ON SOME PROBLEMS SUGGESTED B V
THE
TRACE FORMULA
Janes Arthur
University of Toronto
Toronto, Ontario, Canada
MS5 1A1
In the present theory of autonorphic representations, a riaior coal
is to stabilize the trace formula.
Its realization will have irinortant
conseuuences, among which will be the proof of functorialitv in a
significant number of cases.
However, it will recruire much effort, for
there are a number of difficult problems to be solved first.
Pone of
the problems, especially those concerning orbital integrals, were
studied in [9(e)l.
They arise when one tries to interpret one side of
the trace formula.
The other side of the trace formla leads to a
different set of problems.
Amon? these, for examle, are questions
relatinq to the nontempered autonorphic representations which occur
discretely.
Our purpose here is to describe some of these problems
and to suqqest possible solutions.
Some of the problems have in fact been formulated as conjectures.
They have perhaps been stated in greater detail than is justifieri,
for I have not had sufficient tine to ponder then.
However, thev seem
quite natural to me, and I will be surprised if thev turn out to be
badly off the mark.
Our discussion will be rather informal.
We have tried to keep
thin~sas simple as possible, sometimes at the expense of or'-ittino
pertinent details.
Section 1, which is devoted to real crrouos, contains
a review of known theory, and a description of some problems and related
examples.
Section 2 has a similar format, but is in the global setting.
^Jewould have liked to follow it with a detailed discussion of the
trace formula, as it pertains to the conjecture in Section 2.
for want of time, we will be much briefer.
'Rowever,
After opening with a few
general remarks, we will attempt in Section 3 to motivate the con"jcture
with the trace formula only in the case of
~Sp(4). In so doincr, we
will meet a combinatorial problen which is trivial for D S ~ ( & ) ,hut is
more interesting for general aroups.
I am indebted to R. Kottwitz, D. Shelstad, and D. Voqan for en-
lightenina conversations.
I would also like to thank the University
of Maryland for its hospitality.
$1. A PROBLEM FOR REAL GROUPS
The trace formula, which we will discuss presently, is an eoualitv
1.1.
of invariant distributions.
The study of such distributions leads to
questions in local harmonic analysis.
Ye will beain by lookin'- at one
such question over the real numbers.
G
For the time being, we will take
For simplicity we shall assume that
group defined over
3R.
quasi-split.
IT(<^
(IR))
Let
to be a reductive alo-ebraic
Tten
( G(IR)) )
(reso.
ff
is
denote the set
of equivalence classes of irreducible representations (reso. irreducible
tempered representations) of
G(3R)
the trace formula are functions
f
.
In the data which one feeds into
in
C~G(IP)),
Since the t e r m of
the trace formula are invariant distributions, we need only specifv
f
by its values on all such distributions.
Theorem 1.1.1: The space of invariant distributions on
G(3P)
is
the closed linear span of
where
tr(ir)
stands for the distribution
f
->Â
trir(£)
One can establish this theorem from the characterization
[l(a)l of the image of the Schwartz space of
valued) Fourier transform.
G(3R) under the (operator
We hope to publish the details elsewhere.
Thus, for the trace formula, we need only specify the function
TTtepo(G(F)1 31-6Of
It is clearly important to know what functions on
The elements in
this form.
" ~ T ( G (canBbe) given
) by a finite number
of parameters, some continuous and some discrete.
one can define a Paley-wiener soace on
Via these parameters,
.
L ( R ( P ) )
Tt consists
0-F
functions which, among other thinas, are in the classical palev-YJiener
space in each continuous parameter.
s a c on
to be the inage of
(G(]R))
nap above.
T'7e woulri expect this Paley-Wiener
C"G(P)
under the
)
This fact may well be a consequence o+ recent work of
Clozel and Delorme.
We shall assume it iriplicitly in what follows.
There is one point we should pention before goingr on.
function F
The
can be evaluated on any invariant distribution on
MI?'.
In particular,
is defined for any irreducible representation
tempered one.
If
p =
IT, and not just a
BIT. is a finite sum of irreducible representa-
tions, we set
Now, consider an'induced representation
where
P
=
NM
is a parabolic sub~roupof
is a representation in
ntemD
(IR)
)
(M
, and
representation of the unipotent radical
valued linear function on
of the center of
by translatin9 a
M(n)
,
by
A.
a
,
N(3P)
(defined over
iriN
.
3P)
, a
is the trivial
Let
X
be a convlex
the Lie algebra of the split cor'oonent
and let
Then
G
a
be the representation obtained.
is in general a nonunitarv,
p
*A
-
reducible representation of
G(lR).
Representations of this form are
sometimes called standard representations.
The function
F(p
x
)
,
defined
by the prescription above, can be obtained by analytic continuation from
the purely imaginary values of
tempered.
Suppose that
n
A,
is an arbitrary irreducible, but not neces-
sarily tempered, representation of
tr(n)
can be written
where
p
and
where the induced representation is
G(lR).
It is known (see [15] that
ranges over a finite set of standard representations of
{M(n,p)}
is a uniquely determined set of intecers.
Then
G(X7)
F(n)
is given by
F(T)
=
1 0M(n,p)F(p).
Thus, the problem of determining F(T)
is equivalent to deterpiininr'
the decomposition (1.1.3).
1.2.
Among the invariant distributions are the stable distributions,.
which are of particular interest for global applications.
Shelstar"
has shown [11(c)] that these nay be defined either by orbital inte9i-als
or, as we shall do, bv tempered characters.
We recall the Langlands classification [ 9 ( a ) ] of
@(G/IR)
where
TT(G(IR)).
Let
be the set of admissible maps
\'J
IR
is theP7eilFroupof
is the L-qroup of
up to conjuvacv bv
an L-packet
'['^ ( G (P.)).
4
=
G.
The eleinents in
GO.
TT;
and
F.,
To each
are to he given only
@(G/P)
? @ (O/lP)
Lanqlanc?~associates
consisting of finitely pany representation'=in
He shows that the representations in
fT,
are tempere(? if
and only if the projection of the i v a ~ eof
Let
$temp(G/3R)
<;>
denote the set of all such
Definition 1.2.1:
7i
L ~ Ois bounded.
onto
@.
stable distribution is anv distribution,
necessarily invariant, which lies in the closed linear span
If
for any
4
F
is a function of the form (1.1.2), we can set
@temp (G/3R)
character on
05
G(B)
.
In [ll(c)l Shelstad shows that anv ten~ered
can be expressed in terns of sums of this'form,
but associated to some other qroups of lower dimension.
Riven our
WTO.)
discussion above, this means that any invariant distribution on
may be expressed in terns of stable distributions associated to other
groups.
we shall review some of this theory.
The notion of endoscopic group was introduced in [9(c)] and
studied further in 111 (c)1 .
Let
s
be a semisimple eler'ent in
'"G,
defined modulo
the centralizer of
(over IR)
L~ in
G O .
is a quasi-split prouo in which
""HO =
the connected component of the centralizer of
s
H
in
a split group with trivial center, this specifies H
then
G
H
An endoscopic group
s
Hs For
equals
=
Go.
uniquelv.
H
G
is
For
is a simply connected complex group, in which the centralizer
of any semisimple element is connected ([14], Theorem 2.15).
aroup
If
I?
is then the unique split Froup whose
L
The
oroup is the direct
p r o d u c t of
elenent
in
L ~ Ow i t h
w 6 PIm
PJ_.
a c t on
I n g e n e r a l , it i s r e q u i r e d o n l y t h a t each
L ~ Oby conjurration w i t h soFe element
.
c e n t ( s , ~ ) S i n c e t h e crroup
Cent ( s ,LG 0 )
i s n o t i n creneral con-
n e c t e d , t h e r e miyht be more t h a n one endosconic rrroup f o r a given
a d
.
Two end6scopic groups
equivalent i f t h e r e is a
modulo t h e product of
g â G o
and
H
w i l l be s a i d t o be
H ,
such t h a t
s
g s ' g -1
equals
w i t h t h e connected component of
ZG
s
Z
"s
and t h e map
commutes w i t h t h e a c t i o n of
W .
r e a l l y c o n s i s t s of t h e element s
(Thus, f o r u s an endoscopic group
a s w e l l a s t h e group
s t r i c t l y be c a l l e d an endoscopic datum.
