UBC Theses and Dissertations

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UBC Theses and Dissertations

Semisimple symmetric spaces Rossmann, Wulf 1975

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SEMISIMPLE SYMMETRIC SPACES by Wulf Rossmarm B.A,, University o f Alaska, 1967 A t h e s i s s u b m i t t e d i n p a r t i a l f u l f i l m e n t t h e r e q u i r e m e n t s f o r the degree o f D o c t o r o f P h i l o s o p h y i n the Department o f Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h r e q u i r e d s t a n d a r d The U n i v e r s i t y o f B r i t i s h Columbia September, 19 75 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . „ f M a t h e m a t i c s Department or The U n i v e r s i t y o f B r i t i s h Columbia 20 75 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date O c t o b e r 2nd, 1975 - i -A b s t r a c t . The t h e s i s c o n t a i n s a s t r u c t u r e t h e o r y f o r s e m i s i m p l e symmetric spaces w i t h a p p l i c a t i o n s t o r e l a t e d problems i n a n a l y s i s . I t i s shown how a r o o t system can be a s s o c i a t e d w i t h a s e m i s i m p l e symmetric space ( a b r . " s . s , s p a c e " ) . Analogues o f the C a r t a n , Iwasawa, Bruhat d e c o m p o s i t i o n s o f a s e m i s i m p l e L i e group a r e e s t a b l i s h e d f o r a r b i t r a r y s . s . s p a c e s . The " c o n i c a l d u a l " o f a s . s . space i s i n t r o d u c e d . The r e p r e s e n -t a t i o n o f a s e m i s i m p l e t r a n s f o r m a t i o n group on L ^ - f u n c t i o n s on t h e c o n i c a l d u a l i s decomposed as a d i r e c t i n t e g r a l o f p r i n c i p a l s e r i e s r e p r e s e n t a t i o n s . The a l g e b r a o f i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on the c o n i c a l d u a l i s found t o be commutative and i t s e i g e n f u n c t i o n s a r e d e t e r m i n e d . I n t e r -t w i n i n g o p e r a t o r s between comp a c t l y s u p p o r t e d f u n c t i o n s on a s.s. space and f u n c t i o n s on i t s c o n i c a l d u a l , and v.v., a r e d e f i n e d . The d e f i n i t i o n i s extended t o a l a r g e r c l a s s o f f u n c t i o n s by an i n v e s t i g a t i o n o f t h e convergence p r o p e r t i e s o f t h e s e o p e r a t o r s . An a p p l i c a t i o n t o the harmonic a n a l y s i s on S 0 ( p , q ) o / S 0 ( p , q - l ) o i s i n d i c a t e d . - i i -C o n t e n t s . I n t r o d u c t i o n .. 1 P a r t One. S t r u c t u r e Theory. I . Root space d e c o m p o s i t i o n s . . , 6 I I . • S p h e r i c a l d e c o m p o s i t i o n s t 20 I I I . P a r a b o l i c d e c o m p o s i t i o n s .....31 IV. The c o n i c a l d u a l o f a s e m i s i m p l e symmetric space ... 43 P a r t Two. A p p l i c a t i o n s t o A n a l y s i s . V. A n a l y s i s on the c o n i c a l d u a l 54 V I . The Radon t r a n s f o r m and i t s d u a l 70 B i b l i o g r a p h y 90 - 1 -I n t r o d u c t i o n . A s e m i s i m p l e symmetric space may be c h a r a c t e r i z e d . as a homogeneous space G/H, where G i s a c o n n e c t e d , s e m i s i m p l e , r e a l L i e group, H a subgroup o f G c o n t a i n e d i n the f i x e d p o i n t s e t re automorphism S o f G and S G o f an i n v o l u t i v e automorphism S o f G and c o n t a i n i n g the con-n e c t e d subgroup (G°) 0 o f G°: ( G ° ) o C H C G a . G/H becomes.a pseudo-Riemannian symmetric space under the G - i n v a r i a n t m e t r i c d e r i v e d from the K i l l i n g form o f G; G then, a c t s as the con-n e c t e d group o f i s o m e t r i e s o f G/H. C o n v e r s e l y , a pseudo-Rie-mannian symmetric space w i t h a s e m i s i m p l e group o f i s o m e t r i e s " i s " a s e m i s i m p l e symmetric space* ( 8 ) . The t h e s i s p r o v i d e s a s t r u c t u r e t h e o r y f o r such spaces and a p p l i e s the r e s u l t s t o some problems i n a n a l y s i s r e l a t e d t o them. For o r i e n t a t i o n and m o t