N ow let g â )-T(g)(gEG) be a representation of G by bounded ... 395. § 1. Prelim inary results. Let us introduce some notations and make some general ...
J. Math. Kyoto Univ. (JMKYAZ)
12-2 (1972) 393-411
Som e rem arks o n in v a ria n t eigendistributions o n semisimple L ie groups By Takeshi HIRAI (Received, February 15, 1972)
Introduction L et G be a connected real semisimple Lie group with Lie algebra
Denote by C V G ) the set of all indefinitely differentiable functions o n G w hich vanish outside som e com p act sets. F or a differential operator D on G, w e define its adjoint D * as g.
1GD fi( a 2 ( e d g —
G
h (g ) D *f2(edg ( 1 1 , 12 E C (G )),
where dg is a Haar measure on G . For an y distribution 7r on G, we put (D 7 r ) (f)=- 7 r ( D * f ) ( f C" ( G ) ) . A differential operator o n G is called Laplace operator i f it is invariant under both left and right translations. As usual, let us identify every X E g with a left-invariant differential operator o n G . Then the center 3 o f t h e universal envelopping algeb ra U(g c ) of the complexification g e o f g is the algebra of all Laplace operators on G . The correspondence D—,-D* on U(g c ) is its anti-automorphism generated by X — ›--X (X E g ). A distribution 7r on G is called invariant if it is invariant under any inner automorphism o f G . It is called eigendistribution if there exists a homomorphism A o f 8 into C such that Z7r=-A(Z)7r ( Z E 8 ) . Here A is called the infinitesimal character of 77. Let ZG be the center o f G. -
Takes/il H ira i
394
I f there exists a hom om orphism x of X (Z )7 F (g )
Z G ), 7
Z G
in to C * such that 7r( z g) =
is called Z G -simple.
N o w le t g — )-T (g)(gEG) b e a representation of G by bounded
, operators o n a H ilb e lt s p a c e A . P u t fo r any f E C 7(G) c T ( f = (_T (g)f (g)dg A representation ( T, ,g() is called (topologically) irreducible if ,q ( has no closed invariant subspace except { 0} and cgt itself. A n irreducible representation ( T , 11) is called quasi-simple [2(a), I] if there exist homomorphisms x of Z G into C * and A of 8 into C such that T (z)= X (z)lsc (z E Z G ) , T (Z)=A(Z ) iS ( (Z E 8), 0
where A° is the GArding subspace of A spanned by all T (f)v (f E C ( G ) ,
a n d L o d e n o te s th e identity operator on A ° . T h e character 7 7 of such representation can be defined a s to be the distribution 7r(f) -=-tr( T (f ))(f E C ( G ) ) [2 ( a ) , I I ]. T h e n 7F i s a Z G -sim ple invariant 21E 3
)
eigendistribution corresponding to x and A. C a ll it s im p ly ir r ed u cib le ch a ra cter. Denote by
a(A) (or E(A)) the set of
all invariant eigendistributions
on G (or linear combinations of irreducible characters) with infinitesimal character A. Then i ( A ) D ( A ) . O n e o f th e purposes o f this paper Here is to study th e problem whether '2-1(A) = E(A) for all A or not. we give an elementary proof of existence on S L (n, R )(n > 3) of tempered in varian t eigen d istrib u tio n s w h ich ca n n o t b e ex p ressed a s linear combinations o f irreducible characters. M oreover f o r S L (n, R ), all irreducible characters an d all invariant eigendistributions w ith certain infinitesimal characters A a r e obtained. Therefore w e know exactly th e difference o f a (A ) a n d E (A ) fo r s u c h A . F o r complex classical groups S L (n, C), S O(2n+1, C), S p(n , C ) an d S 0(2n , C), we see that if n < 3 , a(A )= E (A ) fo r any A and that if n > 4, W(A) / ( A ) f o r some A.
In v arian t ei gendistributions o n sem isim ple L ie groups
§ 1.
395
P re lim in a ry re s u lts Let us introduce some notations and make some general statements.
L e t G , g b e as before and
ba
Cartan su b algeb ra o f g . Denote by
P the set of all positive roots o f (gc , be ) with respect to a lexicographic o r d e r . A root a is called real (or im aginary) if it takes only real (or imaginary) values on
b.