An a d m i s s i b l e embeddin?
H
which extends t h e given enbeddincr
projections onto
WR1
HI
and should
See [ 9 ( e ) 1 . )
of an endoscopic froun i s one
c G
L ~ @ , which coninutes v i t h t h e
OF
and f o r which t h e image
L
OF
8
lies in
L
Pie s h a l l suppose from now on t h a t For each endoscopic
I'
L
group we have f i x e d an a d m i s s i b l e enbeddincr
H c GI
such t h a t t h e
C e n t ( s , G).
embeddinas f o r e q u i v a l e n t yroups a r e compatible.
r e s t r i c t i o n t h i s p u t s on
G
is not serious.
i s c u s o i d a l if t h e if-acre of
I,
p a r a b o l i c subgroup of
G.
say t h a t
H
Example 1.2.2:
Let
G = PSp(4)
.
The o n l y c u s p i d a l endoscopic rrroups a r e
s = ('-'-l1)
.
Then
L~
(The a d d i t i o n a l
See [ 9 ( c ) ] . ) We s h a l l
in
d
l i e s i n no Droner
Then
G
and
P.,
with
and
, Hs
PGL(2)
2
x
PGL(2)
.
For e a c h of t h e s e q r o u n s we t a k e t h e o b v i o u s embedclin~o f
If
d)
@ (G/P.) ,
i s any p a r a m e t e r i n
the centralizer in
"P
into
define
<b.
L ~ Oof t h e iriacre o-F
Since t h e h o ~ o ~ o r o h i s i "
(
<b
i s deteripined o n l v up t o
conjuvacv,
'GO
C,
is r e a l l y onlv a
L ~ O . However, v'e c a n i d e n t i q v e a c h
conjucracv c l a s s of s u b f r o u p s o f
of t h e s e subqroups w i t h a f i x e d a b s t r a c t c r o u p , t h e i d e n t i f i c a t i o n
b e i n q c a n o n i c a l up t o an i n n e r automorphism of t h e g i v e n qroun.
where
c
4'
i s t h e i d e n t i t y component of
group which i s
C
4'
known t o be a b e l i a n . ( [ l l ( c )I .
Then
C
4'
Set
is a f i n i t e
See a l s o [ 5 ] . )
I t can
t h e r e f o r e be c a n o n i c a l l y i d e n t i f i e d w i t h a n a b s t r a c t Froup which de-pends o n l y on t h e c l a s s of
For e a c h
on
n4' C$,
x
<b
c
4.
QtepP(G(lR))
,
S h e l s t a d d e f i n e s a n a i r i n ~ <,
>
such t h a t t h e r a n
i s a n i n j e c t i o n from.
i n t o t h e uroun
80)
of c h a r a c t e r s of
Unfortunately, t h e p a i r i n g cannot be defined canonicallv.
6
.
However-
Shelstad shows that there is a function
c
from
cG/zG
to
{?I},
which is invariant on conjuqacy classes, such that
is independent of the pairin?.
C
-s
Here,
is the projection of
This latter function can be used to map functions on
4)'
s
onto
to
G ( p )
functions on endoscopic Froups.
Given a parameter
''
'temp (G/IR)
and a senisinple element
s
â
CG/ZGr one can check that there is a uniaue endoscopic Froun
H
=
H
$I
s
such that
then defines a parameter
parameter in
f
â
c(G(IR)
atemp(H/IR)
,
Otep-p(~133
) .
C
arises in this wav.
H(I!?)
H,
ever17
For anv Function
â C;(H(TR)) , uniaue
H
To do so, it is enouyh to
Shelstad defines a function
up to stable distributions on
For a viven
.
f
specify the value
for every such .I$-,. This is done bv settino
Actuall~,Shelstad defines
fH
bv transferrino orbital intevralsr and
then proves the formula (1.2.3) as a theorem.
the formula as a definition.
Shelstad shows that the riaoriinr-
is canonically defined up to a sicrn.
H
c G
Fowever, we shall take
f -r
H
(It also depends on the enbedrlincr
which we have fixed.) Pie shall fix the sir-ns in anv wav,
asking only that in the case
H
=
GI
f
f
be consistent with the
notation above.
1.3.
That is,
c(l)
=
1.
It is important for the trace formula to understand how the
notions above relate to nontempere6
the pairings
parameters
only for tempered
<>>
<j>,
Shelstaci fiefined
(o.
but it is easv enouoh to
extend the definition to arbitrary parameters.
For one can show that
there is a natural way to decompose anv narameter
(w)
=
(oo (w)(o+ (w)I
so that the images of
'(.(Wn)
tempered whenever
of
(oo
@+
= { 11.
' ^tei-o( W p ) ,
The centralizer in
Ètem
e
@
(M/B) .
must lie n
itself is
of a parabolic subqroup of
'Â¥M so that
(oo
v+
of
L
G,
M(B).
defines an element in
a
There will be a bijection between ~ > n d
, the
0
being the Lanflands quotients obtained froi" the tem-
I]
n n
pered representations in
M(E) .
(o
,
L~ of the imase
n\ will consist of a positive quasi-character
The image of
<))+c ~ ( C / - S IR)
commute, and so that
will be the Levi component M'
+
and
of
and
^O
bv
(o
y o
On the other hand
and the positive ~uasi-character v+
equals
C"
C:
, so we can define the
$0
p
^o
a
c
r on
-IT:
(o
to be the one obtained
x
fro^ the pairin? on
0'
However, simply defining the pairing for nontempered 4
factory.
For it could well happen that the distribution
is not stable if the parameter
(t
is not terioered.
ficulty is that (1.2.3) no lonoer ~ a k e ssense if
parameter for
H.
We shall define a subset
o-F
A related $i^-
41.
a('";/:[")
these difficulties are likely to have nice solutions.
contain
tions of
Let
is not satis-
Ètemp(G/ll?)
G(B)
Y(G/IR)
is not a tempered
-For which
The subset will
and oucrht also to account for the representa-
which are of interest in crlobal annlications.
be the set of
"GO-conjuoacv classes o-F maps
such t h a t t h e r e s t r i c t i o n of
F o r any
to
I/J
raTm
d e f i n e a parameter
ip ? Y ( G / I R )
beloncrs t o
4 '
in
@tern ( R / P 1.
bv
@(G/lR )
Here it i s h e l p f u l t o r e c a l l t h a t
i s t h e map f r o n
17'
to
IR
SL(2,C)
=
L ( ~ r ^ ( 2 0) )
which a s s i g n s t h e t r i v i a l r e p r e s e n t a t i o n t o
PRL(2,P).
Recall a l s o
t h a t t h e u n i p o t e n t conjucracv c l a s s e s i n anv cor'nlex crroun a r e b i j e c t i v e
w i t h t h e conjuoacy c l a s s e s o f m a p s o f
SL(2,C)
i n t o t h e oroun.
"he
u n i p o t e n t c o n j u o a c v c l a s s e s f o r coriolex crroups h a v e been c l a s s i f i e c 1 hv
w e i g h t e d Dynkin diacrrans.
i d e n t i f i e d with a p a i r
a map from
SL(2,C)
(4>,p),
into
Mow a n v
( S e e [I31 .)
C
4)'
i n which
4>
(.
QePn
i t s r e s t r i c t i o n t o t h e diacronal suborouo o f
Thus,
contains
V (G/IR)
@temp( G / R )
(WP)
p
SL(2.C).
can be
Y (R/B )
criven up t o coniur'acv bv
t h e c l a s s i f i c a t i o n of n i l p o t e n t s it follows t h a t
p r o p o s i t i o n 1.3.1:
c
ip
and
C
4' '
p
prom
i s c'eterninec' bv
We o b t a i n
The n a p
c a n be r e g a r d e d a s a s u b s e t o f
a s t h e s e t of
$ = ($I, p )
with
is
@(P/l? )
p
.
It
trivial.
Conjecture 1.3.2:
For any
$ ? f(G/IP)
,
the representations in
are all unitarv.
suppose that
$ =
( 4 ,p)
is an arbitrary parapeter in
CJ ( P / P )
.