i v a t i o n I mention some s p e c i a l c a s e s , which have been i n v e s t i g a t e d i n d e t a i l . (1) Spheres and h y p e r b o l o i d s . The homogeneous spaces S 0 ( p , q ) o / S 0 ( p , q - l ) o a r e s e m i s i m p l e symmetric s p a c e s , which may be r e a l i z e d as ( c o n n e c t e d components.of) t h e h y p e r s u r f a c e s - x 1 2 - x 2 2 . . . - x p 2 + x p + 1 2 . . . + x p + q 2 - 1 i n ] R P + q . I f p = 0 they a r e spheres and the harmonic, a n a l y s i s r educes t o the c l a s s i c a l t h e o r y o f s p h e r i c a l ' , h a rmonics. T h i s t h e o r y has been ex-tended t o a r b i t r a r y v a l u e s o f p and q by R. S t r i c h a r t z ( 9 ) . - 2 -H i s methods r e l y h e a v i l y on the r e a l i z a t i o n o f t h e s e spaces as s u b m a n i f o l d s o f R^"1^, but the r e s u l t s admit an i n t r i n s i c i n t e r -p r e t a t i o n i n terms o f the s t r u c t u r e o f the symmetric spaces S 0 ( p , q ) o / S 0 ( p , q - l ) o t h e m s e l v e s . ( s e e V I . 6 ) . (2) Riemannian spaces. A non-compact.space G / H ; i s Riemannian i f j and o n l y i f , the i n v o l u t i o n S i s a C a r t a n i n v o l u t i o n o f G ( p r e s u p p o s i n g t h a t G a c t s e f f e c t i v e l y on G/H). The b a s i c s t r u c t u -r a l f a c t s c o n c e r n i n g t h e s e spaces a r e c o n t a i n e d i n t h r e e "de-c o m p o s i t i o n theorems": r o o t space d e c o m p o s i t i o n . , C a r t a n decom-p o s i t i o n , Iwasawa d e c o m p o s i t i o n . The r e s u l t s o f c h a p t e r s I , I I , I I I show t h a t a l l t h r e e o f them can be g e n e r a l i z e d t o a r b i t r a r y s e m i s i m p l e symmetric spaces. The a n a l y s i s on Riemannian symmetric spaces i s c o n t a i n e d i n the t h e o r y o f s p h e r i c a l f u n c t i o n , as d e v e l o p e d by H a r i s h -Chandra ( 2 ), S. Helgason ( 4, 6 ), et. a l . (3) L i e gr o u p s . A L i e group G i s d i f f e o m o r p h i c w i t h the homogeneous space (G x GVH, where H i s the d i a g o n a l i n G^G, by the map G X G —>H, ( g ^ )Hf^->g^g2~^" • I n t h i s way a c o n n e c t e d , s e m i s i m p l e L i e group can be r e g a r d e d as a s e m i s i m p l e symmetric space. (The i n v o l u t i o n S o f GKG i s g i v e n by S ( g ^ , g 2 ) = ( g ^ s g ^ ) . ) A l a r g e p a r t o f t h e s t r u c t u r e t h e o r y o f a s e m i s i m p l e L i e group can be i n t e r p r e t e d i n terms o f i t s symmetric space s t r u c t u r e . A case i n p o i n t i s the Bruhat d e c o m p o s i t i o n : i t w i l l become c l e a r t h a t i t p l a y s the same r o l e f o r a L i e group - r e g a r d e d as a symmetric space- w h i c h t h e Iwasawa d e c o m p o s i t i o n p l a y s f o r a - 3 -Riemannian symmetric space (see I I I . 1 1 . ( a ) and ( b ) ) . The e x t e n s i v e s u b j e c t o f a n a l y s i s on s e m i s i m p l e L i e groups need n o t be r e c a l l e d h e r e . I n s t e a d I r e f e r t o H a r i s h - C h a n d r a ' s s u r v e y (3) and t o G. Warner's t r e a t i s e , (10 ). My own i n v e s t i g a t i o n s were m o t i v a t e d by t h e s e examples. In c h a p t e r I , I b e g i n the s t r u c t u r e t h e o r y by showing how a s e m i s i m p l e L i e a l g e b r a w i t h an i n v o l u t i o n g i v e s r i s e t o a r o o t -space d e c o m p o s i t i o n and t o a r o o t system.