D enote by PR (or P 1 ) th e set o f all real (or
im aginary) positive r o o t s . L e t W , b e th e W e y l grou p o f (g e , be ). L e t H be the Cartan subgroup o f G corresponding to the factor group o f th e norm alizer o f
b in
b and W ( H )
G b y the center H o of H .
For any root a, let X a E g, be its non-zero root vector and put A d(h)X a
=e a (h)X , (h (1. 1 )
E
H ) . D efine fo r hEH ,
LI' (h)-- jj p (1 — ,„(h) -
,
LI R' (h) =
p R
(1 — „ h)-1 . (
)
Replacing G, if necessary, by a certain covering group which covers
G finitely m any tim es, we m ay assume that there exists a connected complex sem isim ple L ie group Ge w ith the following two properties. (a ) L et p be the half-sum o f all a E P and H , the Cartan subgroup of
Ge correspon din g to bc . T h e n ep exp X )=eP(x ) (X E be ) d e fin e s a one-valued function on H . (b ) The injection j o f g in to g , can be (
lifted up a h o m o m o rp h ism o f G into G . T h e function ep o f on H is denoted again by ep .
(1. 2)
N o w put
17 (h)=.- e p (h)sign( R (h)) Ll' (h)
(h H )
T h en f o r a n y w E W G (H ) , there exists € (w )= .+ 1 su ch th at 17 (wh)
=E(w )r(h). L et G ' be the set of all regular elements of G and put H ' =H n G', GH =- U gH' g - '. Define fo r any f EC7,(G), a function F f on H ' as geG
(1. 3)
Ff
(h)=V (h) fG / H o f (gh g - l)dg',
where g=gH o, dg is an invariant measure on G /H o and a denotes the complex conjugate o f aE C.
T ak eshi H irai
396
L e t g = r+ p b e a Cartan decomposition o f g a n d a a maximal
abelian subalgebra o f p. Moreover let b° be a Cartan subalgebra of ° ° g such that f) =a+1) nr. Assume that the order in the set of roots o f (g e , I) e ) is compatible with one in the set of roots o f (g e , a s ). Put i te = -
E
C X
a
a e P ,a 1 e 0
,11
11c nQ.
Let K , A and N be the analytic subgroups o f G corresponding to r, a a n d n.
T h e n G =K A N is Iw asaw a decomposition o f G . The
Cartan subgroup corresponding to
b° is denoted by H
° . We see easily
that fo r H =H ° , F f (h) e p (h )
Fl (1N— eachri)f K / Z
ae P I
f
f (k hnk l)dk dn, -
G
where k =k Z G , d k and d n denote appropriate invariant measures on
K /Z G and N respectively. Hence, L e m m a 1 .1 . For H =H ° , the fu n ctio n F f o n H ' can be ex tended
to an in d efin itely d ifferen tia b le fu n ctio n on the w h o le H w ith compact support. Now let /(I) e ) be the subset o f U(t) c ) consisting of all W e -invariant elements.
L e m m a 1 .2 . (See [2(b), p. 1181 a n d [2(c), Th. 3].) unique iso m o rp h ism y=--yt1 o f
(Z
8
onto
I ( c )
T here ex ists s u c h t h a t F z f =y (Z )Ff
8 ) . This y s a t i s f i e s t h a t y (Z * )=-(y(Z))* A homomorphism of / (1) into C is always induced by some
where I): is the dual o f
bc .
Denote this one b y
if and only if pe=cri.e. for some (7
A .
T h e n AF =.11,,
1Fc . We say A t, is regular if a
t
We not equal to the identity. We sometimes identify the homomorphism A of 3 into C and the one Aoy ' of 1 ( h ) . For a fixed A,
for all a E
In v arian t eigendistributions o n semisimple L ie g r o u p s
397
let us consider an analytic function K on H° satisfying for some ,uE(Vc )* such that A=A p oy the following equations: (1. 4)
K(wh)=e(w)K(h)
(1. 5)
(D E I (b(G))
D K = A p (D )K
Define a function for g G
(w E WG (H °), h E H °),
on G from K as follows: for any gEr GH., ir(g)= 0 ;
IT
7
l i
o, 7r(g)=-(1 (h g )) --1 ic(h g ) , where k g E H ° is an element such
that g = g 0 hg g 0 - 1 for some g o E G . Consider the distribution defined as 7 r(f)= < 7 r, f> = f G f (g)7r(g)dg
(f
E
C (G )).
