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 ,
[l(a) ], [ 4 ] ,
[12], [9 (a)l and
C9 (b)I .
1, THE TRACE FORMULA - FIRST VERSION,
Le G be a reductive algebraic group defined over
2).
Let AG be the
split component of the center of G, and set
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
f e
(G(Q)\G (A) 1)
c (G(A) 1) , R(f) is an integral operator on L2 (G(Q)\G
. If
(A)
1)
. 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
which is suitably regular with respect to
that
a(T)
a
is large for each root
of
.
of G ( A ) 1
in the sense
Po,
Po
On
A0 =
\.
The trace formula is then an identity
We describe the left hand side first.
of equivalence classes in
0 denotes the set
in which two elements in
G($),
G ( Q ) are deemed equivalent if their semisimple components are
conjugate.
G(Q)
This relation is just G ( Q )
conjugacy if G
is anisotropic, but it is weaker than conjugacy for general G
If
P
is a parabolic subgroup of
respect to
where
N
Po,
and
o-
e
0,
G
which is standard with
set
is the unipotent radical of
Levi component of
P
.
which contains M
P
.
and
Mp
Then
is the unique
James ARTHUR
is the integral kernel of the operator
convolving f on
R(f).
(-1)dim ( A / A 1
I
P3P0
The function
,
it is
Define
I
(fix,<5x)~p(~p(5x)-~).
K
6eP (Q)\G (Q) p'e
Tp ( H ( - ) - T) ,
a moment, equals
obtained by
L ( N @)M (Q)\G (A)I). If P = G
just the integral kernel of
T
k(x,f)
=
Rp(f)
1
if
whose definition we will recall in
P=G,
and vanishes on a large compact
neighborhood of 1 in G ( A ) I if
T
k&(x, f) i$ obtained by modifying
hood of infinity in G ( @ )\ G ( A )
.
P s G
.
In other words,
KG ,&(x ,y)
in some neighborT
The function k(x,f)
turns
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
Ap - AMp
P I
there is a natural map from
If Q
,
T
P is the
Write a = en.
is a parabolic subgroup that contains
-o<_
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
.
There is associated to each
.
Let
{av : a e
A }
of
in
Let
8
.
a e Ap
*
a "co-ro~t" av
*
be the basis of t^-p/Agwhich is dual to
,Then
<
is the characteristic function of
Hp be the continuous function from G ( A )
to
%
defined
by
T
In the formula for k@(x,f)
above, the point
T
belongs
to a , but it projects naturally onto a point in A .
It
is in this sense that
is defined.
The set
X
which appears on the right hand side of (1.1)
is best motivated by looking at 0.
If
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
W
be the restricted Weyl group of
Jmea ARTHUR
(G
,A).
It is clear that
with the set of
0
is in bijective correspondence
(MB , cB),
Wo-orbits of pairs
standard parabolic subgroup of
conjugacy class in MB(Q).
representations of
and
cB
B
is a
is an elliptic
If we think of the automorphic
G(A)'
as being dual in some sense to the
G(@),
conjugacy classes in
G,
where
we can imagine that the c u s p i d a l
automorphic representations might correspond to elliptic
X
conjugacy classes.
( M ,r ) ,
of pairs
of
G
and
r
where
B
is a standard parabolic subgroup
is an irreducible cuspidal automorphic represen-
tation of MB@)
of groups B
is defined to be the set of W-orbits
.
x
For a given
obtained in this way.
of standard parabolic subgroups.
associated class
Suppose that
L 2 ( N (&)M~(Q)\G (A)'
Pn.
L~ [ N (A)% (Q)\G (A)' )
X
and
,
P
be the set
x
Similarly, we have an
0.
P a
P
0
are given.
Let
be the space of functions @
with the following property.
standard parabolic subgroup B ,
x e G @I)
let
It is an associated class
<ye
for any
x 6
)x
X ,
E
with
B
c
P
in
For every
, and almost all
the projection of the function
I
1
meMg(A) ,
1 (nmxldn ,
% (fa)\%@I
onto the space of cusp forms in
under MB(~)'
(MB I rB)
L2 ( M (Q)
~ \MB@I1
as a sum of representations
is a pair in
x.
rB ,
)
transforms
in which
IÂ there is no such pair in
x,
B , X will be orthogonal to the space of cusp forms on
% (Q)\Mc (A)' . It follows from a basic result in Eisenstein
The trace fornula
series that
group in
L (Np@IMP (Q)\G (A)' )
P
will be zero unless there is a
which is contained in
x
P.