( The r e s u l t s o f t h i s c h a p t e r a r e p r i m i a r i l y i n t e n d e d f o r l a t e r u s e , w h i c h a c c o u n t s f o r the c a t a l o g u e s t y l e o f p r e s e n t a t i o n . ) The n e x t c h a p t e r g i v e s the d e c o m p o s i t i o n o f a s e m i s i m p l e symmetric space G/H i n t o o r b i t s o f a maximal compact subgroup of G. T h i s d e c o m p o s i t i o n may be thought o f as " s p h e r i c a l co-o r d i n a t e s " on G/H; f o r m u l a s f o r i n t e g r a t i o n and d i f f e r e n t i a t i o n i n t h e s e " c o o r d i n a t e s " a r e i n c l u d e d . In c h a p t e r I I I one f i n d s the d e c o m p o s i t i o n o f G/H i n t o a f i n i t e number of open o r b i t s o f a c e r t a i n p a r a b o l i c subgroup of G. The p r o o f o f t h i s d e c o m p o s i t i o n c o n s i s t s i n a r e d u c t i o n t o the s p e c i a l case when the symmetric space i s a L i e group, where i t f o l l o w s from the Bruhat d e c o m p o s i t i o n . C h a p t e r IV i n t r o d u c e s the c o n c e p t o f the " c o n i c a l , d u a l " o f a s e m i s i m p l e symmetric space, a g e n e r a l i z a t i o n o f the "space of h o r o c y c l e s " o f a Riemannian symmetric space. (To i l l u s t r a t e , ? 2 2 the c o n i c a l d u a l o f t h e h y p e r b o l o i d - X i , , , - x + : - x + x , 3 * , 1 p x p + 1 • • • T Xp-l-q _ 4 -can be i d e n t i f i e d w i t h i t s asymptotic cone -x-^ .-. .-x z +.Xp +]_" ... 2 + x , -• 0 . ) A p a r t from p o s s i b l e g e o m e t r i c i n t e r e s t the c o n i -c a l d u a l seems u s e f u l f o r t h e a n a l y s i s on the symmetric space G/H i t s e l f , because i t can s e r v e as a model f o r the r e a l i z a t i o n o f c e r t a i n p r i n c i p a l s e r i e s r e p r e s e n t a i o n s o f G r e l a t e d t o t h e " r e g u l a r " r e p r e s e n t a t i o n o f G on f u n c t i o n s on G/H. These c o n s i d e r a t i o n s l e a d i n t o the a n a l y s i s o f f u n c t i o n s on the c o n i c a l d u a l as a preamble t o t h e a n a l y s i s ' o n t h e symmetric space i t s e l f . T h i s i s c a r r i e d out i n c h a p t e r V, where the "regu-l a r " r e p r e s e n t a t i o n o f G on the c o n i c a l d u a l i s decomposed i n t o p r i c i p a l s e r i e s r e p r e s e n t a t i o n s . The a l g e b r a o f i n v a r i a n t d i f f e r -e n t i a l opera-tors on t h e c o n i c a l d u a l i s found, t o be.;commutative and i t s e i g e n f u n c t i o n s a r e d e t e r m i n e d . To some degree, the r e s u l t s o f the a n a l y s i s on t h e c o n i c a l d u a l can be t r a n s f e r r e d t o the symmetric space. The b a s i c t o o l f o r t h i s i s an i n t e r t w i n i n g o p e r a t o r , the "Radon t r a n s f o r m " , w h i c h i s d i s c u s s e d i n c h a p t e r V. There a r e , however, some r a t h e r s e r i o u s convergence d i f f i c u l t i e s , w h i c h t a k e up a l a r g e p a r t o f t h i s c h a p t e r . I n some s p e c i a l c ases an e x t e n s i o n o f the do-main o f the ( d u a l ) Radon t r a n s f o r m i s p o s s i b l e . F o r Riemannian symmtr-ic' spaces and f o r the ( g e n e r a l l y non-Riemannian) "hyper-b o l o i d s " S 0 ( p , q ) o / S 0 ( p , q - 1 ) Q t h i s l e a d s t o the d i r e c t i n t e g r a l d e c o m p o s i t i o n o f t h e " r e g u l a r " r e p r e s e n t a t i o n o f G on L Z(G/H):. - 5 -In view of the scope of the s u b j e c t i t i s c l e a r t h a t there i l l be no attempt a t completeness. But I do hope t h a t my work w i l l prove u s e f u l i n the f u r t h e r developement of the theory. wi - 6 -P a r t One. S t r u c t u r e Theory. Chapter I . Root space d e c o m p o s i t i o n s . T h i s c h a p t e r c o n t a i n s some v a r i a t i o n s on the theme of " r o o t systems". Some of t h e m a t e r i a l w i l l n e c e s s a r i l y have a f a m i l i a r f l a v o u r . P r o o f s a r e then; g i v e n i n o u t l i n e : f o r m , • i n d i r e c t r e f e r e n c e t o t h e l i t e r a t u r e i s i m p r a c t i c a l L e t G be a s e m i s i m p l e r e a l L i e a l g e b r a , S an i n v o l u t i v e automorphism of G . One knows t h a t t h e r e i s a C a r t a n i n v o -l u t i o n T o f G w h i c h commutes w i t h S ( 8 ) . Moreover, T i s unique up t o c o n j u g a t i o n by an i n n e r automorphism of the form e x p ( a d ( X ) ) , where X i s an element of G f i x e d by S. Set G ± S = {x i n G : S.