Then using the above two lemmas on F f , we obtain
Proposition 1.
d e fin e d a b o v e is an in v arian t eigendistribution o n G w i t h th e in fin ite s im a l c h a r a c te r A=A p oy w h ich v a n is h e s id e n t ic a lly o u t s id e th e c lo s u r e o f GH°. T he distribution
TT
P r o o f. C h ose a Haar measure d h on a C artan subgroup H appropriately, then for any integrable function p o n GH,
. a( ) d .IG H ç g
g
'H
G/hrog)(ghg ')drIV(h)I2dh.
Therefore applying this formula for H = H ° , < 7 r, f> -= f H o Fficdh.
= = L i o F z * f icdh= L o y (Z *)Fficdh
=L y (Z )*F .icdh=f o
s o
f
Ho
F f y ( Z ) K d li
F f.Ap (y(Z))1cdh=A(Z) . Q.E.D.
T a k esh i H ira i
398
D en ote b y WH o(At i ) th e s e t o f a ll invariant eigendistributions 77 obtained from the analytic functions K on H° as above. U sing Lem m a
2.4 in §2, we can prove as in [3(b)] the following proposition (see [3(c)]). But this one is not used to prove that for S L (n , R ) ( n >3 ) , K(A) / ( A ) fo r some A.
P ro p o s itio n 2 . T he set WH o(Ap ) is eq u a l to the set of a ll inv ariant
e i g e n d i s t r i b u t i o n s o n G w i t h i n fi n i t e s i m a l c h a r a c t e r A = A e y w h ich
v a n is h id e n t ic a lly o u t s id e th e c lo s u r e o f GH°. § 2.
R e v ie w o n k n ow n resu lts H ere w e sum m alize some known results in the form of a certain
num ber o f lem m as. Two quasi-sim ple irreducible representations
Ti on S ti(i= 1 , 2) are said to be infinitesimally equivalent [2(a), I, p.230] if th e corresponding representations of U ( ) o n A 7 = E a S i(8 ) (algebraic sum) are algebraically equivalent, where 8 denotes an equivalent class of irreducible representations of K and A i(8) denotes the subspace consisting o f all vectors transformed under T i(k) (k K ) according to
8.
Then,
L e m m a 2 .1 [2 (a ), III].
T w o q u a s i-s im p le ir r e d u cib le r e p r e s e n -
ta tion s of G have the sa m e ch a ra cter if a n d o n ly if th ey are in fin ite s im a lly eq u iva len t. T w o u n i t a r y 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 s h av e th e sa m e c h a r a c t e r i f a n d o n l y i f t h e y are u n ita r y e q u iv a le n t. L e t IV be th e centralizer o f A in K . T a k e ,u,a
4
a n d a finite-
dimensional irreducible representation v o f M . T h e n L =-(p,a , 1.)) defines canonically a representation o f M A N . Indu cing this one from M A N to G , we obtain a representation T L o n a Hilbert space
A L consisting certain vector-valued functions o n K (see e.g., [3(a)]). L e t A a n d A b e tw o clo sed invariant subspaces o f A L such that Jt J ,1{ . If t h e representation induced o n A /A is irredu cible, it i
1
2
2
I
2
In v arian t e ig e n d is tr ib u tio n s o n s e m is im p le L ie g r o u p s
399
is called irreducible constituent of T L . Then we know from Th.4 of [2 (a), I I ] t h e following Lemma 2 .2 . F o r G = S L (n ,R ) o r a c o n n e c t e d c o m p le x s e m is im p le L ie g ro u p , a n y quasi-sim ple ir r e d u c ib le r e p r e s e n ta tio n o f G is in fin ite s im a lly e q u iv a le n t to a n ir r e d u c ib le constituent of s o m e T '.
W e use the follow ing lem m as in § 5. Lemma 2 .3 . L e t T 1 ,
T2, . T d
b e the set of quasi-sim ple ir r e -
d u cib le r e p r e s e n ta tio n s o f G a n y t w o o f w h ic h are n o t in fin ite s im a lly eq u iv a len t. T h e n t h e i r c h a r a c t e r s a re lin e a r ly in d e p e n d e n t. Lemma 2.4 [2 ( d ) ]. A n y in v a ria n t eig en d istr ib u tio n 7 1 - o n G c o in c id e s w it h a lo c a lly s u m m a b le fu n c tio n o n G w h ic h is a n a ly t ic on G '. M o r e o v e r fo r e v e r y C artan s u b g r o u p H , th e function KP-= 1 7 . (77. 1s , ) on H ' n
G ' c a n b e e x te n d e d to a n a n a ly t i c fu n c t i o n o n H '(R )=
{hEH, ZJ (/1)
I
0}.