Moreover, there is an
orthogonal decomposition
Let
Kp
1
R f
x (x,y)
to
be the integral kernel of the restriction of
L (Np(AIMp(Q)\G (A) I )
formula for
^x
.
One can write down a
(x,y) in terms of Eisenstein series.
each side being equal to the integral kernel of
Rp(f).
We have
Xf we
define the modified functions
dim ( A h G 1
kT (x,f) = [
(-1)
p3p0
6eP ( Q ) \G (Q)
I
K ~ l (xfix , fix)Tp (~~(5x1
-T)
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,f) . Of these, the second
integrability of
x
function is the harder one to handle.
To prove its integrability
it is necessary to introduce a truncation operator on
Given
T
G(Q) \G(A)'
as above, the truncation of a continuous function h
on
G(Q)\G@I1
If
x
X ,
is the function
let
ATAT K (x,y) be the function obtained by
1 2
truncating the function
x
in each variable separately.
From properties of the truncation
operator, one shows that
is finite (see [l(d), 511 )
.
One can also show, with some effort,
that
([l(d), Â § 2 ] ) from which one immediately concludes that
.
The trace forrnula
is also finite
( [l(d), Theorem 2.1 ] ) . This was the result that
was required for formula (1.1).
for any
x
e
XI
In the process, one shows that
$he integral of
is zero for sufficiently regular T
([l (d), Lemma 2.41).
In
other words,
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(f)
formula for Jx
.
2,
SOME REMARKS
~t is natural to ask how the terms in the trace formula
(1.1) depend on the point
It is shown in Proposition 2.3
T
of [1(£] that the distributions J&T (£ and J(f)
are polynomial functions of
T
in
<x.
T,
T.
and so can be defined for all points
There turns out to be a natural point
such that the distributions
To
in
%
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 F(I1) be the set of parabolic subgroups of G, now no longer standard,
which are defined over Q and contain M. Then if Q
F(M) ,
also
0
contains M. Let P(M) be the set of groups Q in F(M) such that M equals
M.
Suppose that L e KM)
. We write
e
M
Q
L~ (M) , F~ (M) and F^ (PI)
I
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
(?-
n L(Q) is a disjoint
I
,
The twice formula
union
of equivalence classes in the set, OL
,
associated to L .
can certainly define the distributions JBi
By definition, J
is zero unless
0-
.
on L(A)'
meets L(Q).
If
(9-
we can define a distribution J
Formula (2.1), applied to L
where now
f
,
x
on L@)'
We
set
x
X
,
in a similar way.
yields
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
C;(M@)')
f r o c~(L(A)~) to
m
.
: ~feft)'
where
6Q
fQY(m)
for each
y
E
L&)'
.
If
is given by
is the modular function
Q(A).
Then the formulas
alluded to are
and
tf-c 0 .
where
[l(f)
.
y, c X I
Theorem 3.21
.)
f c c"L(&)')
M
Here 1 wOQ1
elements in the Weyl group of
and
y c L(A).
(see
stands for the number of
( M ,A )
.
As usual,
fY
is the
function
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 will be invariant
and y . the sum vanishes. For example, J^,
if and only if
O'n M(Q)
is empty for each group M e F ~ M )
The trace f o d a
with
M
*
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
x
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).
We shall give a brief discussion
of this construction.
If
H
is a locally compact group, let
H(H)
denote the
set of equivalence classes of irreducible unitary representations
of
Suppose that M
H.
II (M (A)I )
to embed
of
and
M(A)'
II (M (A)I )
on
II (M (A)) ; for
in
AM@?)
0
is any group in
,
0
.
Let
II ( M ( < A ) )
tions in
IItemp
.
We shall agree
M (A) is the direct product
so there is a bijection between
and the representations in
AMOR)
L(M).
(M(/A)
1
)
H (M (A))
which are trivial
be set of tempered representa-
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
.
,
Â5
I we proposed a candidate, l (M ( A ) ) , for the image
Roughly speaking,
l (M (A)I )
of complex valued functions on
is defined to be the space
H t e (M (A)I )
which are ~chwartz
James ARTHUE
functions in all possible parameters.
T
maps
C (M (&) I)
butions on
? (M (&)
') .
T;
of
T
.
~t is also easy
'
1 (M (A) ) ' , the dual
maps
into the space of tempered invariant distri-
I(M(A)),
.