(X) = ± X^ and G I T = {x i n G : T(X)= ) -S -T -S -T ±X) . L e t G ' be the i n t e r s e c t i o n o f G and G , and +S +T d e f i n e G - s i m i l a r l y . Choose a maximal a b e l i a n subspace -S -T A o f G ' . For any ( r e a l v a l u e d ) l i n e a r f u n c t i o n a l r on A s e t G r = (X i n G : ad(Y)X = r ( X ) f o r a l l Y i n A.), . I n A p a r t i c u l a r , GQ = G — , the c e n t r a l i z e r o f A i n G . Set R = [r i n - A * : G R f 0 and r ^ o} . The elements o f R w i l l be c a l l e d "the r o o t s o f the p a i r G, A'j The K i l l i n g form of G w i l l be denoted by 3 . . I a l s o s e t Bg(X,Y) =-B(X,S(Y)) and B (X,Y) = - B ( X , T ( Y ) ) . A l l t h r e e b i -l i n e a r forms on G a r e non-degenerate; B^ , i s p o s i t i v e d e f i n i t e by the d e f i n i t i o n o f a C a r t a n i n v o l u t i o n . - 7 " 1. Lemma, ( a ) G = G- + 5. <£r : < • X d i r e c t sum o f v e c t o r spaces). (b) R spans G as r e a l v e c t o r space. P r o o f . The a d j o i n t of ad(X) w i t h r e s p e c t t o B^ , i s - a d ( T ( X ) ) . Thus ad(A) i s a commutative f a m i l y o f B - s e l f a d j o i n t t r a n s -f o r m a t i o n s o f G, whence the f i r s t a s s e r t i o n . As f o r t h e o t h e r one, i f X i n A s a t i s f i e s r ( X ) = 0 f o r a l l r i n R, t h e n X l i e s i n the c e n t e r o f G, hence i s z e r o . 2 . Lemma, (a) [_Gr,G_s"j G r + S , (b) S ( G r ) = G_ r = T ( G _ r ) ( c ) B(G„,G„) = 0 , u n l e s s r = - s . B T ( G r , G s ) = . 0 = B s(G r,G_ s) , u n l e s s r = s. -S-T P r o o f , ( a ) f o l l o w s from t h e J a c o b i i d e n t i t y , (b) from A G ' and ( c ) from ( a ) and ( b ) . ( F o r more d e t a i l s c f . (*t) p. 141.) 3 . Lemma. For any r i n R, (a) G r ^ ( G r + G - r ) ± S (re.sp.«(GR + G _ r ) ± T ) by t h e map X ^ ( X ± S ( X ) ) (resp.H>(X^± T ( X ) ) ). (b) GJT > ( G r + G _ r ) S > T (resp . ^ ( 'G R + G _ r ) ~ S ' ~ T ) by the map X ^  (X + S ( X ) ) = (X H- T ( X ) ) ( r e s p r > ( X - S ( X ) ) = (X.- ; T ( X ) ) ). - 8 -( c ) C ; S T - ( G r + G_ r) s»'- T ( r e s p . - ( G r + G - r ) S ' ~ T ) by the map X H» (X + S ( X ) ) = (X - T(X)). (resp.V^(X - S ( X ) ) = (X + T ( X ) ) ). P r o o f . From 2.(b) i t i s c l e a r t h a t X h > ( X + S ( X ) ) maps G_r i n t o ( G r + i . G _ r ) S . Moreover i f X + S(X) = 0 , t h e n X i s i n t h e i n t e r s e c t i o n o f G r and G , w h i c h i s z e r o . So t h i s map i s i n j e c t i v e . On the o t h e r . h a n d , i f X i s i n G r, and Y i n G_ r so t h a t S(X + Y ) = (X + Y ) , t h e n Y = S(X), a g a i n because the i n t e r s e c t i o n o f G r and G_ r i s z e r o . T h i s proves one o f t h e a s s e r t i o n s o f ( a ) . The o t h e r s a r e p r o v e d s i m i l a r l y , and (b) and ( c ) f o l l o w from ( a ) s i n c e G_r i s ST i n v a r i a n t so t h a t ST -ST G r = Gr + G . 4 . Lemma. F o r any r i n R, (G_r + G _ r ) ~ = [x i n G~ '~ : a d ( Y ) 2 X = r(Y.) 2X f o r a l l Y i n _ ST P r o o f . L e t . X + S(X) = X + T ( X ) , X i n G_r be an element i n :(G r + G _ r ) S ' T as i n 3.(b). T h e n , f o r any Y i n A, a d ( Y ) ( X +S(X)) = r ( Y ) ( X - S ( X ) ) , and a d ( Y ) z ( X + S ( X ) ) = r ( Y ) (X + S ( X ) ) . S T 2 2 C o n v e r s e l y , suppose X i n G ' s a t i s f i e s ad(Y) X = r(Y) i"X. f o r a l l Y i n A. W r i t e X = X n + T. X. w i t h X<, i n G 0. Then f o r 2 2 any Y i n A, r ( Y ) X = c. s ( Y ) X. Because of l i n e a r independence of the X o T s t h i s i m p l i e s t h a t X = X^ + X . s r - r - 9 -5. Lemma. ( a ) I f X i n A s a t i s f i e s r ( X ) 4 0 f o r a l l r i n R, A ' X then G~ = G , the c e n t r a l i z e r o f X i n G. (b) I f X i n A s a t i s f i e s r ( X ) 4 0 whenever G ^ + 0, then A -h G X'~ S'~ T. P r o o f . ( a ) Suppose X i s i n A and r ( X ) 4- 0 f o r a l l r i n R. I f a d ( Y ) X = 0 f o r some Y . i n G, w r i t e Y = Y n + T- Y w i t h Y n i n G- and i n G r. Then 0 = ad(Y)X = -ad(X)Y = - H r ( X ) Y V . Thus Y R = 0 , f o r a l l r i n R, and Y = YQ l i e s i n G-. S T •(b) S i n c e ad(X)Y l i e s i n G ' whenever X l i e s i n A and —S —T A -S -T X - S —T Y i e s i n G ' , we show t h a t G -' ' = G ' ' as i n ( a ) . A -S -T -S -T But G —' 3 - A, s i n c e A i s maximal a b e l i a n i n G ' . .. . . ±ST 6. Lemma. (a) F o r any X i n G. r r [ X , S ( X ) J = B ( X , S ( X ) ) - Z ] [ X , T ( X ) J = B ( X , T ( X ) ) - Z r where Z r i s the element i n A s a t i s f y i n g B ( Z r , Z ) -- r ( Z ) f o r a l l Z i n A. (b) F o r any X i n (G_r +. G - V ) X S ' ^ ( i , j = +1) t h e r e i s a unique element Y i n ( G r -!