A be the infinitesimal character o f 77' and chose ,atIE I)g such that A=A p tryt). T h e n KP o n H '(R ) satisfies the analogous equations as (1 . 4 ) and (1. 5): Let
(2. 1)
ico(wh)= €(w)K0(h)
(2. 2)
DKP=Aiin(D)0
(w E W (H ), h G
E
H' (R)),
(D E I ( ) ) .
that b=b_+1) + , where b_=1) n f, b+ = t) n p . Then putting H _ = H fl K , H =H _exp '0 + . F o r any connected com ponent F of H '(R ), take h E H o n t h e boundary o f F . A s a s o lu tio n o f (2. 2), KP is expressed as Suppose
0
(2. 3) if X E
KP(ho exp X)= N fic (x ) -
t) is sufficiently sm all
,
,
and
e x p { t t t o x )}
ho exp X
E
F , where p , 's are some
T ak eshi Hirai
400
polynomial functions on
b.
I f K is compact or
Tr
is Z G -simple, all p c
can be taken as not to depend on b_, because F = F exp
Define V•I'(A)
(or W "(A), in case when K is compact) as the subset o f W(A) consisting of such 7r that for any F and H , all p c in the expression (2. 3) can be taken as to be constants (or polynomials with constant terms zero).
§ 3.
Invariant eigendistributions on S L (n , R) I n this section, let G =S L (n ,
R ) a n d H ° its Cartan subgroup
consisting of all diagonal matrices in G . Let us calculate all analytic functions
K
on H° satisfying (1. 4) and (1. 5).
Denote by d(aba2,
the diagonal matrix with diagonal elements a l , a2 ,
d(ai,a2,
an)
F or h =
a,.
,
(3. 1 )
(h )= I11:15 (ai — ai)1
The W ey l group W G(H°), simply denoted by W , is isomorphic to Wc and to the symmetric group S n o f order n as permutation group of at, L e t Ej = ± 1 (1 < j < n ) such that e1e2•••En = 1 and put a2 , .• •, an. En ) .
E2,
D e n o t e b y H °(c) the connected component o f H °
containing d(e i eti, € 2 e1 2, E n e t n ) , where tj
E
R. P u t / k =
2k1, Jk={2k+1, 2k+2, ..., n } and let €(k) be such row
fo r /E I k a n d E j= 1 fo r j E J k . P u t M e= Ho(E(kp)
E
2, • • •,
that €j = —1
A n y H ° (€ ) is
conjugate to some H , under W . It is sufficient to determine the re() j
strictions
K k
of
K
on H k fo r 0 < k < [n / 2 ] because for h E H ° (€ )=w H g
( w E W ) , K(h)=Kk(w - lh).
is isomorphic to S z k X (3. 2)
Any element
The subgroup W k = {w E W ; w H ?,=1 4 }
S n -2 k
and (1 . 4 ) is rewritten as
Kk(wh)=Kk(h)
(b)*
(W E W k ,
is expressed uniquely as
/ 2 2 , • • •,
w here ,u,J E C and pi-Pt52±. • • -hun=0, in such a way that (3. 3)
tz, • •
E p,j tj = (es, t)
16./Gn
(put).
In v arian t e i g e n d i s t r ib u t i o n s on s e m i s i m p l e L ie groups
401
To study the equations (1. 4) and (1. 5), it is convenient to replace
G by the reductive group + G = { gE G L (n , R ); d e t g > 0 } . The results in § 1 can be translated for +G word for w ord. D enote by +bo, +Ho +H°(E), +HZ , +17 , + K and +Kk the analogous objects as V, H °, H°(e), H Z , V K and K k respectively. Then for h = d (a i, a z , a n ) E + H ° , ,
71-1 (3. 4)
+ V
(h )=
(a l a z . . . a n )
2
f l
(a i—
o f)
The W eyl groups are the same for G and G . Denote by t j the differential operator
ap, on + H ° .
