HARMONIC ANALYSIS OF INVARIANT DISTRIBUTIONS
Suppose t h a t
.
.J
algebra @
G
384.
James Arthur*
i s a r e d u c t i v e L i e group, w i t h L i e
To b e s a f e , l e t us assume t h a t
i s t h e set
G
of r e a l p o i n t s of a r e d u c t i v e a l g e b r a i c group d e f i n e d over
IR.
G.
Harish-Chandra h a s d e f i n e d t h e Schwartz space, C ( G ) , on
I t i s a c e r t a i n space of f u n c t i o n s on
G
such t h a t
I n each c a s e t h e i n c l u s i o n i s a continuous map from one space
o n t o a dense subspace of t h e second.
G
A d i s t r i b u t i o n , F , on
i s c a l l e d tempered i f i t e x t e n d s t o a c o n t i n ~ o i !l ~i n e a r
f u n c t i o n a l on
f e c¡(G
EXAMPLE:
,
CiG).
<F,fy>
i s s a i d t o be i n v a r i a n t i f f o r f i x e d
F
i s independent of
Suppose t h a t
Lie a l g e b r a
4.
centralizer is
If
y
T
E
y e G.
Here
i s a Cartan subgroup of
Treg,
G
with
t h e s e t o f p o i n t s i n T whose
T, d e f i n e
Harish-Chandra h a s shown t h a t t h e map which sends
f e C(G)
to
i s w e l l d e f i n e d , and i s a tempered i n v a r i a n t d i s t r i b u t i o n on
G.
The d i s t r i b u t i o n s s o o b t a i n e d , known a s t h e o r b i t a l
i n t e g r a l s of f , p l a y a c e n t r a l r o l e i n t h e harmonic a n a l y s i s
* P a r t i a l l y supported by NSF g r a n t MCS 77-00918
on
They a l s o a r i s e n a t u r a l l y i n t h e s t u d y of
G.
r
where
i s a d i s c r e t e co-compact subgroup o f
PROBLEM:
L~(~'\G),
G.
Define t h e n o t i o n of F o u r i e r t r a n s f o r m f o r an i n -
v a r i a n t tempered d i s t r i b u t i o n .
I n t h i s l e c t u r e we s h a l l describe a s o l u t i o n t o t h i s
W e s h a l l t h e n use o u r s o l u t i o n t o d e f i n e a wide
problem.
c l a s s of i n v a r i a n t tempered d i s t r i b u t i o n s , o f which t h e
o r b i t a l i n t e g r a l s a r e s p e c i a l cases.
Let
be a f i x e d maximal compact subgroup o f
K
t h e set of f i x e d p o i n t s of a Cartan i n v o l u t i o n
Suppose t h a t
and
M = M'
MI and
A , where
N
can be w r i t t e n
m
exp
Since
'
G
= P K , any
n m k ,
x
in
n e N ,
G
meM,
I t i s customary t o d e n o t e t h e v e c t o r
pose t h a t
o
pull
a
back from
Hp(o)
unitary.
M
N\P
to
of f u n c t i o n s on
Any f u n c t i o n i n
its restriction t o
on
K
Then
P = NM,
P , and
i s a v e c t o r group.
A.
K.
Let
e M'
We denote
by
H
keK.
by
HM(m)
Hp(x).
M.
Info) o f
G
Sup-
We can
and t h e n induce t o
P
G.
on a H i l b e r t
G, which i s u n i t a r y i f
Gp(a)
and
can be w r i t t e n
i s an i r r e d u c i b l e r e p r e s e n t a t i o n of
The r e s u l t i s a r e p r e s e n t a t i o n
space
A
I I , where. m
belongs t o f i , t h e L i e a l g e b r a of
H(m).
G.
G.
i s t h e u n i p o t e n t r a d i c a l of
has compact c e n t e r and
M
m e M
Any
is a p a r a b o l i c subgroup of
of
a r e uniquely determined 9 - s t a b l e r e d u c t i v e qroups
A
such t h a t
H
-
P
9
It
G.
o
is
i s uniquely determined by
Hp(a)
o b t a i n e d i n t h i s way, and l e t
be t h e space o f f u n c t i o n s
Ip(o)
b e t h e correspond-
ing representation of
G
on
Hp(o).
The reason for intro-
ducing this latter space is that it does not change if
is twisted by a character of A.
for any quasi-character
of
A
o
In other words, if
a , the
representations
all act on the same space.
A representation v
of
G
is said to be tempered if
its character is a tempered distribution.
for
f
E
This implies that
C (GI ,
is well defined.
We obtain a map from
complex valued functions on
C(G)
to the space of
E(G), the set of classes of
irreducible tempered representations of
G.
The problem is
to characterize the image of this map.
It is known that
IT
is tempered if and only if it is a
subrepresentation of some
Ip(o), where
unitary representation of
M
modulo
A.