- G- r) ^ ' ^ so t h a t f o r a l l Z i n A .ad(Z)X = r ( Z ) Y ad(Z)Y = r ( Z ) X . - 10 -Such elements X and Y s a t i s f y f u r t h e r [x,Y}-= -B(X,X)Z r = B ( Y , Y ) Z r . 1ST A Proof. (a) I f X i s i n GR ( i = +1), then [x,S(X)I i s i n G-and S[X,S(X)] = (S(X),Xj = - ( X , S ( X ) ] , T(x:,S.(X)) = / JT(X),TS(X)] = Q L S ( X ) , i X } =• T]J(,S(X1J. (SO X i s i n G^  = A . Moreover, f o r Z i n A, B(Z, [X,S(XJ ) = B( [z,x] ,S(X)) = r (Z )B(X, S (X) ) ; so [x,S(X)] = B ( X , S ( X ) ) Z r . T h i s proves the f i r s t e quation, and the second one f o l l o w s by m u l t i p l y i n g through by i = +1. •(b) By 3., any X i n (G_R + G _ r ) l S ' J T ( I , J = +1) can be u n i q u e l y w r i t t e n as X =-X + ''iS(X ) = X r +' jT.(X r) w i t h Xr i-jST i n G_R. Set Y = X r - i S ( X r ) = X r - j T ( X r ) . Then Y l i e s i n (GR + G-R)~ L S '"J T and [X,Y] = ( x r + i S ( X r ) , xr -. i S ( X r ) ) = - 2 i [ X r , S ( X r ) ) r B(X,X) = B(X r + i S ( X r ) , X + i S ( X r ) ) ' = 2 i B ( X r , S ( X r ) ) = -2iB(Y,Y). T h i s proves the l a s t a s s e r t i o n i n (b)and the r e s t i s obvious. 7. Lemma. For X i n A set a = exp(ad(X)), the i n n e r automorphism a , i of G c o r r e s p o n d i n g to X. Set G = |Y i n G : a(Y) = Y]. ' Then a(G I S) GJT = G a ' i S ' j T f o r i , j . = ± 1 . -..11 -P r o o f . I t i s c l e a r t h a t the l e f t hand s i d e c o n t a i n s the r i g h t hand s i d e . For the o t h e r i n c l u s i o n , suppose a(£.X r + S ( X r ) ) = £ Y r + T ( Y r ) , where X r , Y r a r e i n G_r and r runs over a subset o f R c o n t a i n i n g e x a c t l y one elemement o f each p a i r +r. Then 1 a r X r + a ~ r S ( X r ) - l Y r + T ( Y r ) , where a r = e x p ( r ( X ) ) . Because of l i n e a r independence t h i s g i v e s a r X r = Y r and a ~ r S ( X r ) = T ( Y r ) f o r a l l r . Thus a " r S ( X r ) = a r T ( X r ) f o r a l l r . S i n c e t h e e i g e n v a l u e s of ST a r e +1, we f i n d t h a t X r = 0 u n l e s s a = 1, and i f a r = 1, t h e n S ( X r ) _= T ( X r ) . T h i s proves t h e lemma. L e t H be the a n a l y t i c subgroup of A u t ( G ) w i t h L i e a l g e b r a q T H = G , K t h e a n a l y t i c subgroup w i t h L i e a l g e b r a K = G ; ( H O K ) Q the c o n n e c t e d subgroup o f Hr\K. 8. P r o p o s i t i o n . G _ S , _ T = (H P I K)Q - A 7 -S -T P r o o f ( c f . (4) ). Choose Z i n A so t h a t A = G ' ' . For any Y i n G ^' 1 the f u n c t i o n k B(k.Y,Z) must have a c r i t i -c a l p o i n t on the compact group (HfyK)o. I f k i s such a c r i t i -S T c a l p o i n t , - t h e n f o r a r b i t r a r y X in(HnK),=-' G ' : 0 = i f B ( e x p ( a d ( t X ) ) k . Y , Z ) ( t = 0 = ,B(ad(X)k.Y,Z) = -B(ad(k.Y)X,Z) = B(X, a d ( k . Y ) Z ) . ,: - 1 2 -S i n c e ad(k.Y)Z i s i n G ^ ' 1 , and s i n c e B i s n e g a t i v e d e f i n i t e on G S ' T , ad(k.Y)Z = 0. Thus k.Y l i e s i n G Z ' " S ' " T = A. -S,-T 9. C o r o l l a r y . Any two maximal a b e l i a n subspaces o f G a r e c o n j u g a t e by an element . i n (H (\K\Q , P r o o f . Suppose A T i s a n o t h e r maximal a b e l i a n subspace of -S -T G ' . For any X i n A' t h e r e i s k i n H K such t h a t k.X i s i n A. Then k . G X , ~ S , ~ T ' G K * X ' ~ S , " T c o n t a i n s A. So i f X -S -T G ' ' = A' , t h e n k.A T c o n t a i n s . _A, hence e q u a l s A, by m a x i m a l i t y . L e t K— be the c e n t r a l i z e r o f A i n K, the n o r m a l i z e r A ~ of A i n K. The q u o t i e n t group W = K^/K— may be c o n s i d e r e d as a group o f l i n e a r t r a n s f o r m a t i o n s o f Ac and o f A*. 