Copying a previous definition we set
and
c
^
The qroup
C
+
C
*
= c G = c / c0 7
H; * ' c
-
Cent ( p (SL(2,(C) ,C ) ) , and in narticular
4'
Therefore, there are natural fans
alwavs eauals
is contained in
C
ill
C
4"
cH;
+
5
It is easy to check that this second man is surjective- In
ill
4'other words, there is an injective man
from the (irreducible) characters on
on
C+.
Fix
$ 6
Y (G/~R) .
C
^
to the irreducible characters
Take one of the pairinfs
discussed above, as well as the associated function
classes of
C / Z .
Me pull back
classes of
C*/
Isle conjecture that the set
. ;2
c
on
< ,>
c
C
^
x
TT4
^
on the coniu~acv
to a function on the conjucac~r
^
can be enlarme?
and the pairin? extended so that all the theorv for ternerecl parameters
holds in this more neneral settinc'.
Conjecture 1.3.3:
representations of
which equals
1
on
There is a finite set
G(3R)
^
which contains
nillirreducible
TT,,
, and an injective nap
OF
a function
f r o "ff
?,
into
$
all uniouelv determined, with the ¥ollowin nro-
perties.
(i)
of "fT,
v
<-,IT> lies in the imaae of C
in
IT
function
if and only if the
belonas to the subset
^
(ii) The invariant distribution
C
^'
IT)<l,ir> tr(x)
is stable.
(If C,
is abelian
which is certainly the case most of
the time, the distribution is
E
which except for the s i ~ n s
in the packet
9'
)
s
^ (TI) is just the sum of the characters
We shall denote the value of this distribution
on the function (1.1.2) bv
(iii) Let
E
^ (71) tr(r) ,
F($).
be a semisi~pleelement in
C,/Z.
Let
E = Hs
be the unique endoscopic group such that
so that, in particular,
 â C;(G(IR)),
and
-
s
)I
defines a
D
is the i m a ~ eof
raneter in
s
in
<Z,TI>
tr it
It is not hard to check the uniaueness assertion of this conjecture.
The third condition states that
x of
depends only on the projection
irreducible character
0
C,,
onto
s
and that For anv
Clb1
in
7f '
(~1t.r
ir(f), if 0 = <-,T>for sore TT?
(1.3.5)
1
-
0, otherwise
.
Assume inductively that the distribution (1.3.4) has been defined and
shown to be stable whenever
group
H
=
H .
is replaced bv a oraoer endoscooic
G
Since the function
f
has alreadv been defined on
-Hs
any stable distribution, the numbers
s
=
x # I, then make sense.
To define
Â¥n is the representation in
($1
and
f(-(f),
such that
A
X(4;,x)( f ) ,
take
with
0 = 1.
IF
<-,nl> eauals
1, we
obtain
The distribution
is then equal to
To complete the inductive definition, it is necessary to show it is
stable.
TT4;
The formula (3.1.5) would then qive the elements in
uniquely, but only as virtual characters.
The repainin? probler? is
to show that the nonzero elements amon? them are linearlv independent,
and that up to a s i ~ n(which would serve as the definition of
e )
f
they are irreducible characters.
The packets ff,should have some other nice properties.
example, one can associate an
R-arou~to any
4;
? Y(G/3R).
For
Define
Rli;
to be the quotient of
C
*
by the g r o w of comoonents in
Cli;/ZC
which act on the identity component bv inner auto~orphis~s.If
is not trivial, the identity component will also not be trivial.
image of
li;
Let
f
be a minimal Levi suboroup of
contains the image of
$.
Then
(M/m
)
.
*
The
will be contained in a Levi component of a nroper narabolic
subaroup of
V
R
G .
^>
R
which
also represents a oarameter in
There is' a short exact seauence
The group
R
li;
should Govern the reducibility of the induced renresenta-
tions
where
P
=
is a parabolic suburour) of
MN
f;.
bJote that
pG
is obtained
by unitary induction from a representation which is in oeneral not
tempered.
Finally, the conjecture should admit extensions in two directions
to real aroups which are not necessarily auasi-split, and to pairs
(G,a),
where
automorphisms).
a
is an autonorphisn of
G
(~oflulothe aroup
OF
inner
Both will eventually be needed to exploit the trace
formula in full qeneralitv.
1.4.
Conjecture 1.3.3 is suaaested by the alobal situation, which we
will come to later.
I do not have much local evidence.
The laraest
Froup for which I have been able to verifv the conjecture c o ~ ~ l e t e l v
is
PSp(4).
However, even this aroup is instructive. We shall look
at three examples which illustrate why it is the parameters ^>,and
not
I
<)>,,
which oovern questions of stability of characters.
qQwill consist of one representation
is not stable.
However, each aroup
C$
ir
such that
Tn each
tr(ir)
will be of order two, and the
-
nipwill consist of
sets
T
and another renresentation.
St is onlv
with these laruer sets that we obtain a nice theorv of stability.
In each example we will consider parameters
such that the projection of
ip
onto
G O
ip
for
G = PSp(4)
factors through the
endoscopic group
with
As we have said,
will consist of one representation
ir.
It will
be the Langlands quotient of a nonunitaril~induced representation
of
VJe shall let
G(lR).
^ n>
TTH
packet
=
representation of
irH
denote the uni-que representation in the
and we shall let
of which
H (I!?)
In order to deal with
know something about the
be the nonunitarilv induced
is the Lann1and.s quotient.
Y-parameters on
C ( R ) ,
we must first
The L-packets
Those with two elements contain discrete
series or limits of discrete series.
where
IT^
pH
@-parameters.
contain one or two elements.
p
They are of the form
has a VJhitaker model, and irhol
is the irreducible
Wh
representation of PSp(4,K) which combines the holoriorphic and antiT
holomorphic (limits of) discrete series for
>
pairinq
<,
on
Z/2Z
C4
on
,
C4 x
and
n4 so that
Sn(4,P ) .
Me take the
< - , T ~ ~ ,is
~ >the trivial character
< - l h o l > is the nontrivial character.
It is
not hard to verify that with this choice of pairing, all Shelsta<?.'s
functions c(s)
may be taken to be
1.
In our examples, we shall
consider only representations with sinaular infinitesimal character,
since these are the most difficult to handle.
will now denote the L-packet in
the lowest limits of discrete series.
series for
I?
which contains
G ( E )
is the lowest discrete
irH
H (3R ) ,
(f)
for any
For this reason,
f C c(G(B)).
- tr nhOl(f) ,
On the other hand, it will be clear in each
example that
with
and
0
pH
as above.
As a distribution on
G ( R )
, this 1.ast
expression is stable.
We will prove the conjecture in each example by lookincr at the
expression (1.3.7) for
in
fp(i()). If
ir
is the unique representation
q ,it will equal
To check the stability of this distribution, we will need. to express
it as a linear combination of standard characters on
construct the packet
*'
G(TP.).
!To then
we will have to rewrite the expression as
a linear combination of irreducible characters.
The term
2 tr irff)
is handled by computina the character formula (1.1.3) for the reoresentation
IT
of
G(lR).
This can be accomplished bv reducinrr to the case
of reaular infinitesimal character throuqh the procedure in [12] an8
then using Voaan's algorithm obtained from the Kazdan-Luszticr conjectures
[15].
We will only quote the answer.
To deal with
f,,($),
we shall
first write the character formula (1.1.3) for the reoresentation
of
H(3R
).
Since
H(3R
)
is isomorphic to
PGL(2,lP
)
x
PGL(2,IP.)
11
IT
,
such formulas are well known.
characters on
H(lR)
Example 1.4.1:
T'7e will then lift the resultinc' standard
to characters on
f
Let
G(E)
usina the renarks above.
be given bv the diaorain
in which the vertical arrow on the left is the parapeter -For PGL(2,TR)
which corresponds to the lowest discrete series, and the inacre of
in
SL(2,C)
is contained in f el}.
1-1
The centralizers are qiven as
follows.
We write
IT
1-1
for the representation in
As we have aareec1,
then denotes the standard representation of which IT
is the
1-1
H
quotient, and IT^ and p v denote the corresnondina representations
O1-1
P
of
H(E).