M @)I
HYPOTHESIS 3.1:
1 (M (A) )
continuously into
to see that the transpose
space of
It is easy to show that
For each
M
e
(Mo), TM maps
C (M(A)')
onto
Moreover, the image of the transpose,
.
is the space of aH tempered invariant distributions on M@)'
This hypothesis will be in force for the rest of 53.
is any tempered invariant distribution on M @ )
I
be the unique element in
7 (M ( A ))
,
such that
If
I
we will let
I
A
TM (I) = I
.
Important examples of tempered invariant distributions are
the orbital integrals.
valuations on
S
and that for each
Q,
maximal torils of
Suppose that
M
defined over
is a finite set of
v
Q .
T; = ( TT T ( Q ) ) n M W
in
S,
T
is a
Set
1
,
ves
and let T
'
be the set of elements -y
S treg
centralizer in
in T;
whose
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 %
is the function on
which is usually put in as a
T;
normalizing factor.)
on
C (M (!A)) .
?Y
on
I
is a tempered invariant distribution
Y
By Hypothesis 3.1 it corresponds to a distribution
2 (M @A)'
) . Now suppose that
Â
has compact support.
Then the map
has bounded support; that is, the support in
closure n
4
.
T
2 (M (a))
Let
)
-
maps
1 (M(A) I).
Ts
,
C
:
(M (&)I )
has compact
the function
There is a natural topology on
continuously into
HYPOTHESIS 3.2 : For each group H
onto
S ,reg
be the set of functions
such that for every group
has bounded support.
such that. T
I (M@)'
T1
c
L (M,) ,
I (~($4))
1 (M @I1
maps
C:
Moreover, the image of the transpose,
).
(M @A) )
James ARTHUR
.
1
is the space of all invariant distributions on'M(A)
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.
be
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
c
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:(L(A)
1
)
+ (E
,
L
6
L(M~),
is a family of distributions such that
The trace formula
for a
h L
f
1) and y e L(A)'
c~(L(A)
. Then there is a unique
family
of invariant distributions such that for every L and f,
PROOF
:
Assume inductively that I
for all groups
for any f
M e
L(Mo) with M
c_(L(A) 1) . Then
5
has been defined and satisfies (3.1)
L. Define
.
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.
n
According to (2.3) and (2.3) , we can apply the proposition to
the families { JL} and {JL1. We obtain invariant
x
James ARTHUR
distributions
and
]
:
I
{
which s a t i s f y t h e analogues o f
{I;}
(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
L=G
of t h e f o l l o w i n g theorem.
THEOREM 3.5:
PROOF:
For any group
L
in
i(Mo),
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
M e I ( M )
with
M
;L .
For any such
f o r any
$ e I ( M @ ) ) . Then
by ( 2 . 2 )
and ( 3 . 1 ) .
M
we a l s o have
The trace formula
4, INNER
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 _ ,
I@, Jx
and
I
explicitly.
x
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
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
0.
orbital integral of
express
I(f)
([1 (c), (8.7)1 )
f
GL.)
It is possible to then
(see [l(f), 5141 for the case
is not unramified, we would expect to express
o-
as some kind of limit of weighted orbital integrals.
J(f)
Then
If
.
as a certain invariant distribution associated
to a weighted orbital integral.
of
G(Q).
I*(£
would be a limit of the corresponding invariant
distributions.
Je(f)
and
In any case, the lack of explicit formulas for
I&(f),
with
o-
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 .)
x
Unlike with the classes
however, it seems to be essential to have a formula for all
in order to apply the trace formula.
@
,
x
We shall devote the rest
of this paper to a description of such a formula.
Suppose that
A (P)
P
e
F(M)
is a parabolic subgroup. Let
be the space of square-integrable automorphic forms on
N (AIMp( Q ) \G iA)
whose restriction to Mp (A)'
There is an Eisenstein series for each
is square integrable.
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
*
x
X e¥^p,a- If
+
A2
(P) be the space of vectors
XI
A (P) which have the following two properties.
IT
6
n ( l > ~ ( A ) ) , let
e X
2
(i) The restriction of
6
to
G(A)
belongs to
.
L~(N~È)M~(Q)\G(A)~
(ii) For every
m
x
o> (mx)
->-
I
transforms under
Let
G (A), the function
in
m
6
fL> (A),
Mp(<A.) according to
i2
(P) be the completion of A2
(PI with respect to the
XI'"
xrn
inner product
*
For each
of
IT.
