G (Q) p e

(fix,<5x)~p(~p(5x)-~).

Tp ( H ( - ) - T) ,
1 if

whose definition we will recall in and vanishes on a large compact P s G

a moment, equals

P=G,

neighborhood of 1 in G ( A ) I if T k&(x, f) i$ obtained by modifying hood of infinity in G ( @ )\ G ( A )

In other words,

in some neighborT The function k(x,f) turns

( y) KG ,& x ,

out to be integrable, and the distributions on the left of the trace formula are defined by

If P

is a standard parabolic subgroup,

characteristic function of a certain chamber. and P I Ap - AMp If Q

P is the Write a = en.

, T

is a parabolic subgroup that contains

-o<_

there is a natural map from

onto

a . Q'

We shall

write

for its kernel. (P , A ) .

Let

Ap

denote the set of simple roots of

It is naturally embedded in the dual space

The trace formula

of in

There is associated to each Let

a e Ap

a "co-ro~t" av

be the basis of t^-p/Agwhich is dual to ,Then

{av : a e

A }

<

is the characteristic function of

Let by

Hp be the continuous function from G ( A )

to

defined

T In the formula for k@(x,f)

above, the point

belongs It

to a , but it projects naturally onto a point in A . is in this sense that

is defined. The set

X

which appears on the right hand side of (1.1) If

0-

is best motivated by looking at 0.

0,

consider those

standard parabolic subgroups B which are minimal with respect to the property that union of M ( Q )

a- meets M B . Then

e n

MB

is a finite

conjugacy classes which are elliptic, in the

sense that they meet no proper parabolic subgroup of MB which is defined over

Q.

Let

be the restricted Weyl group of

Jmea ARTHUR

(G

,A).

It is clear that

is in bijective correspondence (MB , cB), and cB where B is a

with the set of

Wo-orbits of pairs G,

standard parabolic subgroup of conjugacy class in MB(Q). representations of GA' () G(@),

is an elliptic

If we think of the automorphic as being dual in some sense to the we can imagine that the c u s p i d a l

conjugacy classes in

automorphic representations might correspond to elliptic conjugacy classes. of pairs of G and ( M ,r ) , r

X

is defined to be the set of W-orbits where B is a standard parabolic subgroup

is an irreducible cuspidal automorphic represen-

tation of MB@) of groups B

For a given

X ,

let

be the set

obtained in this way.

It is an associated class

of standard parabolic subgroups. associated class Suppose that

L 2 ( N (&)M~(Q)\G (A)'

Similarly, we have an

Pn.

for any

<ye P a

0.
P
0

x 6 )x

and

are given.

Let in For every

be the space of functions @ with the following property. with B

c

L~ [ N (A)% (Q)\G () ) A'

standard parabolic subgroup B ,

x e G @I)

, and almost all

the projection of the function

I
under MB(~)' (MB I rB)

1 (nmxldn ,

1 meMg(A) ,

% (fa)\%@I
L2 ( M (Q)\MB@I1 ~

onto the space of cusp forms in

transforms in which

as a sum of representations

rB ,

is a pair in

x.

I there is no such pair in

x,

B , X will be orthogonal to the space of cusp forms on % (Q)\Mc () . It follows from a basic result in Eisenstein A'

The trace fornula

series that group in

L (Np@IMP (Q)\G (A)' )

will be zero unless there is a P. Moreover, there is an

which is contained in

orthogonal decomposition

Let R f

Kp
1

x (x,y) ^x

be the integral kernel of the restriction of

to

L (Np(AIMp(Q) (A) I ) \G

One can write down a We have

formula for

(x,y) in terms of Eisenstein series.

each side being equal to the integral kernel of define the modified functions dim ( A h G 1 kT (x,f) = [ (-1) p3p0 6eP ( Q ) \G (Q)

Rp(f).

Xf we

K ~ l (fix , fix) (~~(5x1T ) x Tp -

we immediately obtain an identity

T It turns out that the functions kx(x,f) are also integrable.

In fact, the sums on each side of the identity are absolutely integrable. The distributions on the right of the trace formula are defined by

James ARTHUR

The trace formula (1.1) follows. Most of the work in proving (1.1) comes in establishing the kT (x, f) and kT (x, . Of these, the second f) integrability of

function is the harder one to handle.