The character formula (1.1.3) is easily shown to be
On the other hand, from the well known character for~'.ulafor
obtain
From our formula for
tr
IT
u
( f)
we see that this equals
I T
v
we
18
t r ~ ( f +) t r
IT
hol ( f )
on one hand e o u a l s
but can a l s o be w r i t t e n a s
From t h e second e x p r e s s i o n we s e e t h a t i t i s s t a b l e .
From t h e f i r s t
e x p r e s s i o n we s e e t h a t t h e o t h e r a s s e r t i o n s of t h e c o n j e c t u r e hold if
we d e f i n e
and
<.,
TT
P
>
Oe could have d e f i n e d
=
@
1,
<-
,
ThOl>
would have been t h e same exceot t h a t
C(TP).
PRL(2,1P).
Define
@
Evervthincr
These exar-oles a r e t h e l o c a l
PSp(4)
(See a l s o [ 9 ( d ) ,$ 3 1 . )
Example 1.4.2:
.
{ I T , ~ , I T
would
~ ~ ~s t}a n 6 f o r a
a n a l o ~ u e sof t h e n o n t e m e r e d cusp forris of
Kurakawa [ 7 ] .
-1
s o t h a t t h e v e r t i c a l arrow on t h e l e f t
corresponded t o a hio-her d i s c r e t e s e r i e s of
p a i r of d i s c r e t e s e r i e s of
=
by t h e diaqram
discovered by
in which the inacres of
pl
+
p2.
pl
and
pa are contained in {±1} and
The centralizers are
for the reoresentation in T T , and Follow
^,^2
the notation above. The character formula (1.1.3) can be calculated
We write
tr
IT
(f) =
IT
tr
^1^2
(f)
P
- tr
(f)
p
^1^2
Ul
-
tr
On the other hand., fro^ the character formula for
t
r
p
(f) - tr p
^11^2
Thus, the distribution
Ul
(f) - tr
p
^2
(f) + tr )
f
(
,
T
I
P
(5)
+ tr
~,,^(f)
^2
I
T
'
'
\-reobtain
Ul1^2
- tr IThol(f)
on one hand equals
but can also be written as
(f) - tr
tr P
.
ull^
(f)
P
- tr
^1
0
(f)
+
(tr vTATh(?)+ tr nhOl(?))
.
p2
From the second expression we see that it is stable.
Fror? the first
exoression we obtain the other assertions of the conjecture if we
define
and
This example is the local analogue of the nontei"nerer1 cusn forms r'iscovered bv
Howe and Piatetski-Shapiro [3].
Example 1.4.3:
p,
= p2 =
u.
Define
as in the last examie, excent now take
This example is perhaps the nost striking.
from the previous two in that
U;
of a proper parabolic subcrroup of
subgroup P
L L
= M M
up in the fact that
factors throucrh a Levi subnroun '^M
"G.
(It is the naxinal parabolic
whose unipotent radical is abelian.)
C
^
It is different
is infinite.
This shows
T'7e write
IT
notation above.
The character formula for
ed of the three to compute.
tr
IT
VIP
5 , and follow the
for the representation in
V-rlJ
(f)
lJ ll'
is the most con~licat-
It is
tr plJ
=
IT
,11
( f ) - tr p (f)
v
- tr
IT
ho1
.
(f)
We also have
=
trnH
'J,v
From our formula for
(fH)
tr
IT
'J 1 lJ
(f) and the formula for
tr
TT
v
( f)
in
Example 1.4.1, we see that this equals
tr
IT
v1lJ
(f)
- tr
IT
v
(f)
.
Thus, the distribution
f($)
=
2 tr
IT
PIP
(f)
-
fg($)
on one hand effuals
but can also be written as
From the second expression we see that it is stable.
From the first
excression we obtain the other assertions of the conjecture for the
endoscocic aroups
G
and
H
if we define
22
€$hrv
1
=
=
E
* (IT
U
)
and
In this example we have a third endoscopic crroun to consider
the Levi subcrroup MI which we can identify with
factors throuqh
M ,
it defines a parameter in
mJi2}. Since
Y(M/R).
-
+
TO complete
the verification of the conjecture we must show that
The packets
nt
n*
M
and
both consist of one elementl the represen-
tation
The definitions of Shelstad are set
is dual to induction.
UD
so that the nau
Therefore, we will be done i-F we can show that
the induced representation
pu
is the direct sun of
character of
M(lR
).
standard character on
GL(2,IR).
=
Ind pG OR)
(E)^ '^ id?.,)
and TT . Now, u is a nonternuerec1 unitarv
v,vV.
It is the difference between a nontemnered
v
GL(2,IR)
and a lowest discrete series on
The induced character
tr(p)
is the difference between thp
correspondin? two induced standard characters.
The first is just
tr(pPrv). The second is a tempered character on
ducible; its constituents are
character equals
IT
Wh
and
vhol.
G(IR)
which is re-
Therefore, our induced
It follows that
Therefore the order of the
R
qroup is errual to the number of irrerluc-
ible constituents of the induced representation
as we would hope.
parameter
>,
Observe that the analowe of the
is trivial.
R
froun ^or the
Thus, we see a further exarnnle o'f:behaviour
which is tied to the parameter
rather than
$
> 11'
This suoaests a concrete problem.
Problem 1.4.4:
of the center of
morphic to
R
Ip'
trivial element.
M(IR).
Let
zM be the Lie aloebra of the svlit copponent
Let
w
The Peyl oroup of
%M
be a representative in
G(39)
of its non-
It is known that the corresnonciina intertvinincr
operator between
and
can be normalized accorrlinri to the
prescription in [9(b),Appendix 111.
Let
normalized intertwinin? operator at
'X = 0.
whose square is
is in this case iso-
1.
M(w)
be the value o^ the
It is a unitarv onerator
Its definition is canonical up to a choice of the
representative w
in
G(3R
)
.
The problem is to show that
~ ( w ) is
not a scalar, and more precisely, to show that if the deterninant of
w
is positive, then
2.
A GLOBAL CONJECTURE
2.1.
The conjecture we have just stated can be made for anv local
field
F.
If
F
is non-Archipiedean, however, the Weil
CTOUP
must be
replaced by the crroup
introduced in [9(d)l.
over
F,
@(G/F)
If
G
is a reductive auasi-split aroun r'efined
must be taken to be the set of ecmivalence classes
of maps
while
to
1
Gtemp(G/F)
will be the subset of those pans whose restriction
has bounded image, when projected onto
define the parameters
I) we must add on another
~'(G/F) to be the set of
belonas to
In or6.er to
SJ, (2,C)
.
Fe t?.?e
L ~ Oconjuaacy classes of naps
such that the restriction of
SL(2,C)
'AGO.
@
Qtenp(G/F).
to the nroduct of b 1
For anv such
ip,
with the first
the parameter
belongs to
@
(G/F) .
The conjecture also has a global analogue.
and let
field with adgle ring A,
If
G
G
Let
F
be a global
be a reductive group over
F.
is not split, there are minor complications in the definitions
related to endoscopic groups.
we shall simply take
(See [9(e)].)
to be split.
G
To avoid discussing them
Then the global definitions con-
nected with endoscopic groups follow exactly the local ones we have qiven.
The conjecture will describe the automor~hicreoresentations which
are "tempered" in the global sense; that is, representations which occur
in the direct integral decomposition of
G (A)
on
2
L (G( F ) \G ( A ))
.
However, we cannot use the global Weil group if we want to account for
all such representations.
For even
GL(2)
has many cuspidal automorphic
representations which will not be attached to two dimensional representations of the Well group.
The simplest way to state the global conjecture
is to use the conjectural Tannaka group, discussed in I9(d)l.
properties hold for the representations of
If certain
GL(n), Lanqlands points out
that there will be a complex, reductive pro-algebraic group
G
I t e m n (F)
whose n-dimensional (complex analytic) representations parametrize the
automurphic representations of
place.
For each place
algebraic group
v,
GL(n,A) which are tempered at each
there will also be a complex, reductive pro-
G
equipped with a map
Itemp(v'
1
whose n-ciimensional representations parametrize the temoered representations of
The composition of this map with an n-dirensional
GL(n,F).
representations of
G
( )
will give the F -constituent of the corntemp
v
responding automorphic representation.