Then
z(-Ftg), considered as the algebra )
of differential operators on +H°, is nothing but the symmetric polynomials -1p z , p , n ) E ( Fi)g)* and D ( t ) I ( b°), A p(D (t))=D (i,). We restrict ourselves to treat p, such that 111+142+
For any
o f t 1 , t 2 , ..., tn .
p. = ( e i
... +p, n = 0 . Then for any such p,, there exists a one-one correspondence between the set of all solutions and that o f +
K
+
K
of the equations (1. 4), (1. 5) on H°
K
of the corresponding equations on + H ° , by restricting
o n H ° . Therefore it is sufficient for u s to study the following
equations: for 0 < k < [n/2],
(3. 5)
+Kk\w..., h) =+Kk(h)
(h E +H Z , w E Wk),
(3. 6)
D(t)+Kk=-D(-t)+Kk
(D E /( 114)).
W ; rp . = p } - and a = a W (p ,) f o r a E W . Any , solution of (3. 6) is expressed uniquely as follows: for h---=d(f i et , , 28 t2, Put W ( ) =
e n et.)E +HZ , (3. 7) where
+Kk(h)= f f e , R
co
pz,-(t) exp ((up, t ) ) ,
pr, 's are some polynomials o f t = ( t b t 2 ,
(3. 5) and (3. 6) in terms of p o 's .
where w p a ( t ) = p a (
-
lt).
Let us rewrite
The equation (3. 5) is written as
(w E W k ,
(3. 8)
tn).
a
W),
Take a complete system E of representatives
T ak eshi H irai
402
of the double coset space W k \W /W (i.t). Then (3. 8) means that it is sufficient to determin p a f o r cr
(3. 9)
E and that fo r any crE E, E
wPa=P6
be the set of different numbers in
N o w let a l , a2 ,
put
p, n , a n d r= { j ;
Wk fl w (g /.)).
D efin e c r ( i ) a s (a l iu)i=p, c (i).
A r — {j; 1 1 .1=a74 •
-
Then
F o r any subset A o f {1, 2, ..., n } , put
1,01 = ar}
(A ) = lw
D m(A )= tr, W
E
W ; w A =A , w (i)=i for any i A }
Using the same method as in [3(b), §9], we can prove the following
Lemma 3.1. T he system of equations (3.5) an d (3 .6 )is expressed i n term s of pa ( c i E ) as (zo
wPd =Po
(3. 10)
W k nW (o.A r ), 1 < r < N ) , ( m > 1 , 1 < r< N ) .
D m (o-A r )p j =0
F ix crE E and r and put A =aA
A
nB =0
r
n /k , B =crA
r
fl hic,
p=p, then
and the above equations fo r g and r are
(A )
w p =p ( w E W (3. 11)
1 D m (A
nB)p--0
if A or B=56, the polynomial
nW (B)),
(m > 1 ).
p does not contain the variables t i( jE A
B ) explicitely (see [3(b), §9]).
If A
I 56and B I
n
0, the equation (3. H )
has the following solution:
(3. 12)
.P (t)= 4 A )-1 jeE.4 ti
—
( B)
-1
E
JEB
w h ere O A denotes th e number o f elements in A . Restricting this solution
p ( t) from
+H
t oto H , we always obtain non-zero function.
N o w d e n o te b y WH o(A) an d VI'l .;0(A) th e sets afro(A) nW (A) and
In v arian t eigendistributions o n sem isim ple L ie groups
403
T h e n w e o b ta in fro m t h e a b o ve arguments the
9111 o(A) n V.i"(A). following
Proposition 3. F o r S L (n,
R ),
.( A ) = V - V s .( A ) + % / 1 ;.( A )
(d ire ct
s u m ) . W hen n =2 , alw ay s W'i o(A).={0 } an d 9IH.(A )=240(A ). W hen n
>3, % ;0(A )={ 0} o r * 1 0 } according as A is re g u lar or not. Irredu cible characters o f S L (n,R )
§ 4.