10. P r o p o s i t i o n . R i s a r o o t system i n A* w i t h Weyl group W. The p r o o f proceedes by t h r e e lemmas. 11. Lemma. W l e a v e s R i n v a r i a n t . F o r each r i n R, W c o n t a i n s a ( n e c e s s a r i l y u n i q u e ) r e f l e c t i o n i n r . - 1 3 -P r o o f . The f i r s t a s s e r t i o n i s c l e a r . F o r t h e o t h e r one, g i v e n r i n R, choose X and Y as i n ' 6., b o t h non z e r o . ( T h i s i s c l e a r l y p o s s i b l e . ) E x a c l y one o f them, say X, l i e s i n K. Then exp(l.ad(X)) l e a v e s t h e p l a n e spanned by Y and Z r i n v a r i a n t and p r e s e r v e s t h e p o s i t i v e de-f i n i t e b i l i n e a r form B^. T h e r e f o r e e x p ( t a d ( X ) ) Z r = - Z r f o r a p p r o p r i a t e t i n E.. (Indeed, i f we n o r m a l i z e X by Bj.(X,X) = 1, one f i n d s by s e r i e s c o m p u t a t i o n s t h a t e x p ( a d ( t X ) ) Z r = c o s ( t ( Z r ( ) Z r + s i n ( t \ Z . \ ) Y 1 r where l Z r l = B ( Z r , Z r ) 7 . So t =-fl" 1 Z r f 1 wi11 do.) Moreover, s i n c e r ( Z ) • = 0 i m p l i e s ad(X)Z = - r ( Z ) Y = 0, the h y p e r p l a n e 'r(Z) = 0 i s p o i n t w i s e f i x e d by e x p ( a d ( t X ) ) , w h i c h t h e r e f o r e g i v e s the r e q u i r e d r e f l e c t i o n i n r . I t s u n i q u e n e s s f o l l o w s from A = R Z r + k e r ( r ) . 12. Lemma. For any r , s i n R, 2 B ( Z r , Z g ) / B ( Z r , Z' ) i s an i n t e g e r . P r o o f . G i v e n r i n R, choose X and Y as above. Then i n t h e c o m p l e x i f i e d L i e a l g e b r a G Q o f G e x p ( a d ( i t Z ) ) X = c o s ( t r ( Z ) ) X + i . s i n ( t r ( Z ) ) Y f o r a l l Z i n A and a l l r e a l t . ( T h i s i s a g a i n v e r i f i e d by s e r i e s c o m p u t a t i o n s . ) I n p a r t i c u l a r , i f r ( Z ) = 2 , e x p ( a d ( i Z ) ) X = X, so e x p(ad(iZ))"commutes w i t h e x p ( a d ( t X ) ) f o r a l l r e a l t , and - 14 -hence w i t h the r e f l e c t i o n w r i n r found above. I t f o l l o w s t h a t e x p ( a d ( i Z ) ) = e x p ( a d ( i w r . Z ) ) . On the o t h e r hand w .Z = Z - (2r(Z))Z„ r r = Z - (4 / r ( Z r ) ) . So i f X o i s i n G , t h e n s —s X S = l . X S '•= e x p ( a d ( i ( Z - w . Z ) ) ) X x* s •= exp(4 i . s ( Z r ) / r ( Z R ) ) X S , i . e . 2 s ( Z r ) . / r ( Z r ) = 2 B ( Z r ,Z g ) / B ( Z r ,Z S) i s an i n t e g e r . I t f o l l o w s from 11.- and 12. t h a t R i s a r o o t system i n i A*, and t h a t W c o n t a i n s the Weyl grotip W of t h i s r o o t system. S i n c e W a c t s t r a n s i t i v e l y on the Weyl chambers of R, the a s s e r t i o n W = W' i s a consequence o f 13. Lemma. W a c t s f r e e l y on the Weyl chambers o f R. P r o o f . The r a t h e r s t a n d a r d argument runs as f o l l o w s : Suppose w i n W maps a Weyl chamber C i n t o i t s e l f . S i n c e C i s convex, i t c o n t a i n s an element X Q f i x e d by w: we may t a k e X Q = 1/n ( X + wX + ... + w N X ) , where n i s the o r d e r o f w. A r e p r s e n t a t i v e m i n f o r w t h e n c e n t r a l i z e s the c l o s u r e T Q o f e x p ( a d ( i l X ) ) i n Aut(G_Q). S i n c e To i s a t o r u s i n the ((compact) T -T a n a l y t i c subgroup of A u t ( G ^ ) w i t h L i e a l g e b r a G + i G ., the c e n t r a l i z e r o f To i n t h i s subgroup i s connected , ( c f . ( 4 ) , Cor.2.8 - 15 p.247). So m = e x p ( Y ) f o r a p p r o p r i a t e Y i n ( G T +1G~ T) X°. By 5. ( a ) t h i s i m p l i e s t h a t m c e n t r a l i z e s A. Hence- w i s the i d e n t i t y i n W. ST L e t R A be the s e t o f r o o t s r i n R f o r w h i c h G ^ 0. U — r I n g e n e r a l R^ i s n o t a r o o t system i n A*, s i n c e i t need not span A*. I n s p i t e o f t h i s I d e f i n e the "Weyl g r o u p 1 W Q o f R " t o be t h e subgroup o f W,. g e n e r a t e d by t h e r e f l e c t i o n s i n t h e r o o t s i n R Q , and a "Weyl chamber o f R Q " t o be a c o n n e c t e d component o f {x i n A : r ( X ) 4- 0 f o r a l l r i n . These Weyl chambers o f R Q a r e s t i l l convex, and the u s u a l argument ( c f . ( ^ i - ) , Theorem 2.12, p. 248) shows t h a t W Q permutes the Weyl chambers o f R^ t r a n s i t i v e l y . 