The sets
as the set of
y(G/F)
which we have defined could also be described
L ~ Oconjugacy classes of maps
The centralizer in
G
of the image of
flrv
is the same as the central-
izer of the image of the corresponding parameter associated to the weil
group.
In other words,
and
Assumincr the existence of the
We make the same definitions ulobally.
aroups
G
and
(F)
'
t
e
m
p
TTtenp
let
(V'
Y(G/F)
be the set of
conjugacy classes of maps
G
@:
If
G
il)
(:
Y(G/F)
i
Gn
tenp
F)
X
SL(2,C)
+
G'
.
is any such global parameter, set
*
c
=
cent($(%
temp (F)
x
SL(2,C))
, '"GO) ,
and
The composition of the map
with
)I
gives a parameter
$v E Y(G/Fv).
There are natural maps
and
Assume that the analogue of the local Conjecture 1..3.3 holds.
for each field
Fix
F .
a function
finite set n * v ,
Then for any place
Y(G/F).
ij; â
on
E
*v
and a function
global packet'
of
Bv nV
IT =
c
*
G(A)
TTv
we have a
v
, a pairing
on the conju~acyclasses of
C
/Zc
Define the
v
to be the set of irreducible representations
such that for each
vI
I T
belongs to
Define the global pairing
and the global function
for
IT =
4
rV
in
n+ and
x
in
C+
with i ~ a n e xv
Almost all the terms in each product should equal
to expect that for any element
s
C
Cq/ZG
1.
in
C
\
.
It is reasonable
with image
s
in
C
/Z_
I
*v
If this is so, the qlobal pairin? will be canonical.
Conjecture 2.1.1:
(A)
The representations of
in the spectral decomposition of
metrized by Y(G/F).
$
L (G (F)\f;( @ ) )
R(iP)
which occur
occur in packets
~3-r2-
The representations in the packet corresponding to
*
will occur in the discrete spectrum if and only if C
is finite.
(B)
integer
Suppose that
d^
^
C
is finite.
Then there is a, positive
and a homomorphism
such that the multiplicity with which anv
in
I*(G(F)\G{&) )
In particular, if
is
IT
2.2.
^
d
TT
C
n+ occurs discretelv
equals
and each
C
if the character
C
*v
equals
<-,IT>
Some comments are in order.
are abelian, the multinlicitv of'
S,,
and is zero otherwise.
First of all, the introduction
the Tannaka groups would seem to put the conjecture on a rather shakv
foundation.
set
C
Y(G/F)
However, evervthincr nay be formulated without them.
is the same as the collection of pairs
@temp(G/F)
conjugacy by
and
C
0'
fiven un to
0'
Included in the conjecture (and also i~qlicitin
p
is a map from
[9(d)I) is the assertion that
SL(2,C)
atenP(G/F)
classes of autonorphic representations of
everv place.
into
(alp), where
C
is the set of
G(/A)
1,
eauiv?lence
which are tempered at
We could simply take this as the definition
O-F
atepP(O/F).
To avoid mentioning the Tannaka group at all, we would need to define
^
for each
in oternP(~/~),For then C
would just be the cen0
tralizer of the image of p in C
If one qrants the existence of cer0
tain liftings, one can show that C
is equal to the centralizer in
0
G o of an embedded L-group in G .
C
-
Notice that the conjecture does not specify whether an automorphic
representation which occurs in the discrete spectrum is cuspidal or not.
Indeed, it is quite possible for a global packet
TT^
to contain one re-
presentation which is cuspidal and another which occurs in the residual
discrete spectrum.
(See 121 and also Example 2.4.1 below.)
I do not
know whether there will be a simple explanation for such behaviour.
Multiplicity formulas of the sort we conjecture first appeared in
The integer d
[8].
was needed there, even for subgroups of
Res
(GL(2)), to account for distinct global parameters which were
E/F
everywhere locally equivalent. The sign characters S q are more mvsterious.
Suppose that
morphism of
is the set of fixed points of an outer auto-
G
GL(n,C).'
Then one can observe the existence of such charac-
ters from the anticipated properties of the twisted trace formula for
GL(n).
with
The character will be
1 if
corresponds to a pair
ip
trivial; that is, if the representations in
p
at each local place.
1/2
are tenpered
£. will not be trivial, and
In general, however,
will be built out of the orders at
Ti
(4),p)
of certain L-functions of
4).
Incidentally, in the examples I have looked at, both local and alobal,
C,
the groups
C,
have all been abelian.
is no more than a guess.
tions <
The extrapolation to nonabelian
In fact if
C*
is nonabelian, the func-
may turn out to be only class functions on
,7r>
C
r and not
irreducible characters."
2.3.
Let us look at a few examples.
Consider first the group
The centralizer of any reductive subgroup of
*
This means that the packet
so that a parameter
a pap of
SL(2,C)
\f)
L ~ O = ~ ~ ( n is
, ~connected.
)
(both local and global) should each con-
tain only one representation. The groups
GL(n,,C)
G=GL(n).
x..
.x
C
4)
GL(n,C)
will be of the form
,
will consist of the ter'perec' parameter
into this Froup.
The representations in
4)
an(?
TL
should belonq to the discrete spectrum (nodulo the center of
G(p) )
e=uals Cx. This will be the case nreciselv when
*
GL(nl,C) and 0 is the irreducible n, dinensional re-
if and only if
C
'
l
,
C
4)
equals
presentation of
SL(2,C).
Then
nl
will necessarilv divide
n,
n = n m, and 4) will be identified with a cusnidal autornornhic re1
presentation of GL(m,A) , enbedded diaoonallv in GI,(n). *his
prescription for the discrete spectrum of
is exactly what is excected.
(See [ 4 1 . )
GL(n,A)
(modulo the center)
It is onlv -For f;L(n)
(and closely related froups such as
SL(n)) that the distinction
between the cuspidal spectrum and the residual discrete snectrur" will
be so clear.
The multiplicitv forpula of the conjecture is comatihle \.~iththe
results of Labesse and Lanolands [ 8 ] For SL(2).
More recentlv, flicker
[2] has studied the ouasi-split unitarv crroup in three variables.
"^he
conjecture, or rather its analogue for non-split groups, is comodtible
with his results.
Langlands has shows [9(b), Appendix 31 that for the split group
of type
G2
there is an interesting automorphic representation which
occurs in the discrete noncuspidal spectrum.
Its Archimedean component
is infinite dimensional, of class one and is not tempered.
of such a representation is predicted by our conjecture.
the complex group of type
G2.
The existence
L ~ Ois just
It has three unipotent conjugacy classes
which meet no proper Levi subgroup.
These correspond to the principal
unipotent classes of the embedded subgroups
where
and
O
3
Let
\
and
pi
=
z
SL(3,C)
.
(<}>,pi)
be the parameter in
such that
Y(G/F)
<(>
is trivial
is the composition
SL(2,C)
+
L 0
Hi
+
LG0
r
in which the map on the left is the one which corresponds to the
principal unipotent class in
*1
'"~0. The packet
G@).
ment, the trivial representation of
contains one ele-
It is the packet
which should contain the representation discovered by Lanqlands.
remaining representations in
which occur in the discrete spectrum,
, are presumably cuspidal.
as well as all such representations in
2.4.
PSn(4) of the three
Finally, consider the global analoques for
examples we discussed in
51.
4'2
The
The global conjecture cannot be proved
yet for this group, for there remain unsolved local problems.
However,
Piatetski-Shapiro has proved the multiplicity formulas of the first two
examples below by different methods.
(See [lo(a)I ,
[lo(b)1,
[lo(c)I
.)
Using L-functions and the Weil representation, he reduced the proof to
a problem which had been solved by Waldspurger [16].
In each exari~le 4' will be fiven bv the fliaoram For the corres-
ond din^
local example in
$1 excent that
r.7
TR
is to be replaced hv
G
the Tannaka qroup
or, as suffices in these exannles, bv
TTtenp (')
the global ThTeil group Fp. Each p will be a Grossencharacter of
order
1
or
G
2,
and
!-T
TTtemp (')
integer d
4'
since the one dimensional renresentations of
will be
Example 2.4.1:
~ ~ \ e 'all
. . co-incide.