AN
Let
be any second order left and right in-
Then if
such that for all such
o(AM), a priori a scalar, is actually posi-
the operator
tive.
is an irreducible
which is square integrable
variant differential operator on M
0
o
IT
is any subrepresentation of
Ip (0), define
It i s e a s y t o show t h a t t h i s number depends o n l y on t h e
tempered r e p r e s e n t a t i o n
a e ET(M),
P , and any
pose t h a t
s
and n o t on
?I
we can d e f i n e
P
a e ET (M)
,
then
f i n i t e sum of tempered r e p r e s e n t a t i o n s o f
additively.
If
D = DA
a.
ET(G).
If
Ip ( 0 )
is a
n
Hom(ci.,iIR),
is a p o s i t i v e i n t e g e r , define
I I s I I ~ , ~=
-.
l(6)
Let
Otherwise,
be t h e set of a11 f u n c t i o n s
such t h a t
llslL,n <
for a l l
P, D
and
semi-norms
[ ,
THEOREM
1:
-
The map
n.
A
becomes a F r e c h e t space.
i s a continuous s u r j e c t i o n from
COROLLARY:
-
With t h e topology d e f i n e d by t h e
I (G)
The t r a n s p o s e
P
is a d i f f e r e n t i a l operator
i f a l l t h e d e r i v a t i v e s i n t h i s formula e x i s t .
set
Sup-
We d e f i n e
G.
of c o n s t a n t c o e f f i c i e n t s on t h e r e a l v e c t o r space
and
For any
[loll t h e same way.
i s a complex valued f u n c t i o n on
i s a p a r a b o l i c subgroup, and
s ( I p( o ) )
and
C (G)
onto
I
(6).
i s a continuous i n j e c t i o n .
THEOREM 2:
The image o f
i s e x a c t l y t h e space of i n v a r -
ch'
i a n t tempered d i s t r i b u t i o n s on
For
COROLLARY:
(a)
f
ch,, ( f ) = 0
(b)
E
Then
suppose t h a t e i t h e r
f o r every
TT E E ( G ) ,
I(£
= 0
ET(G)
where
and a l l
f o r every i n v a r i a n t tempered d i s t r i b u t i o n
Therefore,
f o r some
Recall t h a t i f
i s a s e t o f r e p r e s e n t a t i v e s f o r conjugacy c l a s s e s
{TI
T.
I.
,
o f Cartan subgroups of
on
T
'reg
The c o r o l l a r y i s e a s y t o prove.
IT E
or
f o r every Cartan subgroup
= 0
Y
C(G)
G.
T
G , and
t ( y )
(b) i m p l i e s ( a ) .
S E I'(G).
i s a bounded f u n c t i o n
But
Therefore ( a ) i m p l i e s t h a t
I ( Â £ = 0.
The p r o o f s of Theorems 1 and 2 a r e more d i f f i c u l t and w i l l
appear elsewhere.
blem.
If
I
However, t h e y give a s o l u t i o n t o o u r pro-
i s a tempered i n v a r i a n t d i s t r i b u t i o n on
d e f i n e i t s Fourier transform t o be ( c h ' l l ( l ) .
element i n t h e d u a l space o f
I
(2).
G, we
I t is an
Suppose t h a t
Let
9A = A.
Â¥tfi
M
i s a v e c t o r subgroup o f
A
be t h e c e n t r a l i z e r o f
be t h e L i e a l g e b r a o f
A.
such t h a t
G
and l e t
i s c a l l e d a s p e c i a l subgroup
A
i f it i s t h e s p l i t component o f a p a r a b o l i c subgroup
of
G
P.
Suppose t h a t t h i s i s t h e c a s e and l e t
of a l l such
r o o t s of
P..
(P,A).
Â¥ai. =
P
If
P e P(A), l e t
P(A)
be t h e set
t p be t h e s e t of simple
Then
{H€
a(H) > 0 ,
i s c a l l e d t h e p o s i t i v e chamber o f
out
in
A
G
a e tp)
P.
As
P
v a r i e s through-
t h e a s s o c i a t e d p o s i t i v e chambers a r e d i s j o i n t and
?(A),
dense i n Ã
.
Groups
P
and
P'
in
P (A) a r e s a i d t o be
a d j a c e n t i f t h e i r chambers s h a r e a common w a l l .
L e t u s i d e n t i f y -^t- w i t h i t s d u a l s p a c e by means of a
s u i t a b l e p o s i t i v e d e f i n i t e b i l i n e a r form
<
,>
on .a.
Suppose t h a t
{cp(A): P e P ( A ) ,
A eds}
i s a family of a n a l y t i c f u n c t i o n s .