To prove its integrability G(Q) \ ( ) GA'

it is necessary to introduce a truncation operator on Given on T

as above, the truncation of a continuous function h is the function

G(Q)\G@I1

1 2 truncating the function

If

X ,

let

ATAT K (x,y) be the function obtained by

in each variable separately. operator, one shows that

From properties of the truncation

is finite (see [l(d), 511 ) that

One can also show, with some effort,

([l(d), 2 ] ) from which one immediately concludes that

The trace forrnula

is also finite

( [l(d), Theorem 2.1 ] ) . This was the result that

In the process, one shows that $he integral of

was required for formula (1.1). for any

XI

is zero for sufficiently regular T other words,

( l (d), Lemma 2.41). [

In

T This second formula for Jx(f)

is an important bonus.

We shall

see that it is the starting point for obtaining a more explicit T formula for Jx ( f )

2,

SOME REMARKS
~t is natural to ask how the terms in the trace formula T.

It is shown in Proposition 2.3 T of [1(] that the distributions JT ( and J(f) are poly& nomial functions of
T

(1.1) depend on the point

T,

and so can be defined for all points To in

in

<x.

There turns out to be a natural point

such that the distributions

James ARTHUR

and

are independent of the minimal parabolic subgroup Po. For a better version of the trace formula, we can set T = To to obtain

(incidentally, To is strongly dependent on the maximal compact subgroup K For example, if G = G L and Ho is the group of diagonal matrices, . T0 will equal zero if K is the standard maximal compact subgroup of GI,@). However, if K is a conjugate of this group by Mo(A.), To might not be zero). The distributions J
Y

and J

x will

still depend on No and

K. Moreover, they are not invariant. There are in fact simple formulas to measure how much they fail to be invariant.

Let L(M~)be the set of subgroups of G, defined over Q, which contain Mo and are Levi components of parabolic subgroups of G. Suppose that M e L(Mo). Let L(M) be the set of groups in L(M~)which contain M. Let FI) be the set of parabolic subgroups of G, now no longer standard, (1 which are defined over Q and contain M. Then if Q also 0 contains M. Let P(M) be the set of groups Q in F(M) such that M equals
e
M

F(M) ,

M.

Suppose that L e KM)

. We write

L~ (M) , F~ (M) and F^ (PI)

n L(Q) is a disjoint

for the analogues of the sets 1 (M), F(M) and P(M) when G is replaced by L. Now suppos'e that 0 is a class in 0. Then '
(?-

The twice formula

union

of equivalence classes in the set, OL

associated to L . on LA' ()

We

can certainly define the distributions JBi

set

By definition, J
(9-

is zero unless

0-

meets L(Q).

If

we can define a distribution J Formula (2.1), applied to L

on L) @' yields

in a similar way.

where now

is any function in C ~ ( (A)' ) L

The formulas which measure the noninvariance of our distributions depend on a certain family of smooth functions

indexed by the groups L e I ( M ) and define these functions here. map

Q e F (M)

We will not

They are used to define a continuous

James ARTHUR

f r o c~(L(A)~) to m
: ~feft)'

C(@' ;M))
is given by

for each

L) &'

If

fQY(m)

where

6Q

is the modular function

Q(A).

Then the formulas

alluded to are

and

where [l(f)

tf-c 0 .

y c X I ,

Theorem 3.21

.)

f c cL&' "()) M Here 1 wOQ1

and

y c L(A).

(see

stands for the number of

elements in the Weyl group of function

( M ,A )

As usual,

fY

is the

f will equal f by definition. We therefore QrY obtain a formula for the value of each distribution at fy - f as a sum of terms indexed by the groups Q c FL (Mo), with Q * L . If Q = L r The distribution will be invariant if and only if for each f L and y . the sum vanishes. For example, J^, will be invariant if and only if

O'n M(Q)

is empty for each group M e F ~ M )

The trace f o d a

with

L.

3, T H E T R A C E F O R M U L A IK I N V A R I A N T F O R M
There is a natural way to modify the distributions J-, and
J

so that they are invariant.

This was done in the paper

[I()] under some natural hypotheses on the harmonic analysis of the local groups G(Q). of this construction. If H is a locally compact group, let
H(H)

We shall give a brief discussion

denote the

set of equivalence classes of irreducible unitary representations of

H.

Suppose that M

is any group in

L(M).