~
In each example the
1.
This is the example
0-F
Kurakawa.
Take the
(3iaoram in ~xanple1.4.1, letting the vertical arrow on the le^t
~ a r a ~ e t r i zae cuspidal automorphic representation
of
PGL(2,A).
The character
T
=
%
v
T
v
As in the local case, we have
E4'
should be
1
or
-1
according to whether the
order at
s
=
1/2
of the standard
1, Gunction
Our conjecture states that a representation
L(s,T)
is even or oftf
in the packet
TT
occurs in the discrete spectrum if and onlv if the character
on
C,
order
equals
2
PGL(2,F)
T
or
1
%J .
The local centralizer vrouo
C
'
V
will be of
dependin* on whether the reoresentation
belongs to the local discrete series or not.
belongs to the local discrete series at
Then the ~ l o b a lpocket
will contain
11
r
2r
11
< T I - >
of
v
cuonose that
T
rlifferent places.
reoresentations.
Exactly half of them will occur in the discrete snectruri of
L ( c ; (F)\G ( A) )
.
(If r
=
0,
the one reoresentation in
occur in the discrete spectruri if and onlv if
For a given complex number
of
PGL(2,A) x
sub~roupof
G
A ~ .It
s,
nll
will
%'
= l-)
consider the representation
is an autonorphic reoresentation of a Levi
which is cuspidal modulo the center.
induced representation of
R(A)
^he associate?
will have a global intertwining
operator, for which we can anticipate a clobal normalizing factor
equal to
From the theory of Eisenstein series and the exnecteg properties o^
the local normalized intertwining operators, onecan show that
TT11
will have a representation in the residual discrete snectrui" if and
only if the function above has a pole at
case precisely when
L(l/2,r)
s
=
1.
does not vanish.
This will be the
Thus, the nuriber of
nll
cuspidal automorphic representations in the packet
should effual
2r-1
or 2r-1 - 1, denendinc on whether 11(1/2,T) vanishes or not.
Exanple 2.4.2:
This is the example of Howe and Diatetski-Shapiro.
Take the diaoram in Exapple 1.4.2 with
E,
The character
representation
should always be
6
IT,,,will occur
onlv if the character
1.
#
v
3. Then
Our conjecture states that a
in the discrete s~ectrupif and.
eouals
1.
Each local centralizer m o w
will be isomorphic to B/2Z. It follows that the packet TT
*v
11'
will contain infinitelv nanv reoresentations, and infinitelv rranv
2
should occur discretely in L ( G(F)\G ( A ) )
C
.
Exam~le2.4.3:
Take the diavrar" in Example 1.4.2 with
pl
=
}in.
Then
Each local centralizer proup
*
the oacket
*
since C
will be isomorphic to '57i/2%, so
'J'V
will contain infinitelv rianv representations. ITowever
is infinite, the conjecture states that none of then win.
'-1
occur discretely in
3.
3.1.
C
L^ ( G( F ) \G ( A) )
.
THE TRACE FORMULA
The conjecture of
$ 2 can be motivated by the trace fornula, if
one is willing to errant the solutions of several local nrohlens.
T,le
hope to do this properly on sope future occasion, but at the nonent
even this is too larne a task.
T'7e shall be content here to discuss
?
few ~roblemsconnected with the trace formula, and to relate ther".to
the conjecture in the example we have been lookinc' at
PSp(4).
- the crrou?
For a more detailed description of the trace forr'-ula,see the
paper [l(b)] and the references listed there.
Let
G
be as in
$2, but for six?.plicity, take
field of rational nur'.bers 0 .
F
to be the
The trace formula can be recarded as an
equality
of invariant distributions on
P(A)
.
The distributions on the left
are parametrized by the serpisirnwle coniucracv classes in
while
0(iC),
those on the ricrht are oarainetrized bv cuspidal autoinorohic renresentations associated to Levi components of parabolic subcrrouns of
Included in the terms on the left are orbital intearals on
G.
P(/B)
(the distributions in which the semisimple conjuuacy class in
G(0)
is regular elliptic) and on the rioht are the characters of cnsnirial
automorphic representations of
Levi subgroup is
G
itself).
(the distributions in which the
G(@)
In oeneral the terms on the left are in-
variant distributions which are obtained natural-lv from weighted orhita
intearals on
G(A)
.
The terms on the riuht are siinnler, an3 can he
given by a reasonably simple explicit forinula.
(See [l(b)l)
.
The ooal of [9(c)] was to beain an attack on a fundanental
problem
G
- to stabilize the trace formula. The endoscopic croups for
are auasi-split groups defined over
(p:
enc'oscopic groups over the completions f t
suppose that for each endoscopic group
errbedding
H
c G
H
thev can be recrard-ec'.as
of
0. As in
$1, we
we have fixed an admissible
which is compatible with equivalence.
We also
assume that the theory of Shelstad for real crroups has been extenrW
to an arbitrary local field.
and any endoscopic Froup
in
c
(H(A))
.
H
Then for any function
f <=
C ( P ( @ ))
we will be able to define a function
For example, if
f
fÃ
is of the torn qVfv, we siinnlv
set
However,
f
tributions on
will be determined onlv up to evaluation on stable disH(A).
To exploit the trace fornula, it will be
necessary to express the invariant distributions which occur in terms
of stable distributions on the various rrroups
IKB.).
Kottwitz [ 6 1 has introduced a natural eouivalence relation, called.
stable conjuqacy, on the set of conjuoacv classes in
reqular semisimple classes.
jugacy classes in
G(Q),
-
If
0
0. For any
If
is an endosco~ic~ o u p
for
H
â
is the set of all serisimnle con-
let ?5 be the set o^ stable conju~acvclass-
es in
o
~ ( 8 on
) the
7,
set
it can be shown that there is
GI
a natural map
from the semisim~lestable conjuoacv classes of
One of the main results of [9(e)1 was a formula
G(<D).
for any
f
â
C;(G(A)
elliptic elements.
and
H(<Ti) to those of
S-
)
and any class
o â
0 consistino of reoular
For each endoscopic croun
is a stable distribution on
H(A)
E,
.
,(i";,K)
is a constant
The sum over
IJ
OH
(as well as all such sums below) is taken over the eouivalence classes
G.
of cuspidal endoscopic croups for
Problem 3.1.3:
Show that the fornula (3.1.2) hole's ^or an
arbitrary stable conjuqacv class
0
in
7.
This problem is similar in spirit to that posed bv Conjecture
1.3.3. It is not necessary to construct the stable distributions
H
S- . One would assume inductivelv that thev had been defined for anv
OH
H # G.
(of course we could not continue to work within the limited
category we have adopted for this exposition
group with embeddings H
CG.)
- namely,
G
is a solit
The problem would then amount to showing
that the invariant distribution
was stable.
However, this assertion is still likely to be quite diffi-
The problem does not seem tractable, in qenera1,without a good.
cult.
knowledge of the Fourier transforms of the distributions I-.
0
In any case, assume Problem 3.1.3 has been solved.
I(£
=
1^f)
I
=
I(f)
Off)
Define
r
0
and
for any
f
â
c
(G(A) )
.
The expression for
each side of the trace formula (3.1.1).
absolutely.
I (f) is just equal to
It is clear that it converoes
The same cannot be said of the expression for
The problem is discusses in [9(e),VIII.5].
that there are only finitely many
8.12 of [9(e)l.)
H
must ~ a k ethe assumtion
such that f
This is certainly true if
S(-F).
G
# 0.
(See L e m a
is adjoint
for then there are onlv finitely panv endoscopic nrouns (un to ecruivalence, of course).
Since the constant
l(C;,G) ecruals
1, we obtain
if we assume inductivelv that the expression used to define
verges absolutely whenever
~'(£
H
#
G.
converyes absolutely, and
3 ~ '
con-
It follows that the expression -For
sG is a stable distribution on
C(@)
.
Moreover,
3.2.
An identity (3.1.4) could be used to yield interesting inforpation
about the discrete spectrum of
GI
since there is an explicit formula
for
The formula is given as a sum of inteorals over vector spaces
a
/ a G'
*
where
over
0 , AM
nent
M
of
P =
m
is a parabolic subcrroun of
(t'efined
G
is the split component of the center o-F the Levi
.