W e s h a l l c a l l it a
p a t i b l e family i f whenever
P'
P
and
i s i n t h e i r common w a l l ,
cp(A) = c
LEMMA 1:
If
(A).
t h e family i s compatible,
g-
a r e a d j a c e n t and
A
is an a n a l y t i c function o f
PROOF:
The o n l y p o s s i b l e s i n g u l a r i t i e s o f t h e f u n c t i o n a r e
<B,A> = 6
for roots
mands f o r which
pairs, P
{B).
If
A.
and
0
<(<,A>
of
(G,A).
Given
0
is a s i n g u l a r i t y occur i n adjacent
= 0
Moreover, w e can assume
PI.
The c o n t r i b u t i o n i n t h e above sum from
i s i n t h e common w a l l o f
A
i n the curly brackets is
t h e o n l y sum-
0.
P
and
P',
ap \^J ( - Q p s )
(P,P')
Fix
It follows t h a t
<6,A> = 0
I t i s e a s y t o chock t h a t
is a compatible family.
If
m
Therefore
E
M,
so t h a t
Suppose t h a t
y
' "reg.
is
[I
x e G.
is well defined.
is
t h e expression
not a singularity.
EXAMPLE:
=
T
i s a C a r t a n subgroup i n
I t c a n b e shown t h a t
M, and
i s a tempered d i s t r i b u t i o n .
However, it i s n o t i n v a r i a n t .
We would l i k e t o modify it t o o b t a i n an i n v a r i a n t d i s t r i b u tion.
LEMMA
-- 2:
a
Suppose t h a t
sentation of
R
~
M.
Il P
i s an i r r e d u c i b l e u n i t a r y r e p r e -
Then t h e r e e x i s t u n i t a r y o p e r a t o r s
+ H
(0): H ( o )
P'
P,P'
(01,
E
P(A),
such t h a t
(iii)
If
P
and
P'
common w a l l ,
Suppose t h a t
a r e a d j a c e n t and
R
P'l P
Po
E
A
(uA)=R
(0).
P'l P
P(A), f
E
C ( G ) , and
Then
i s a compatible family.
$'*(Fro)
= l i m
A+0
i s well defined.
LEMMA 3:
Therefore
5 c p ( A ) ( n ae 'tp
<a,A>)
I t i s independent of
-1
Po.
The map
o+i(iA(fno),
lies in their
U E E ~ ( M ) ,
a
E
E(M).
I t follows from Theorem 1 t h a t t h e r e i s a f u n c t i o n
it,(f)
in
suchthat
C(M)
ch(^(f)) = k(f,o)
f o r every
a e E(M).
$(f)
i s n o t uniquely d e f i n e d , b u t
i t f o l l o w s from Theorem 2 t h a t i f
tempered d i s t r i b u t i o n on
y e Treã
Suppose t h a t
M,
i s any i n v a r i a n t
I
i s uniquely d e f i n e d .
I(,$,(f))
We d e f i n e a number
~ ~ ' ~ ' ^ ( ^ , f )
i n d u c t i v e l y by
z
{A* :A*CA }
M*,T,A
F
( Y , ( I ) ( ~ ) )= I D ( Y ) 1
T G
f(x-'yx)v(~dx.
The sum on t h e l e f t i s over t h e f i n i t e s e t of s p e c i a l subgroups which a r e c o n t a i n e d i n
A.
W e a r e assuming i n d u c t i v e l y
t h a t we have d e f i n e d a l l t h e summands e x c e p t t h e one f o r
which
A*
i s t h e s p l i t component o f t h e c e n t e r o f
G.
In
t h i s l a t t e r c a s e , of c o u r s e ,
M*,T,A
P
Notice t h a t i f
A
(y,fn*'f)'
= F
GrT,A
i s t h e s p l i t component of t h e c e n t e r o f
G
t h e r e i s o n l y one term i n t h e sum.
F
~
'
(Y,f).
In t h i s case
~f ) ~ i s~ j u(s t ~t h e, o r b i t a l i n t e g r a l
earlier.
THEOREM 3 :
The map
G,T,A
F
(Y):
G,T,A
f + F
(Y,f)
F ~ ' ~ ,(f )Y
defined
i s an i n v a r i a n t tempered d i s t r i b u t i o n on
G.
Notice t h a t t h i s theorem i s n e c e s s a r y f o r o u r i n d u c t i v e
d e f i n i t i o n t o work.
i s proved i n [ l l .
The theorem, a l o n g w i t h Lemmas 2 and 3 ,
The d i s t r i b u t i o n s
~
~
'
~ seem
~ ~ t o( bey
t h e n a t u r a l replacements o f t h e o r b i t a l i n t e g r a l s when one
t r i e s t o study
L'(~\G)
and
r\G
i s assumed o n l y t o have
f i n i t e volume.
Reference
[ll
J. Arthur, On t h e i n v a r i a n t d i s t r i b u t i o n a s s o c i a t e d t o
weighted o r b i t a l i n t e g r a l s , p r e p r i n t .
DUKE UNIVERSITY
DURHAM, NORTH CAROLINA
)