We shall agree

to embed of
M(A)'

II (M (A) ) I
and
AM@?)

in
0

II (M (A) ; for )

M (A) is the direct product

so there is a bijection between

H (M (A) )

II (M (A) ) I
on
AMOR)
0

and the representations in

which are trivial

Let

IItemp

(M(/A)

be set of tempered representa-

tions in

II ( M ( < A ) )

From Harish-Chandra's work we know that

there is a natural definition for the Schwartz space, C ( M l & ) ) , of functions on

C (M
1 (a))

M(A)'

There is also a linear map

7 "

from

to the space of complex valued functions on

%emp ( M @ ) ~ ) , ' given by

In [l ( f) of
T

I we proposed a candidate, l (M ( A ) ) , for the image

l (M (A)I )

Roughly speaking,

is defined to be the space which are ~chwartz

of complex valued functions on

I H t e (M (A) )

James ARTHUE

functions in all possible parameters. T maps

It is easy to show that 1 (M (A) ) maps

C (M (&) I)

continuously into

~t is also easy

to see that the transpose space of butions on I(M(A)), M @I )

T ;

of

1 (M (A) ) ' , the dual

'

into the space of tempered invariant distri-

.
For each M
e

HYPOTHESIS 3.1:
? (M (&)

(Mo), TM maps

C (()) MA'

onto

') .

Moreover, the image of the transpose,

is the space of aH tempered invariant distributions on M) @'

.
I

This hypothesis will be in force for the rest of 53. is any tempered invariant distribution on M @ )

If

we will let TM (I) = I

I
A

be the unique element in

7 (M ( A ))

such that

Important examples of tempered invariant distributions are the orbital integrals. valuations on
Q,

Suppose that

S v

is a finite set of in Set

1

and that for each M defined over Q .

S,

is a

maximal torils of

T = ( TT T ( Q ) ) n M W ;
ves

,
in T ; whose

and let T ' be the set of elements -y St reg centralizer in

The trace formula

equals

T ;

Given

f e C (M~A)')

and

1 y e TSrreg,

the orbital

integral can be defined by

ID(^) 1 %
on

is the function on

T ;

which is usually put in as a

normalizing factor.)

C (M (!A)) .
on
2 (M @ ) A'

I is a tempered invariant distribution Y By Hypothesis 3.1 it corresponds to a distribution

?Y

) . Now suppose that

has compact support.

Then the map

has bounded support; that is, the support in closure n

T

T1

S ,reg

has compact

Let

I (@' M)

be the set of functions Ts

2 (M (a))

such that for every group

the function

has bounded support. such that. T

There is a natural topology on

C :

I (~($4))
).
(M @A) )

maps

(M () ) &I

continuously into

1 (M @I1

HYPOTHESIS 3.2 : For each group H

onto

L (, , M)

maps

C :

1 (M(A) I . )

Moreover, the image of the transpose,

James ARTHUR

is the space of all invariant distributions on'M(A)

.
be

If I is an arbitrary invariant distribution on M(A) l , let 1 the unique element in I (M(A) ) such that TA(1) = I.
A

Much of the paper Cl(f)l is devoted to proving the following theorem

THEOREME 3.3

There is a continuous map

for every pair of groups M

L in L(MO), such that

and

We will not discuss the proof of this theorem, which is quite difficult. Given the theorem, however, it is easy to see how to put the trace formula into invariant form. PROPOSITION 3.4
:

Suppose that

J~ : C(() :LA

+ (E

L(M~),

is a family of distributions such that

The trace formula

for a

h L

1 c~(L(A)) and y e LA' ()

. Then there is a unique

family

of invariant distributions such that for every L and f,

PROOF

Assume inductively that I

M e

has been defined and satisfies (3.1) L. Define

for all groups

L(Mo) with M

for any f

c_(L(A) 1) . Then

is certainly a distribution on L(A) 1

The only thing to prove is its invariance. We must show for any
y e L (A)

that IL (fy) equals IL (f). This follows from the fornula for

~ ~ ( f the) formula for 1(l(fy), and our induction assumption. ~ ,

According to (2.3) and (2.3) , we can apply the proposition to L the families { J } and {JL1. We obtain invariant

James ARTHUR

distributions

] : I {

and

{; I}

which s a t i s f y t h e analogues o f

(3.1).
The t r a c e formula i n i n v a r i a n t form i s t h e c a s e t h a t of t h e f o l l o w i n g theorem.
L=G

T E R M 3.5: HOE

For any group

in

i(Mo),

PROOF:

Assume i n d u c t i v e l y t h a t t h e theorem h o l d s f o r a l l groups with

M e I ( M )

;L .