The most
M is the Lie aloebra OF Ap"(lQ)
interesting part of the formula is the terr' for which the integral is
PI
and
COFTIO-
a
actually discrete; in other words, for which
p
It is onlv this
= 17.
term that we shall describe.
Suppose that
P = MN
is a parabolic suboroup and that
irreducible unitary representation of
M(/A)
.
Let
po
a is an
be the inc'ucert
representation
where
N( A ) ,
idN
is the trivial representation of the unipotent radical
and
.,.L
2
0
(A(3R) M(Q) \M(&) )o
of the subrepresentation of
decomposes discretely.
M(p)
Let W ( a )
on
is the o-prinarv component
L2 ( A M ( l R ) 0 M ( Q l \M(;A))
be the Weyl croup of
a ,
which
and let
^
reo be the subset of elements in Ta7(a!K) whose snace 06 Fixefi.
vectors is a,,.
For any w in ItT(a ) let T ( w ) be the (unnorralizeri)
M
global intertwining operator from Po to Pwo- For anv function
f S c(G(A)),
define
where the first sum is over pairs
G(Q)
conjuqacv.
Then
formula for (3.2.1).
I
as above, with
(M,u)
M
given up to
is the "discrete part" of the explicit
Here we have obscured a technical coriolication
for the sake of simplicity.
It is not known that the sup over
5
(3.2.2) converges absolutely (althouoh one expects it to do so).
in
In
order to insure absolute convercence, one should reallv nroun the snpmands in (3.2.2) withothex components of
I(f)
in a wav that takes
account of the decomposition on the riuht hand side of (3.2.1).
We expect to be able to isolate the various contributions of
(3.1.4) to the distribution
(for every
for any
f ?
G)
c
(G( A )
The distribution
H
distributions S
for
This would mean that we could find
a stable distribution
would be stable.
H
I .
I .
)
.
G
S+
on
G(p)
such that
Said another way, the distribution
Now this is actually a rather concrete assertion.
I+
is certainly oiven bv a concrete formula, and the
are defined inductivelv in terns of the fornulas
Moreover, Kottwitz has recently evaluated the constants
,(G,H).
We will not qive the general formula, but if
are both split groups,
where
,(G,H)
H = H
eauals
~ o r r n ( s ~ ~ , ~denotes
G~)
the aroup of elements
normalize the coset
an8
i";
in
0
'GO
which
sZG.
A formula like (3.2.3) will have interestin? implications -For the
discrete spectrum of
Consider the one dinensional autororwhic
G.
representations of the various endoscopic aroups
PSp(4,lR)
suqqest that for
H # G,
F.
Our exa~nlesFor
the contributions of such one di-
mensional representations to the riaht hand side of (3.2.3) will not
be stable distributions of
f.
Thev wil1"have to corresnonr' to sor'ethinn
in the formula (3.2.2) for
I+(f).
Suwpose that sore one dimensional
representations cannot be accounted for bv anv terms in (3.2.2) indexed
by (M,o), with
M = G.
M # G.
Then thev will have to correspond to t e r m with
In other words, they ought to qive rise to interestino nonter'per-
ed automorphic representations of
G(A)
which occur in the discrete
spectrum.
It is implicit in our conjecture that we should index the one climensional automorphic representations of
in which the image of
of
SL(2,C)
W
(P
in
H 0
H(D)
bv paps
commutes with
!I0
and the ipiaae
corresponds to the principal unipotent in
H
.
(For the
correspondence between unipotent conjuaacv classes and representations
of
SL(2,C),
see [13].)
It is of course easv to do this.
clear is why we should do it.
dimensional representations of
without it?
Whv introduce an
H(A)
SL(2,C)
What is not
when the one
can be describe$ oerfectlv well
According to the conjecture, the
SL(2,C)
factor will he
essential in describing the correspondin9 automorphic representations
of
G ( b )
of
H(A)
.
In particular, a one dimensional autoip.orohic representation
should aive rise to automoprhic representations of
which occur discretely (modulo the center of
the iraaqe of
W
15
x
SL(2,C)
G
L ~ .
shall examine this auestion
PSp(4).
Consider the example of
3.3.
Q,
if and onlv if
under composition
lies in no proper Levi subqroup of
for
G(A) )
G ( A )
G
G = PSp(4).
As a reductive Group over
has only two cuspidal endoscopic qroups (up to equivalence)
-
itself, and
with
Let us look at the formula (3.2.3) in this case.
equals
1.
has order
Since
The constant
1 (G,G)
The Froup
2,
the nontrivial element beinci the coset of the ri-atrix
we have
The group
'S
+
has no proper cuspidal endoscopic oroun.
H
eauals
1
;
.
This peans that
and so is civen bv the formula (3.2.2).
Forpula
(3.2.3) is then euuivalent to the assertion that the distribution
is stable.
Since the distribution
is neither stable nor tempered, the assertion would r'ive interestinr'
information about the discrete spectrui".of
R.
The one dimensional autoroorphic representations of
where
in
p,
{+I}.
where
w1
and each
and
u2
F.
are lust
are Grossencharacters whose imar'es are container'
For anv such representation define
is the projection of
w
onto the connutator quotient of
pi(wl) is identified with a central elenent in
As we did for real groups, we define a r"ao
SL(2,rT).
W
Q '
as the composition of the map
with
^
fTF
Then the global L-nacket
it).
eouals
exactly one element. the representation (3.3.1).
the natural embedding
of
W
SL(2,C)
x
ft
into
c G ,
H
G .
fTi,
9v
we identify each
it)
an? contains
comvosinc"'with
with a ~anninr'
In this wav we obtain parameters in
They are just the ones considered in Examples 2.4.2 and 2.4.7.
Y(G/Q).
The contribution of
equals the product of
$
it)
and
to the richt hand si8.e o* (3.2.3)
H
with the character of the representation
(3.3.1) evaluated at
for
G(3R)
parameters
f . Assume that the Examples 1.4.2 and 1.4.3
carry over to each local croup W P ) . Then to the local
$ I
.
rackets
?
Y(G/<r>), obtained fror'
we have the local
I
)
.
On these packets. the sians
are all
E
1.
1-F
*v
the contribution of
where
sv
)I
and
is the image of
H
s
to (3.2.3) is just
in
'
C
V
onto
.
C
This becomes
V
if we assume the nroduct formula
/ZG
and
-
sv
is its nroiection
Suppose that
ul
= p2 = p.
The conjecture reauires that (3.3.2)
should be cancelled by a term in (3.2.2) indexed bv
M
# G.
The proiection of the inaae of
$
onto
(Pro) with
is conjucrate to
a subaroup of
where
and
But
M
Pi0
of
Then
G
o
is the identity component o-F the L-crroup of a Levi suboroun
which is isomor~hicto
GL(2).
Set
can be reqarded as an automorphic representation of
occurs discretely (~odulothe center of
M(A)).
which
M
It is the pair
(Pilo) whose contribution to (3.2.2) we will compare with (3.3.2).
Let
w
be a representative in
of the Weyl qroup
W(a).
of the nontrivial element
The representation
of an automorphic representation of
The contribution of
G(0)
PGL(2).
o
is a lift to
It is fixed by
(M,o) to the formula (3.2.2) for
I+(f)
GL(2)
ad(w).
is
(3.3.4)
since
We can expect a decomposition
of
T(w)
p. 2821.)
into local normalized intertwining operators.
If
<f>-,
is the three dimensional reoresentation of
tained by composing $$
(3.3.3), and
m(w)
I$-,
(See [ Q ( b ) ,
ob9
with the adjoint representation of the crroup
71
is its contraaradient, the global nornalizincr factor
equals
One checks that it equals
where
uv
Therefore, (3.3.4) equals
1.
is the character
y(det ( - ) )
on
local component of the Grossencharacter
p.
RL(2,Qv), with
pv
the
"ith a resolution to
Problem 1.4.4, or rather its analoc'ue for each nlace
v,
the expression
would become
This is just (3.3.2).