G(A)
&P,C
on
x2XI
there is an induced representation
Ox,
(PI, defined by
IT
The representation is unitary if
X
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
T
be a p o i n t i n
f f L
which
is s u i t a b l y regular with respect t o
We s h a l l begin by
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
Kx(xly)
kernel
where
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
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
To o b t a i n
A K (x,y),
x
X^
Then
JT(f)
x
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
.
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
x
expression f o r
P
Po
,
( [ l ( d ) , Theorem 3.21)
II ( ~ $A)
p ) , and
T
nTXI^ ( P I ^ )
T
J (£
on
A?
XI^
(P)
f o r any p a i r of v e c t o r s
becomes t h e formula
(P).
w e j u s t t r u n c a t e each o f t h e two
E i s e n s t e i n s e r i e s i n t h e formula.
x=y
A?
A
c
ioi-i
,
.
Given
d e f i n e an o p e r a t o r
by s e t t i n g
and
@
in
A
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).
x
Moreover, we know that
T
J(£
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
(P,A) is not very explicit. However, there
XlT
is a more concrete expression for QT (PIA) , due to Langlands.
X^
(see [8(a), 591 .) It is valid in the special case that P
of the operator
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
and
P
are groups
F ( M ) and s belongs to W (CT.,a z
isomorphisms from
elements in W
M
P
pllp
(s,A).
Ofp
to .<7~,,
Forany
is defined to be
onto
) , the set of
1
obtained by restricting
"PI
then there is an important function
$ed2(p),
The trace formula
X
The integral converges only for the real part of
chamber, but
Mpl,;(s,A)
can be analytically continued to a
-
meromorphic function of
a
maps from
representation in
in a certain
A e uc
with values in the space of
PIC
A (PI). Suppose that T is a
2
A (P) to
II ( M (A))
*
.
2
(P) to AxrsTT(Pl). If
XI*
value at A' = A of
A*
Then M
lp
(s,A)
maps the subspace
A e i<^ , let uT
XI*
(P,A) be the
where
Here,
Z(A )
pl
G
is the lattice in .<rt-p
generated by
1
{a":
a e A } .
1
.
T
( P I A)
is an operator on A
(P)
X Ã ˆ
XI*
Lanqlands' formula amounts to the assertion that if
T
belongs to P
the operators Q~ (PIA ) and u
(PIA)
Then
OJ
xr
XI*
x IT
P
are
equal. This makes the right hand side of (4.3) considerably more
explicit.
T
However, J (f) is given in (4.2) by a sum over all
James ARTHUR
s t a n d a r d p a r a b o l i c subgroups
P
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
asymptotic w i t h r e s p e c t t o
T
P
.
Set
We s h a l l s a y t h a t
to
Po
T
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
1 1 ~approaches
~ ~
if
infinity, but
T
remains w i t h i n a
region
6 > 0.
f o r some
THEOREM 4 . 1 :
If
d>
and
4'
a r e vectors i n
A^
x,v
(P) ,
the
difference
approaches z e r o a s
to
Po.
T
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
The convergence i s uniform f o r
X
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
OF A PALEY-WIENER THEOREM,
The asymptotic formula of Theorem 4.1 is only uniform for
A
in compact sets.
However, the formula (4.2) entails
*
*
\
if
Therefore Theorem 4.1 apparently cannot be exploited.
P
.'
G.
over the space
i/f-/ict. ,
integrating
which is noncompact
Our rescue is provided by a multiplier theorem, which was
proved in [l(g)] as a consequence of the Paley-Wiener theorem
The multiplier theorem concerns C ( G OR) ,
for real groups.
the algebra of smooth, compactly supported functions on
~,r,),
GQR)
which are left and right finite under the maximal compact subgroup of
G(R).
Set
where
/k-
is the Lie algebra of some maximal real split torus in
M (R)
and
JhrK
Then
of
/k-n
G(C),
is the Lie algebra of a maximal torus in
is a Cartan subalgebra of
and
(qÃ, $C) .
,k-
Let
distributions on
q à £ the Lie algebra
is invariant under the Weyl group, W
E(
$)
4
C:
c
(G OR)
(G fJR) , \,)
, %) ,
which are invariant under
y
t
Il (G OR) )
,
then
W
.