For any such

we a l s o have

f o r any

$ e I ( M @ ) ) . Then

by ( 2 . 2 )

and ( 3 . 1 ) .

The trace formula

4, INNER
I@, Jx

PRODUCT OF TRUNCATED E I S E N S T E I N S E R I E S ,

An obvious problem is to evaluate the distributions J _ , and I

explicitly.

How explicitly is not clear, but

we would at least like to be able to decompose the distributions as sums of products of distributions on the local groups G(Q).

0.

In [l(c)] w e d e fined the notion of an u n r k i f i e d class in If 6 - is unramified, JT f ) can be expressed as a weighted ( f ([1 (c), (8.7)1 )

orbital integral of express I(f)

It is possible to then

as a certain invariant distribution associated (see [l(f), 5141 for the case

to a weighted orbital integral. of GL.) If

o-

is not unramified, we would expect to express

J(f) Then

as some kind of limit of weighted orbital integrals. I*( would be a limit of the corresponding invariant In any case, the lack of explicit formulas for with
o-

distributions. Je(f) and

I&(f),

ramified, should not be an

insurmountable impediment to applying the trace formula. One can also define an unramified class in
X .

For any

such class, it is also not hard to give an explicit formula for ~ ( 5 ) . (see [l(d), p. 1191 .)

Unlike with the classes

however, it seems to be essential to have a formula for all in order to apply the trace formula.

We shall devote the rest

of this paper to a description of such a formula. Suppose that

A (P)

F(M)

is a parabolic subgroup. Let

be the space of square-integrable automorphic forms on whose restriction to Mp (A)' is square integrable.

N (AIMp( Q ) \G iA)

There is an Eisenstein series for each

6 e A (P) given by

James ARTHUR

~t converges for

Re(X)

in a certain chamber, and continues

analytically to a meromorphic function of and in

IT
6

X e^p,a- If

e X

(P) be the space of vectors XI A (P) which have the following two properties.
2

n ( l > ~ ( A ) ) , let

A2

(i) The restriction of

to

G(A)

belongs to

L~(N~)M~(Q)\G(A)~ (ii) For every m

->-

in
I

G (A) the function , m

6

o> (mx)

fL> (A),
IT.

transforms under Let

Mp(<A.) according to

xrn inner product

i2 be the completion of A2 (P)

(PI with respect to the XI'"

For each of G(A) on

*
&P,C

there is an induced representation (PI, defined by

x2 XI

Ox,

IT

The representation is unitary if

is purely imaginary.

Now, suppose again that a minimal parabolic'subgroup

The trace formula

Po e P ( M )

h a s been f i x e d .

Let

be a p o i n t i n

f f L

which

W s h a l l begin by e Po. d e s c r i b i n g t h e r i g h t hand s i d e of (1.2) more p r e c i s e l y . The kernel Kx(xly) can be e x p r e s s e d i n terms of E i s e n s t e i n s e r i e s

is s u i t a b l y regular with respect t o

where

i s summed o v e r a s u i t a b l e ortho-normal b a s i s of

A?

X^

(P).

To o b t a i n

A K (x,y),

w e j u s t t r u n c a t e each o f t h e two
Then JT(f)

E i s e n s t e i n s e r i e s i n t h e formula. x=y

i s given by s e t t i n g

i n t h e r e s u l t i n g e x p r e s s i o n , and i n t e g r a t i n g o v e r

G ( Q ) \G @)I

.
x

I t t u r n s o u t t h a t t h e i n t e g r a l over

G ( Q ) \G(A)'

may be t a k e n i n s i d e a l l t h e sums and i n t e g r a l s i n t h e formula T T f o r A,A K ( x , x ) . T h i s p r o v i d e s a s l i g h t l y more convenient expression f o r

P

J (

( [ l ( d ) , Theorem 3.21)

Given

Po

II ( ~ $A)) , and p
on
A?

ioi-i

d e f i n e an o p e r a t o r

nT ( P I ^ ) XI^

XI^

(P)

by s e t t i n g

f o r any p a i r of v e c t o r s becomes t h e formula

and

in

X,v

(PI.