Thus, when
pl - p2
=
p,
group, the contribution of
$
so that
and
H
$
factors through a Levi sub-
to (3.2.3) v~ouldbe cor'wletelv
cancelled by a term in ( 3 . 2 . 2 )
T$
with
f
#
C.
This suncrests that such
contribute nothino to the discrete spectrum of
G(A)
,
as nredicted
by the conjecture.
In order for the two terns above to cancel, it was essential that
3.4.
1 (GIH)
=
l ~ ( a )
1
the oomnon value, we recall, bein@
.
This fact mav be interpreted
as a combinatorial property of the complex croup
*
c
=
The generalization of this ~ropertywill be a kev to affectincr sir'ilar
cancellations for arbitrary croups.
Let
C
be the set of co~plexwoints of a copi~lexreductive
algebraic qroup.
We do not assurse that
the identity component of
and let W
Me shall describe it.
C.
Let
be the normalizer of
C
is connected.
Let
c
he
0
be a Cartan subgroup of C ,
0
in C, modulo T . Then W is
To
To
an extension of
the Weyl Troup of
Let
"rec
value.
If
where
n(w)
(C0,T0) .
It acts on
be the set of elements in T'J
w
is any element in
1-7,
To
and on its Lie alr-ehra.
for which
1
is not an eioen-
set
equals the number of positive roots of
0
( C ,'?
0
)
which are
mapped by v to necrative roots. ( E (w) is independent o^ how the
positive roots are chosen.) For each connected corponent x of C
we define
where W
in
x.
(x) is the set of elements in Wreq induced1 frori noints
reg
The number i(x) is a sort of scalar analogue of the invariant
distribution (3.2.2).
For each component x
0
C -orbits of elements in
of
x
C,
let
Orb(C0 ,x)
be the set
o-F
for which the acljoint riap (as a linear
operator on the Lie algebra of
C )
is seriisiriple.
I?
s
belongs
to any of the orbits,*the crrouo
satisfies the same hypothesis as
C.
Its conjuc'acv class in
C'
pends only on the orbit of
s.
The number
of connected components in
C
also depends only on the orbit of
de-
It is possible to define uniquely a number .a(C), for every group
C, which depends only on C 0, and vanishes unless the center of
s.
c0
is finite, such that
for every group C. Indeed, there are only finitely rianv orbits s in
0 0
Orb(C ,C ) such -that the center of Cs is finite, so we can define
o(C)
inductivelv bv this last eauation.
We see inductively that it
r1
depends only on
c".
The numbers
o(C)
are scalar analogues of the
stable distribution defined by (3.2.3).
Theorem 3.4.2:
Plith the possible exclusion OF the case that
'
C
has exceptional simple factors, we have
for every component
x
of
C.
The details will avpear in [l(c)].
to look at the exceptional qroups.)
(I have not yet had a chance
Equations (1.3.6).
(3.1.21 and (3.4.11 are all in the same spirit.
They each provide an inductive definition for a set of objects (stable
distributions, for example) in tern's of oiven objects (such as invariant distributions).
The inductive definition in each case is bv a
over indices which are closely related to endoscopic crroups.
(1.3.6) and
( 3.1.2)
should have twisted analocrues.
SLIP
Equations
These should be
true identities, involvinq the objects defined by the oricrinal equations.
The twisted analo-e
of (3.4.1) we have just encountere$.
It is the
formula (3.4.3).
REFERENCES
Arthur, J.,
(a) & theorem on the Schwartz svaee of a re6uctive L<e arouoProc. Nat. Acad. Sci. 72 (1975), 4718-4719.
(b) The trace formula for reductive arouos, to annear in Dubl.
Math. de l'university Paris VII.
(c) In preparation.
--
Flicker, Y., L-packets
&
liftings for U(3),
preorint.
Howe, R., and Piatetski-Shaoiro, I., A counter-example to the
"generalized Tamanujan conjecture" f
o
r
errouns,
Proc. Svmoos. Pure Hath., Vol. 33, Part 1, Amer.M.ath,?oc.,
Jacquet, H., article in these proceedinos.
Kna?p, A., Commutativity of intertwining operators
groups, Comp. Math. 46 ( 1 9 8 2 ) , 33-84.
Kottwitz, R . ,
preprint. .
Rational conjuqacy classes
Kurakawa, N., Examples
semisimnle
reductive arouns,
eiaenvalues of Hecke o~eratorson Fieoel.
Inventiones Hath., 49 (19781, 149-165.
Labesse, J. P., and Lanalands E.P.,
L-indistinauishabilitv--For
SL(2), Can. J. Math. 31 (1979), 726-785.
.
Lanalands.
>-. 3. P.
On the classification of irreducible representations of
real alqebraic qroups, mimeoaranhed notes, Institute q r
Advanced Study, 1973.
On the Functional Eouations SatisFied
Fisenstein cerie5,
Lecture Notes in Iflath., 544 (1976).
Stable conjuaacy: definitions 5nd lem.as, Canad. J. Yath.
3 7 9 1 . .. 700-725.
Automorphic representations, Shinura varieties, and motives.
Ein Marchen. ?roc. Symos. Dure Math., Vol. 33, part 2,
Amer. Math. Soc., Providence R.I., (1979), 205-246.
Les debuts d'une formule des traces stables, publ. Vath.
de l'universite
Paris ~11,oo.1983).
-
-
7
10.
Piatestski-Shaoiro, I.,
On the ~aito-~urakawa
lifting, orenrint.
(a) (b) Special Automor~hic forms on ?GSn(4),~reyrint.
(c) Article in these uroceedings.
11.
Shelstad, D.,
(a) Notes on L-indistinguishability (based on a lecture of
R.P. Langlands), ^roc. Symoos. Pure Math., Vol. 33, ^art 2 ,
Amer. Math. Soc., Providence ?..I., (1979), 193-203.
(b) Embeddings of L-grouos, Canad. J. Math. 33 (1981). 513-558.
(c) L-indistinguishability -For real urouns, Math. Ann. 259 (I0""')
385-430,.
(d) Orbital inteqrals, endoscooic arouos and L-indistinnuishahj1ity for real grouos, to appear in Publ. Math. de 11universit6
---P a n s VII.
12.
Speh. B., and Voqan, D., Reducibility of ueneralized orinci~al
series representations, Acta. Vath. 145 (1980), 227-299.
13.
Springer, T., and Steinberq, R., Conjuuacy classes, Lecture Motes
in Math., 131, 1970, 167-266.
14.
Steinberg, R . , Torsion $ reductive uroups, Advances in Tfath.
15 (1975), 63-32.
15.
Vo~an.D., Complex aeometry
groups, preprint .
16.
Waldspurger, J.L., Correspondence
Preprint.
Note added in proof:
representations 0
2
reductive
Shimura
auaternions.
in the local Conjecture
£4
in the global Conjecture 2.1.1 should
The sign function
s
1.3.3 and the sign character
both have simple formulas.
Suppose that
is given as in Conjecture 1.3.3.
*'
belongs to the centralizer C
Then
E,
Then
Let
-
b
*
be the image of
should be given in terms of the pairing on
e (IT) =
^1
<A,,lT>,
In particular, if the ur-ipotent element
C
*
x
A
IT
4)
4)
in
C.,
by
T e n
4)'
in
G O
is even, the function
Suppose that
F
is olobal and
is qiven as in Conjecture 2.1.1.
be the Lie algebra of
will be identically 1.
E,
Assume that
*
C
is finite.
Let g
G O , and define a finite dimensional representa-
tion
for
c â ‚
ID '
w ? G n
and
g
â
temp ( F )
SL(2 ,S) .
Then there is a
decomposition
where
ti,
a given
\
and
pi
are irreducible (finite-dimensional) representa-
i, the represents-tion
@
is equivalent to its contragredient.
Then from the anticipated functional equation of the L-function
L(b,^),
Let
we see that
I- be the set of such indices
'!'
equals
i
such that
1
E ( ~ , @ Jactually
~)
-1, and such that in addition, the dimension of
pi
is even.
Then the sign character should be given by
Such a formula (assuminq it is true) is rather intruiginq. It ties
1
the values of €-facto at
in an essential way to multiplicities
of cusp forms, and it also suggests that the adjoint representation of
the L-group might play some role in questions of L-indistinguishability.