El$)'
there is a unique function
with the following property.
representation in
,
of
be the algebra of compactly supported
multiplier theorem states that for any
f_
nM
The
and
%,y
If IIÃ
in
is any
(El.
James ARTHUR
{v%}
where
is the
W-orbit in
associated to the
A
infinitesimal character of
transform of
y
.
and
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 ~ (A)G' , K) , the algebra of K
support of
finite functions in C
)
~ (A)G'
is a natural surjective map
kernel of
on
fyl .
G @)
h .
For each group
4
h :
Suppose that
y e
.
- + Ã ˆ
Let
P
3
Po
.bl
,
there
be the
E ( Vis~actually supported
1
f e c ~ G @ ) , K)
Any function
is the restriction to
of a finite sum
where each
f
restriction to
depends only on
and
.
TT
e
is a
m
finite function in C (G ( Q ))
.
The
of the function
G@)'
f
K
.
We denote it by
II (M- (A)). Then the operator
p X r T (P
A
{vn}
in
infinitesimal character of
ii_.
a scalar multiple of
there is a Weyl orbit
f
.
pxl
TT
Suppose that
(P
,A
f )
Y
f) . For if
associated to the
Then
P
3
Po
will be
The trace formula
We shall now try to sketch how the multiplier theorem can
be applied to the study of
x.
J~
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
f
in
C:
.
T
, K)
(G (A)
Now this formula is only valid
which are suitably regular in a sense that depends
N > 0
If
T.
,
let G(:C
(A)1
, K)
be the space of functions
which are supported on
is the usual kind of function used to describe estimates on
Gift).
See [l(c),
constant C
Â
11
.I
Then it turns out that there is a
such that for any
N
,
and any
f e c~(G@)
formula (4.2) holds whenever
(see [l(i), Proposition 2.21 .)
~f
f
belongsto
supported on
C;(G(A)~,K)
and
lc&($)'
is
1
, K),
James ARTHUR
then
f
Y
CN+Ny (G (A)'
will belong to
into the right hand side of (4.2).
, K) .
We substitute f
Y
We obtain
This equals
where
for
and
H
IT e
1
II ( M (A)
~
only on the projection of
finitely many
function of
IT.
H
H
1.
The function
on A .
H c
f^ .
depends
It vanishes for all but
It can also be shown to be a smooth bounded
.
The expression (5.1) is a polynomial in
F
ifi^T (H)
~ e t-y
T
be the Dirac measure on
whenever
.$
at the
The trace formula
point
Then
and set
H ,
N
Y
=
[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)
.
0
Suppose we could integrate (5.2) against an arbitrary
Schwartz function of
H e
4'.
By the Plancherel theorem on
the resulting inner product could be replaced by an inner
$
*
*
product on i i
. If the original Schwartz function were
/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
H
.
James ARTHUR
vT
be the decomposition of
any nonzero point
into real and imaginary parts.
Then
will cause (5.2) to be nontempered.
X
Nevertheless, it is still possible to treat (5.2) as if all the
points
X
were zero.
This can be justified by an elementary
but rather complicated lemma on polynomials.
We shall forgo
the details, and be content to state only the final result.
Let
i
*
*
/iaG
and
ir
<:
S (ib*/ici.)w
be the space of Schwartz functions on
which are invariant under
II ( M (A) l),
W
.
If
It is a Schwartz function on
is a unique polynomial
to
A
T
pT(B)
*
*
*
in
T
s (i.b*/i<)'
there
such that
approaches infinity strongly with respect
Po.
(ii) Suppose that
where
*
r ifip/i^tG '
i<x/i^ G "
(i) For every function B r
approaches zero as
*
set
B (A) = B ( i Y + A) ,
THEOREM 5.1:
*
B r S (i,b /iaG)
B(0) = 1
.
Then
The trace f o m d a
See [l(i), Theorem 6.31.
If the function B
0
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
approaches infinity
Po.
See [1(i) Theorem 7.11
5,
T
.
AN EXPLICIT FORMULA,
Theorems- 5.1 and 5.2 provide a two step procedure for
T
if f is any function in C ~ (A)
GI) which
calculating Jx f )
is
K
finite.
One first calculates
as the polynomial which is asymptotic to
James ARTHUR
rlJi^p/imG
~r(~x,n(P,A)~x,~(~,A,f))~T(~)d
.