I n th is a n d th e n e x t sections, we calculate all irreducible characters of G =S L (n ,R ) with certain infinitesimal characters A. Let us apply Lem's 2.1 and 2.2. P u t a = h ° and K =S 0 ( n ) , then M =
{d(e i ,e 2 , ..., en )} and M A = H ° . Take 11—(p, 1 ,
p,n)E(t)g)* and let
7.=-(1 1,1 2, • .., vn ) be a row o f y i= 0 o r 1 . Then y determines a character ,
,
,
o f M and the pair (IL, y) determines a character (4. 1)
X '(h )
w here h=d(ai, (22,
T[" T
of H ° =M A as
,
Ii
i3, TI"(A)= {0} or I {0} according a s A is re g u lar or not. W e p ro v e d i n §4 and §5 th a t o n G = SL(n, R ) E(Ati ) = %'(At i ) *
fo r any imaginary pE(bg) .
§ 6. The case of complex semi-simple Lie groups In this section let G be a connected complex semisimple L ie group and H = H ° its Cartan su bgrou p. Th en w e can apply Prop 's 1 and
2.
For any root a of (g, b), define H a E b as a (X )= < H
w here < , >
a
, X > (X El)),
denotes th e K illin g fo rm of g. L e t X—›-iY(X E b) be
the conjugation o f b w ith respect to the real subalgebra spanned by
H a ( a E P ) . D en o te b y b* th e dual space o f
b
o v e r C . Then any
character x o f H can be expressed uniquely as
x(exp X)-=exp Ip (X )+ q (X )} ( X
El)),
where p, q E b * . (N ote that H is connected.) Denote x by (p, q) and consider it also as an element o f b * . L e t W b e th e W ey l group o f
(g, b).
It o p e r a te s o n x = (p , q ) a s w x = -(w p , w q )(w E W ). The
W eyl group W, o f (g c , be ) is isomorphic to Wx W in such a way that
a=(w, , w')(w, w 'E W ) o p e ra te s o n x = (p , q ) as crx=(wp, w' q). Let T x be the induced representation o f x on a H ilbert space S ix defined in §2 and vx its character. Let
Ab ) be the set of W-invariant analytic e
differential operators on H . T h e n /(be ) is gen erated b y j(1),) and W e see from these facts that Zw-x=Ax (Z)7rx (Z E 3 ), and that
j(b e ).
7rx=wx (or A = A ' ) if a n d only i f x'=wx (or=ax) fo r some wE W(or aE W e ). A study o f t h e equations (1. 4) a n d (1. 5) g iv e s us the /
following
Lemma 6.1.
F o r a n y c h a ra c t e r x = (p , q ), TgAx ) = TI'(Ax)±
In v arian t eigendistributions o n sem isim ple L ie g r o u p s
409
TI"(Ax) (direct sum ) and %'(Ax) is spnned b y { Trx' ; x' — (p,wq), wE W } .
w h e n c e E(Ax)Da'(Ax). W e w a n t to p ro ve E(Ax)=5?1'(Ax). Meanwhile we obtain from
[3(b), A p p . I I ] ( * ) the following Proposition 9. L et G b e a n y o f S L (n, C ), S O (2n+1,C ), S p(n, C ) an d S O ( 2 n ,C ) . W h en n = 2 o r 3, "(A)= {O} f o r a n y A. W hen n > 4 , t h e r e a l w a y s e x i s t s o m e A f o r w h i c h %"(A) { 0 } . M o reo v er
Tr(A )=-{ 0} f o r a n y A=A x w it h im a g in a r y x =( p , q)EW. A s a corollary of the last assertion of this proposition, we obtain T h eo rem 2 . F o r a n y c o m p l e x c l a s s i c a l g r o u p G , a tem p ered
in v arian t eigendistribution o f G is a lw a y s a lin e a r co m b in a tio n of the
c h a r a c t e r s o f its ir r e d u c ib le u n ita r y r e p r e s e n ta tio n s . Now , to determine E(A), we apply Lem's 2.1 and 2.2 and some results o f D. P . Zhelobenko in [4(a), (b)].
Suppose, for simplicity,
that G is simply connected. Then a pair of p, q E I)* defines a character of H if and only if p a —qa is integer for any a E P, where p a = 2 < p , a > I
.
A character x =(p , q ) is called discretely positive if for any
aE P, p a an d qa a re not negative integers at the sam e tim e. D . P. Zhelobenko [4(a), §11] defined for any discretely positive character
x , "th e m in im al r e p r e s e n ta tio n ,u,(x)" as the restriction of T x on an invariant subspace glx o f Slx with a stronger topology than the one induced from Stx and proved the following facts. L e m m a 6.2.