Then (1.2)

James ARTHUR

where

We hasten to point out that (4.2) does not represent an T explicit formula for Jx() It does not allow us to see how to decompose J

into distributions on the local groups G(Q). J(

T

Moreover, we know that

is a polynomial function of

T.

However, this is certainly not clear from the right hand side of (4.2). The most immediate weakness of (4.2) is that the definition of the operator (P,A) is not very explicit. However, there XlT A) is a more concrete expression for QT (PI , due to Langlands. X^ (see [8(a), 591 .) It is valid in the special case that P
Q

belongs to the associated class

Px ;

that is, when the

Eisenstein series on the right hand side of (4.1) are cuspidal. To describe it, we first recall that if in P and

are groups

F ( M ) and s belongs to W (CT. a z ,

Ofp onto

isomorphisms from elements in W

M

) , the set of 1 obtained by restricting

to .<7~,,

"PI then there is an important function

pllp

(s,A).

Forany

$ed2(p),

is defined to be

The trace formula

The integral converges only for the real part of chamber, but Mpl,;(s,A)
-

in a certain

can be analytically continued to a

meromorphic function of a maps from

A e uc

A (P) to
I ( M (A) I )

with values in the space of PIC A (PI). Suppose that T is a

2

representation in

Then M

l p

(s, A)

maps the subspace XI* (P,A) be the

2 (P) to AxrsTT(Pl). If XI* value at A = A of '

A*

A e i<^ , let uT

where

Here,

Z(A ) pl

G is the lattice in .<rt-p

generated by 1

{a":
Then
OJ

a e A } .

1
T ( P I A) is an operator on A (P) X XI* Lanqlands' formula amounts to the assertion that if T belongs to P the operators Q~ (PI ) and u A (PI A)

P are

xr

XI*

x IT

equal. This makes the right hand side of (4.3) considerably more explicit. However, J (f) is given in (4.2) by a sum over all
T

James ARTHUR

s t a n d a r d p a r a b o l i c subgroups belong t o

x'

n o t be e q u a l .

U n f o r t u n a t e l y , i f P does n o t T t h e o p e r a t o r s CIT ( P , l ) and w (PIX) may XI^ x,Tf The b e s t w e can s a l v a g e i s a formula which i s
P

asymptotic w i t h r e s p e c t t o Set

W s h a l l say t h a t e
to
Po

approaches i n f i n i t y s t r o n g l y u i t h r e s p e c t infinity, but

T

if

1 1 ~approaches ~ ~

remains w i t h i n a

region

f o r some

6 > 0.

T E R M 4.1: HOE difference

If

d>

and

4'

a r e vectors i n

A^

x,v

(P) ,

the

approaches z e r o a s to
Po.

approaches i n f i n i t y s t r o n g l y w i t h r e s p e c t
X

The convergence i s uniform f o r

i n compact s u b s e t s

T h i s theorem i s t h e main r e s u l t of [ l ( h ) l . t h e formula of Langlands a s a s t a r t i n g p o i n t .

The proof u s e s

11

The trace formula

5, CONSEQUENCES
A

OF A PALEY-WIENER THEOREM,

The asymptotic formula of Theorem 4.1 is only uniform for in compact sets.
\

However, the formula (4.2) entails i/f-/ict. ,

integrating if P
. '

over the space

which is noncompact

G.

Therefore Theorem 4.1 apparently cannot be exploited.

Our rescue is provided by a multiplier theorem, which was proved in [l(g)] as a consequence of the Paley-Wiener theorem for real groups. The multiplier theorem concerns C ( G OR) ,

~,r,) ,

the algebra of smooth, compactly supported functions on

GQR)

which are left and right finite under the maximal compact subgroup of G(R). Set

where
M (R)

/kand

is the Lie algebra of some maximal real split torus in

Then of

/k-n

JhrK
and Let

is the Lie algebra of a maximal torus in

nM

(El.

is a Cartan subalgebra of

G(C),

,kE(

q the Lie algebra

,
of W

is invariant under the Weyl group, W

(q, $C) .

distributions on

$)

be the algebra of compactly supported

which are invariant under

y
t

The and

multiplier theorem states that for any

f_
C :

E) l' $
%,y

(G OR)

, %) ,

there is a unique function

(G fJR) , \ , )

with the following property.