* *
I I p I)
PaPo~eII( ~ (A)
One then chooses any
T
J (f) by
T
P M ~ )
B
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)
A fG
, M) family is a set of smooth functions
indexed by the groups
Q
compatibility condition.
groups in
P(M)
and
A
in
P(M),
c (A) = c
Q
Q'
(A).
{c (A)} is a
Q
which satisfy a certain
Namely, if
Q
and
Q'
are adjacent
lies in the hyperplane spanned by the
common wall of the chambers of
if
Suppose that M e L (No) .
family, introduced in [I(£ I.
(G ,M)
and
Q'
in
*
ifflthen
M'
A basic result (Lemma 6.2 of [l(f 1 ) asserts that
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)
{d (A) } is another (G ,M) family.
Q
associated to the (G ,M) family
families.
Suppose that
Then the function (6.1)
is given by
([l(f), Lemma 6 . 3 1 ) .
For any
(Ms ,M)
(6.2) associated to the
and
s
cM(A)
S e F(M),
family
l
c(A)
(G , M)
For any
*
i<xL
A
family
is associated a natural
to lie in
iffl-
is a certain smooth function on
only on the projection of
onto
and choose
*
iaM
{ c (A)}
( G IL)
is the function
s
.
M
,
Ql e P(L).
L e L (M) ,
and any
family. Let
which depends
A
be constrained
The compatibility
condition implies that the function
is independent of
Q.
We denote it by
c
(A).
1
there
Then
James ARTHUR
(G , L)-family.
is a
We write
for the corresponding function (6 2).
A = 0
value at
We sometimes denote its
-
simply by
cL
A typical example of a (G ,M)
family is given by
Q c P(M), A
{ yQ
where
: Q c
P (M)}
is a family of points in
4
%
M
compatibility condition requires that for adjacent Q
where
a
A
is the root in
wall of the chambers of
Q
ia,
t
-
The
and
Q'
,
which is orthogonal to the common
Q
and
.
Q'
If each
c
is actually
positive, the function ct4(A) admits a geometric interpretation.
It is the Fourier transform of the characteristic function in
OLM
c
of the convex hull of
= cM(0)
{yQ : Q e P
(M)1 .
The number
is just the volume of this convex hull.
(G , M)
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
X
in
*
i m .
P
e
F(M)
For any
and let M = M
Q e P(M),
put
.
Fix
The trace formula
Then
M P
A = M
Q
*
i s a f u n c t i o n on
on
A* ( P ) .
is a
I
Q p
imM
(A)-'MQiP(A+R) ,
A
6
ioi-
with values i n t h e space of operators
I t c a n b e shown t h a t
(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
Given
P
ii)
a n d t W P , f t .
any r e p r e s e n t a t i v e
w
t
of
t
in
),
set
G(Q).
Q=wq
t l pIwt '
fox
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
*
MI
James ASTHUR
I t can a l s o be shown t h a t
equals
MQJp(A)
where
YQ(T)
- IM Q l P ( s , W
e
(SP-\}
%
is t h e projection onto
l ( T - T o )
+
(Y(T))
of t h e p o i n t
To.
Theref o r e ,
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
s
W(-ffl.
I&P)
Formula ( 6 . 3 ) s u g g e s t s a way t o h a n d l e (6.4)
and d e f i n e
A(Y(T))
c (A) = e
Q
.
of
We s e t
The trace fommla
and
d (A) = t r (Q~~ P ( X ) l ~ Q j p (
r s
*')P x r 71 IP,X,~))
Q
for any
Q
{ d (A))
are
P(M),.
It is not hard to show that
(G ,M )
families.
S
e
and
The function (6.4) equals
an expression to which we can apply (6.3).
of terms indexed by groups
{cQ(A) }
F(M).
The result is a sum
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
required polynomial
pT ( B )
.
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
,M
)
family.
If L
( G , M ) families, so it is itself a
is any group in
L (Mp),
T
ML (PIX) = lim
1
M~ (PfX,A)8 (A)"'
A+O QcF(L) Q~
I
is defined.
It is a polynomial in
of operators on
(P)
.
T with values in the space
James ARTHUR
If L
M
are any two groups in
L ( M ), let
be the set of elements in W (mM,aM) for which
W1'
UC
reg
is the
space of fixed vectors.
THEOREM 6.1:
P
2
P,
,
The polynomial
n ( M È
~ '),
pT (B)
equals the sum over
L e L ( M ~ )and
s
e $(@ãp)re
of
the product of
with
n
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
x
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
x
It would be better if we could remove it.
f
is still required to be
finite.