T h e r e p r e s e n ta tio n p (x ) i s c o m p le t e ly i r r e d u c i b le
in th e sence of R . Godement an d th e tw o ,u(x) an d ii,(x') are eq u iv a len t if a n d o n l y i f t h e r e e x i s t s s o m e w E W s u c h t h a t x'
§11].
A n y quasi-sim ple ir r e d u c ib le r e p r e s e n t a t io n o f G i s i n fi n i t e s i m a lly e q u iv a le n t t o s o m e ,u(x ) [4(b), T h . 7].
T a k esh i H ira i
410
Define the character of
. t ( x ) as
th at of the restriction of Tx on the
closure of 32x in S lx and denote it by ii(x). Then it follows from Lem. 6.2 that for any A, the set of all irreducible characters with infinitesimal character A does consist o f all d ifferen t r,c(x) w ith d iscretely positive x su c h th a t Ax = A . T h is g iv es us the dim ension of (A). On the other hand, by Lem . 6.1, the dimension of T (A) is equal to the number o f different 7T X s u c h th a t A = A . T h u s w e s e e th a t d im (A )= d im TI'(A), w h e n c e (A)=-21'(A).
T h e o re m 3. L et G be a co n n ected co m p lex semisimple L ie group. F o r a n y A, Ti(A)=TV(A)+TI"(A) ( d ire c t s u m ) an d (A)=T„i'(A). This theorem and Prop. 9 g iv e us the following
T h e o re m 4.
F o r S L (n, C ), S O (2n+1,C ), S p(n,C ) an d SO(2n,
C ), if n > 4 , th er e a lw a y s ex ist inv ariant eigendistributions on it w h ich c a n n o t b e e x p r e s s e d a s lin e a r co m b in a tio n s o f ir r e d u cib le ch a r a cte r s . N o s u c h distribution is te m p e r e d . (* ) E rrata. In [3(b), A p p . I I ]; p . 60, the 2nd line from below
sh o u ld b e "p(t ; p ) = p(T 't
;T
'p r ) (p E W , T E Z (c), T 'E Z (d ))"; p . 63,
th e r ig h t h a n d s id e o f (17') sh o u ld b e m u ltip lied b y
p i
(
ezi_e— zi
)
;
p . 66, th e 4 th a n d 5 th lin e s fro m b e lo w sh o u ld b e " i n another
p ( t) is sym m etric w ith respect to the u nion of t j ( j At n Bt) a n d — t i( jE A in B Ï ) and w ith respect to th e u n ion of t i (iEA -k n B T ) and —ti (i , 4 i n B t ) " .
cases,
KYOTO UNIVERSITY
References [1]
I. M . Gel'fand and M I. Graev: Unitary representations of the real unimodular groups (principal non-degenerate series). Izv. AN SSSR, Ser. M ath ., 17, 189-248 (1953).
In v arian t ei g en d istrib u tio n s o n s e m is im p le L ie g r o u p s
411
[ 2 ] Harish-Chandra: Representations o f semisimple L ie groups. I , I I , III. Trans. Am er. Math. Soc., 75, 185-243 (1953), 76, 26-65,234-253 (1954). (b) Th e characters of semisimple L ie groups. Ibid., 83, 98-163 (1956). (c) A formula for semisimple Lie groups. Amer. J . Math., 79, 733-760 (1957). (d) Invariant eigendistributions on a semisimple Lie group. Trans. Amer. Math. Soc., 119, 457-508 (1965). (e) Discrete series for semisimple L ie groups. II. Acta M ath., 116, 1-111 (1966). (a)
[3]
T . Hirai: (a) (b)
The characters of some induced representations of semisimple Lie groups. J . Math. K yoto U niv., 8, 313-363 (1968). Invariant eigendistributions of Laplace operators on real sim p le L ie groups. I. Japan. J. Math., 39, 1-68 (1970).
(c)
[4]
Characters o f representations and invariant eigendistributions on real semisimple L ie groups (in Japanese). Sugaku, 23, 241-260 (1971). D. P . Zhelobenko: (a) Operational culculus o n a complex semisimple L ie gro u p . Izv. A N SSSR, Ser. Math., 33, 931-973 (1969).
(13)
Classification of extremely irreducible and normally irreducible representations of a complex semisimple Lie group. Ibid., 35, 573-599 (1971).