If II

in is any

representation in

Il (G OR) )

then

James ARTHUR

where

{v%}

is the

W-orbit in and

associated to the
A

infinitesimal character of transform of support of

y

is the Fourier-Laplace

The theorem also provides a bound for the

in terms of the support of y and of f . & %,Y We apply the theorem to C ~ ( ) , K) , the algebra of K AG '
~ () AG '

finite functions in C

For each group

is a natural surjective map kernel of on h .

h :
y e

4
1

Po

- +

Suppose that

E ( Vis~actually supported
is the restriction to

Let

.bl

there

be the

fyl .

Any function

f e c ~ G @ ) , K)

G @)

of a finite sum

where each

is a
G) @'

finite function in C (G ( Q ))

The

restriction to

of the function

depends only on and

TT

We denote it by

f
pxl

.
TT

Suppose that
(P

Po

II (M- (A)). Then the operator

p X r T (P

,A

f ) Y

will be

a scalar multiple of

f) . For if

there is a Weyl orbit

{vn}

in

associated to the Then

infinitesimal character of

ii_.

The trace formula

We shall now try to sketch how the multiplier theorem can be applied to the study of J~

x.

The key is the formula (4.2) ,

and in particular, the fact that the left hand side of (4.2) is a polynomial function of for points on in f
C :

T.

Now this formula is only valid

which are suitably regular in a sense that depends

If

N > 0

let G ( : C

1 (A)

, K)

be the space of functions

(G (A)

, K)

which are supported on

is the usual kind of function used to describe estimates on Gift). See [l(c),

11

.I

Then it turns out that there is a N

constant C

such that for any

and any

f e c~(G@)

, K),

formula (4.2) holds whenever

(see [l(i), Proposition 2.21 .) ~f

f

belongsto

C;(G(A)~,K)

and

lc&($)'

is

supported on

James ARTHUR

then

f Y

will belong to

A' CN+Ny (G ()

, K) .

We substitute f

into the right hand side of (4.2).

We obtain

This equals

where

for

and

IT e

II ( M (A) ~
H

1.

The function

T ifi^ (H)

depends

only on the projection of finitely many function of

H
IT.

on A .

It vanishes for all but

It can also be shown to be a smooth bounded

.
T whenever

The expression (5.1) is a polynomial in

H c

f^ .

~ e t-y

be the Dirac measure on

.$

at the

The trace formula

point

H ,

and set

Then

[IH[~ .

The expression (5.1) equals

This function is a polynomial in

T whenever

Its value at H = 0 is just the right hand side of (4.21, which equals J T (f) as long as dp (TI is greater than C (1+ N)

Suppose we could integrate (5.2) against an arbitrary Schwartz function of

H e

4. '

By the Plancherel theorem on

inner $ the resulting* inner product could be replaced by an were * product on i i . If the original Schwartz function
/ikG

taken from the usual Paley-Wiener space on

.bl we would be ,

able to replace (4.2) by a formula in which all the integrals were over compact sets. Unfortunately this step cannot be taken

immediately, because (5.2) may not be a tempered function of For each

1 e

I1 ( M ( A ))

, let

James ARTHUR

be the decomposition of any nonzero point

X

vT

into real and imaginary parts.

Then

will cause (5.2) to be nontempered.

Nevertheless, it is still possible to treat (5.2) as if all the points

X

were zero.

This can be justified by an elementary We shall forgo

but rather complicated lemma on polynomials.

the details, and be content to state only the final result. Let i

S (ib*/ici.)w

be the space of Schwartz functions on W

/iaG
i r

*
<:

which are invariant under

If

B r S (i,b /iaG)

and

II ( M (A) l),

set

B (A) = B ( i Y + A) ,

r ifip/i^t ' G

It is a Schwartz function on

i<x/i^ G "

THEOREM 5.1:

(i) For every function B r pT(B) in

T

s (i.b*/i<)'

there

is a unique polynomial

such that

approaches zero as to
Po.

approaches infinity strongly with respect

(ii) Suppose that

B(0) = 1

Then

where

The trace f o m d a

See [l(i), Theorem 6.31.