K
ought to apply to an arbitrary function in C;(G@)
Moreover,
Our formula
I).
The
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
groups should not be too difficult to prove.
In any case, we
assume the existence of such normalizations for the following
theorem.
THEOREM 6.2.
Suppose that
the sum over M e L ( M I , L
f e
C ~ ( (A)
G l)
L (M),
of the product of
with
e
.
Then
n (M(A) l)
J (f)
and
equals
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).
x
Let D (f) be the sum of the terms in the expression for J (i) for
x
x
which
and s
In particular, the distribution D
of J
x
is invariant. As the "discrete part"
it will play a special role in the applications of the trace
XI
formula.
BIBLIOGRAPHY
1. ARTHUR,
J. ,
( a ) The S e l b e r g t r a c e f o r m u l a f o r g r o u p s o f F - r a n k one,
Ann. o f M a t h . 100 ( 1 9 7 4 ) , 236-385.
( b ) The c h a r a c t e r s o f d i s c r e t e s e r i e s a s o r b i t a l i n t e g r a l s ,
I n v . M a t h . 32 ( 1 9 7 6 ) , 2 0 5 - 2 6 1 .
( c ) A t r a c e formula f o r r e d u c t i v e groups I : terms a s s o c i a t e d
45 ( 19781, 9 1 1 - 9 5 2 .
t o c l a s s e s i n G(Q), Duke M a t h . J . ( d ) A t r a c e formula f o r r e d u c t i v e groups I 1 : a p p l i c a t i o n s o f
a t r u n c a t i o n o p e r a t o r , Comp. M a t h . 40 ( 1 9 8 0 ) , 8 7 - 1 2 1 .
( e ) E i s e n s t e i n s e r i e s and t h e t r a c e f o r m u l a , P r o c . Sympos.
P u r e M a t h . , v o l . 3 3 , P a r t I, Amer. M a t h . Soc., P r o v i d e n c e
R.I.
( 1 9 7 9 ) , 253-274.
-
( f ) The t r a c e f o r m u l a i n i n v a r i a n t f o r m , Ann. o f M a t h . 113
1 9 8 1 ) 3 1-14.
( g ) A Paley-Wiener theorem f o r r e a l r e d u c t i v e groups, p r e p r i n t .
( h ) On t h e i n n e r p r o d u c t o f t r u n c a t e d E i s e n s t e i n s e r i e s , t o
a p p e a r i n Duke Math. J.
(i)
On 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 s e 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 .
The trace formula
(j) On 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 series
I1 : E x p l i c i t f o r m u l a s , p r e p r i n t .
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P r o c . Sympos. P u r e M a t h . , v o l . 3 3 , P a r t I , Amer. Math. S o c . ,
P r o v i d e n c e , R. I. ( 1 9 7 9 ) , 179-184.
3. FLICKER, Y . ,
The a d j o i n t l i f t i n g f r o m S L ( 2 ) t o P G L ( 3 ) , p r e p r i n t .
4 . GELBART, S . , a n d JACQUET, H . , Forms o f GL(2) f r o m t h e a n a l y t i c
p o i n t o f viewg, P r o c . Sympos. P u r e Math., v o l . 3 3 , P a r t I , A m e r .
Math. S o c . , P r o v i d e n c e , R . I .
( 1 9 7 9 ) , 213-252.
5 . JACQUET, H., a n d LANGLANDS, R.P., A u t o m o r p h i c Forms o n G L ( 2 ) , Lect u r e N o t e s i n M a t h . , 114 ( 1 9 7 0 ) .
6 . KNAPP, A.W., and STEIN, E.M., I n t e r t w i n i n g o p e r a t o r s f o r s e m i s i m p l e
g r o u p s 11, I n v . Math. 60 ( 1 9 8 0 ) , 9-84.
7. LABESSE, J . P . , a n d LANGLANDS, R.P.,
Can. J . Math. 31 ( 1 9 7 9 1 , 726-785.
L-indistinguishability
8 . LANGLANDS, R.P.,
(a) E i s e n s t e i n series, P r o c . Sympos. P u r e M a t h . , v o l .
Math., S o c . , P r o v i d e n c e , R . I .
( 1 9 6 6 ) , 235-252.
f o r SL(2),
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
l t U n i v e r s i t 6 P a r i s V I I , volume 1 3 ,
9. OSBORNE, M.S., a n d WARNER, G.,
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