If the function B

happens to be compactly supported, the

The first statesame will be true'of all the functions B . ment of Theorem 5.1 can be combined with Theorem 4.1 to give

THEOREM 5.2:

Suppose that

B c cW(i.b*/i~Q)
c

Then

P (B)

is the unique polynomial which differs from

by an expression which approaches zero as strongly with respect to

Po.

approaches infinity

See [1(i) Theorem 7 1 .1

5,

AN EXPLICIT FORMULA,

Theorems- 5.1 and 5.2 provide a two step procedure for T if f is any function in C ~ (A) I) which G calculating Jx f ) is
K

finite.

One first calculates

as the polynomial which is asymptotic to

James ARTHUR

I I( pI PaPo~eII ~ (A) )
One then chooses any T J (f) by

P M ~ )

rlJ * * ~r(~x,n(P,A)~x,~(~,A,f))~T(~)d . i^p/imG

T

such that

B(0) = 1 ,

and calculates

The second step will follow immediately from the first. first step, however, is more difficult.

The

It gives rise to some

combinatorial problems which are best handled with the notion of a (G , M) family, introduced in [I( I. Suppose that M e L (No) .

A fG

, M) family is a set of smooth functions

indexed by the groups

in

P(M),

which satisfy a certain Q and

Q'

compatibility condition. groups in P(M) and

Namely, if

are adjacent

lies in the hyperplane spanned by the

Q

common wall of the chambers of c (A) = c

Q'

(A).

iffl then M' A basic result (Lemma 6.2 of [l(f 1 ) asserts that
Q'
(G ,M)

and

in

if

{c (A) is a }
Q

family, then

extends to a smooth function on

iNM.

A second result, which

is what is used to deal with the combinatorial problems we

The tvaee f o d a

mentioned, concerns products of

( G IM)

families.

Suppose that

{d (A) } is another (G ,M) family. Q associated to the (G ,M) family

Then the function (6.1)

is given by

([l(f), Lemma 6 . 3 1 ) .

For any

S e F(M),

s cM(A)

is the function

(6.2) associated to the

(Ms ,M)

family

and

c(A)

is a certain smooth function on

iffl

only on the projection of For any

(G , M)

onto

iaM

family

{ c (A)}

and any

which depends

L e L (M) ,

there

is associated a natural to lie in

i<xL

( G IL)

family. Let

be constrained

and choose

Ql e P(L).

The compatibility

condition implies that the function

is independent of

Q.

We denote it by

c
1

(A).

Then

James ARTHUR

is a

(G , L)-family.

We write

for the corresponding function (6 2). value at cL A typical example of a (G ,M)

A = 0

We sometimes denote its

simply by

family is given by

Q c P(M), A

ia,

where

M compatibility condition requires that for adjacent Q

{ yQ

: Q c

P (M)}

is a family of points in

4 %

The and Q'

where

is the root in

A
Q

which is orthogonal to the common and Q'

wall of the chambers of

If each

is actually

positive, the function ct4(A) admits a geometric interpretation. It is the Fourier transform of the characteristic function in
OLM

of the convex hull of

= cM(0)

{yQ : Q e P

(M)1 .

The number (G , M)

is just the volume of this convex hull.

families of this sort are needed to describe the distributions

Jn, and
IT

in the cases where explicit formulas exist.

(see

11(c), 571, [l(f), 5141 and also [ K b )I .) For another example, fix also a point

F(M)

and let M = M put

in

i m .

Fix

For any

Q e P(M),

The trace formula

Then

M P
Q

A = M

Q p

(A)-'MQiP(A+R) ,

ioi-

MI

i s a f u n c t i o n on
on

imM

with values i n t h e space of operators

A* ( P ) .

I t c a n b e shown t h a t

is a

(G , M )

family of

(vector valued) functions.

I n o r d e r t o d e a l w i t h ( 6 . 1 ) we must l o o k back a t t h e T ( P , i ) i n 54. The e x p r e s s i o n (4'.4) c a n be XI^ w r i t t e n a s t h e sum o v e r s e W (mP ,K P ) o f d e f i n i t i o n of

ii)

Given

a n d t W P , f t . w t of

),

set

Q = w q l pIwt ' t

fox

any r e p r e s e n t a t i v e

in

G(Q).

Then

i s a b i j e c t i o n between p a i r s which o c c u r i n t h e sum above and

groups
Q e P(M).

Notice t h a t

James ASTHUR

I t can a l s o be shown t h a t

equals

MQJp(A)

- IM Q l P ( s , W

(SP-\}

(Y(T))

where

YQ(T)

is t h e projection onto

of t h e p o i n t

l ( T - T o ) Theref o r e ,

To.

t h e f u n c t i o n which must be s u b s t i t u t e d i n t o ( 6 . 1 ) , can b e o b t a i n e d by s e t t i n g

A' = A

i n t h e sum o v e r

W(-ffl.

I&P)

of

Formula ( 6 . 3 ) s u g g e s t s a way t o h a n d l e (6.4)

W set e

and d e f i n e
A(Y(T))

c (A) = e

The trace fommla

and

d (A) = t r (Q ~ P ( X ) l ~ Q j p ( s ~ r *')P x r 71 IP,X,~)) Q for any { d (A))

Q
are

P(M),. (G ,M )

It is not hard to show that families.

{cQ(A) }

and

The function (6.4) equals

an expression to which we can apply (6.3). of terms indexed by groups

S
e

The result is a sum

F(M).

The contribution to

(6.1) of each such term can be shown to be asymptotic to a polynomial in T

The sum of all these polynomials will be the pT ( B )

required polynomial

Once again , we will skip the

details and state only the final result. In the notation above, set

This is a product of two

(G

( G , M ) families, so it is itself a is any group in

L (Mp),

,M

family.

If L

ML (PIX) = lim 1 M~ (PfX,A)8 ( ) ' A" A+O QcF(L) Q~ I

is defined.

It is a polynomial in
(P)

T with values in the space

of operators on

James ARTHUR

If L

are any two groups in

L ( M ), let
U C

W1'

reg

be the set of elements in W (mM aM) for which , space of fixed vectors.

is the

THEOREM 6.1:
P
2

The polynomial

pT (B)

equals the sum over

P,

n ( M '), ~

L e L ( M ~ )and

e $(@p)re

of

the product of

with

See [ l ( j ) , Theorem 4.11.

The theorem provides an explicit formula for pT (B) From T this we can obtain a formula for J (f) and, in particular, for

It i s easy to show that

Then

Jx( can be obtained from the formula of the theorem by

simply suppressing T .

The trace formula

The formula for function B . the function

J ( will still depend on the test

It would be better if we could remove it. f is still required to be

K

Moreover,

finite.
I. )

Our formula The

ought to apply to an arbitrary function in C(@ ;G)

dominated convergence theorem will permit these improvements provided that a certain multiple integral can be shown to converge absolutely. The proof of such absolute convergence turns out

to rest on the ability to normalize the intertwining operators between induced representations on the local groups G(Q). At

first this may seem like a tall order, but it is not necessary to have the precise normalizations proposed in [8(b), Appendix 111. We require only a general kind of normalization of the sort established in [ 6 ] for real groups. The analogue for p-adic In any case, we

groups should not be too difficult to prove.

assume the existence of such normalizations for the following theorem.

THEOREM 6.2.

Suppose that

f e

C ~ ( (A) G l)

Then

J (f)

equals

the sum over M e L ( M I , L

L (M),

n (M(A) l)

and

of the product of

with

J a m e s ARTHUR

This is Theorem 8.2 of C l ( j ) I. Implicit in the statement if the absolute convergence of the expression for J (f).

Let D (f) be the sum of the terms in the expression for J (i) for

which and s

In particular, the distribution D of J

XI

is invariant. As the "discrete part"

it will play a special role in the applications of the trace

formula.

BIBLIOGRAPHY
1. ARTHUR,

J. ,

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(i) a f a m i l y o f d i s t r i b u t i o n s o b t a i n e d f r o m E i s e n s t e i n seOn r i e s I : A p p l i c a t i o n o f t h e Paley-Wiener theorem, p r e p r i n t .

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9, Amer.

( b ) On t h e F u n c t i o n a l E q u a t i o n s S a t i s f i e d b y E i s e n s t e i n S e r i e s , (1976). L e c t u r e Notes i n Math.,

(c) B a s e Change f o r G L ( 2 ) , A n n a l s o f Math. S t u d i e s , 1980. ( d ) L e s d e b u t s d ' u n e f o r m u l e des t r a c e s s t a b l e s , P u b l i c a t i o n d e

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t o appear i n t h e s e proceedings.

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