14. P r o p o s i t i o n . WQ = ( H n K f c ) / ( H H K ^ - and permutes t h e Weyl chambers o f R^ s i m p l y t r a n s i t i v e l y . ST , c T P r o o f . G r f 0 i f . , and o n l y i f , (G^. + G_ r) ' =f 0 ( s e e 3 . ( b ) ) . R e f e r i n g t o t h e p r o o f o f 11., we see t h a t f o r r i n R^ we may S T choose t h e elements X and Y so t h a t X i s i n ( G r + £- r) ' and Y i n (G + G _ r ) " S ' ~ T . Then exp(tT\ZX \ X) l i e s i n ( H O K ) 0 , I t f o l l o w s t h a t (Hr^ K ) O A / ( H O K ) - c o n t a i n s WQ. T r i v i a l m o d i f i c a t i o n s o f the p r o o f o f 13. show t h a t ( H r \ K ) 3 A / ( H n K £ p a c t s f r e e l y on the Weyl chambers o f R Q. S i n c e W permutes t h e s e chambers t r a n s i t i v e l y , the p r o p o s i t i o n - 16 -f o l l o w s . ST S T 15. Remark. R Q i s t h e r e s t r i c t e d " r o o t system" o f G = G '' + ~S -T ST G ' . Of c o u r s e , G need n o t be s e m i s i m p l e , b u t a l w a y s i s r e d u c t i v e (even r e d u c t i v e i n G, b e i n g i n v a r i a n t under,a C a r t a n i n v o l u t i o n ) 16. Examples. ( a ) L e t G' be a s e m i s i m p l e r e a l L i e a l g e b r a , G = G T X G' t h e d i r e c t p r o d u c t . D e f i n e i n v o l u t i o n s S and T o f G by S(X,Y) = (Y,X) and T(X,Y) - ( T ' ( X ) , T ' ( Y ) ) , where T' i s any C a r t a n i n -v o l u t i o n of G'. T i s a C a r t a n i n v o l u t i o n o f G commuting w i t h S. I f A' i s a maximal a b e l i a n subspace o f G' , t h e n A. = {(X, -X) : X i n A') -S,-T i s a maximal a b e l i a n . s u b s p a c e o f G , w h i c h I s h a l l : now • • i d e n t i f y w i t h A ' i n t h e o b v i o u s way. W i t h t h i s conven-t i o n : G r = • G ^ X G r R = R Q = R' K- = K X K ' — K A = { ( k l > k 2 ) - : . k 1 k 2 1 i n K'~> k l 5 k 2 i n K'} W = W Q = W ' . 17 -( b > G > s o ( P , o ) . L e t e ( 1 . s t a n d a r d b a s i s o f *n 1 ' = P + q) be the ' ° e f l n e a m a t r i x s by S e i - f o r i M i 2 i i s e i = e f Q r i = s e „ = _p n e n„ n-1 and d e f i n e an i n v o l u t i o n S o f G by S(X) = - s X t s ( X t t h e t r a n s -pose o f X ) . T (X) = -X*" d e f i n e s a C a r t a n i n v o l u t i o n w h i c h commutes w i t h S. One e a s i l y v e r i f i e s t h a t • H = G S = [X i n G : X e n .= 0 i = SO(p,q-.l) K = G T = {X i n G : X(Re1 + . . . E ' e p ) £ . E e i + . . . +*e \ = SO(p) K SO(q). -S -T A maximal a b e l i a n subspace o f G : ' i s : - S 0 ( 1 , L ) . Set G = SO(p,q)Q, w h i c h may be r e g a r d e d as a subgroup o f A u t ( G ) by t h e a d j o i n t r e p r e s e n t a t i o n . ( R e c a l l t h a t SO(p,q) i s the subgroup o f GL(n) w h i c h l e a v e s t h e q u a d r a t i c form 2 2 2 2 2 ' ~x^ -x^ - ... -Xp + X p + i +...+xn i n v a r i a n t and whose elements have an upper l e f t hand pxp s u b m a t r i x o f p o s i t i v e d e t e r m i n a n t . ) Then: G ^ ~{g i n G •-etexnaW o t h e ° 1 M d C h e - 18 -K A = i g l n G : g 6 l = - 6 1 ' g 6 n = - e n ' a n d g ^ e 1 + . ' * ' +le ) C 1 P + . . . +E.e { P 1 P J . ( H n K ) = f g i n G :• g e L = +e],, g e n = e n , and ' g ( R e 1 + . . . + E e ) C R e 1 + . . _ . +Re p^j. T h e r e f o r e : W ^ z2 WQ = W, u n l e s s p = 1, i n w h i c h case WQ i s t r i v i a l . F o r the l a s t a s s e r t i o n one has t o observe t h a t f o r p = 1 e v e r y element o f (HCiK) f i x e s e, , because o f the r e s t r i c t i o n on A 1 the s u b d e t e r m i n a n t s of elements i n SO(p,q)Q. There a r e o n l y two r o o t s , R = [ + r | ; t h e c o r r e s p o n d i n g r o o t spaces a r e G + r |X i n G : X(e± + e n ) = 0, and f o r i 4 l , n Xe ^ e t ( e i + e n ) j V i - : -( c ) G = S L ( n ) . D e f i n e a m a t r i x s by se^ = -ej_ f o r i = 1,2, ...p, se^ = e~ f o r i - p+1, ....p+q = n and d e f i n e an i n v o l u t i o n S o f G by S(X) =-sX cs. A C a r t a n i n v o l u t i o n commuting w i t h S i s g i v e n by T(X) = -X t. . The d i a g o n a l m a t r i c e s i n G form a maximal a b e l i a n sub--S -T space o f G ' , i n d e e d even a maximal a b e l i a n subspace o f -T G . R t h e r e f o r e c o i n c i d e s w i t h the u s u a l r e s t r i c t e d r o o t system o f G. We f i n d : •'• - 19 ;-H G S - SO(p,q) K = G T = SO(n) c o n s i s t s o f m a t r i c e s o f the form e^ i e i t where iH->i'- runs over t h e p e r m u t a t i o n s o f 1,2,...n. ( H A K ) ^ c o n s i s t s m a t r i c e s i n • c o r r e s p o n d i n g t o p e r m u t a t i o n s w h i c h l e a v e the s u b s e t s l , 2 , . . . p and p+l,...p+q=n i n v a r i a n t . T h e r e f o r e • w - s p + q w 0 ~ S 0 K S n . - 2 0 -Chapter I I . S p h e r i c a l d e c o m p o s i t i o n s . The C a r t a n d e c o m p o s i t i o n "G = KAK" of a connected,semi-s i m p l e , r e a l L i e group G g i v e s r i s e t o " s p h e r i c a l c o o r d i n a t e s " on t h e Riemannian symmetric space G/K. I n t h i s c h a p t e r >-. analogous c o o r d i n a t e s on a l l s e m i s i m p l e symmetric spaces w i l l be d e r i v e d from a v e r y g e n e r a l f i b r a t i o n theorem f o r symmetric spaces ( 8 ) . -L e t G be a c o n n e c t e d , s e m i s i m p l e , r e a l L i e group, S an i n v o l u t i v e automorphism of.'G, T a C a r t a n i n v o l u t i o n o f G com-S muting w i t h S. L e t H be a subgroup o f the f i x e d p o i n t s e t G S S o f S c o n t a i n i n g (G ) Q , the connected subgroup of G . The T f i x e d p o i n t s e t G = K o f T i s connected*.and compact. mod.;the z e n t e r of G. The i n v o l u t i o n s of the L i e a l g e b r a G o f G c o r r e s p o n d i n g t o S and T w i l l be denoted by the same l e t t e r s , and the no-t a t i o n a l c o n v e n t i o n s o f c h a p t e r I . w i l l r e m a i n i n f o r c e . The f i b r a t i o n theorem r e f e r r e d t o above a s s e r t s t h a t the map KX.G~ S'~ TXH —-> G, (k,X,h) W » k e x p ( X ) h , i s a s u r j e c t i o n w i t h f i b e r [ ( k g " 1 , A d ( g ) X , g h ) : g i n Hr\K above k e x p ( X ) h ) . Now ~ S — T 1 a c c o r d i n g t o 1.8, G ' =.{Ad(g)A : g i n HOK), S O i f A = exp(A) i s the a n a l y t i c subgroup o f G w i t h L i e a l g e b r a A, th e n the m u t i p l i c a t i o n map A x H G i s a s u r j e c t i o n w i t h f i b e r [ ( k g \ g a g \ g h ) : g i n Hr\ K so t h a t g a g ~ l i s i n A} above kah. 21 -Moreover, i f a 4 1 whenever r i s a r o o t i n R Q , t h e n G ' ' = A, by 1.5. ( b ) , and t h e n the f i b e r above kah i s | ( k g 1 , g a g \ g h ) g i n (HnK) = the n o r m a l i z e r o f A i n H 0 K | . Set A' = (a i n A : a r ^ 0 whenever r i s i n R Q j . Then th e s e o b s e r v a t i o n s a r e summerized i n 1. P r o p o s i t i o n . The map K x A X H ->G, (k,a,h) Ws> k a h , i s s u r -j e c t i v e and a submersion w i t h f i b e r {(kg \ g a g ^,gh) : g i n (Hr\K) A5 above kah on K X A ' X H. P r o o f . I t remains t o show t h a t t h i s map i s a submersion on K X A ' X H . For t h i s i t s u f f i c e s t o compute i t s t a n g e n t map a t the p o i n t s (e,a,e) w i t h a I n A r. I d e n t i f y i n g the t a n g e n t space of K-X A XH at: (e,a,e) w i t h K X A X H, t h i s t a n g e n t map i s g i v e n by (X,Y,Z) r~> Ad(a)X + Y + Z. S i n c e Ad (a )KO H =' a A (HO K) = (HOtK)- (by 1.7. ), c o m p a r i s o n of dimensions shows t h a t Ad(a)K+ A + H = G, so t h a t the t a n g e n t map i s i n d e e d s u r j e c t i v e . Set W = (HnK) / ( H r \ K ) A . ' WT may be c o n s i d e r e d as a • A A A subgroup o f W = K /K^ c o n t a i n i n g W^ . = (Hf) K ) Q a / . ( H D K ) Q . Choose a fundamental domain f o r t h e a c t i o n o f WT on A and l e t -A+ be i t s i n t e r i o r . ( I t i s c l e a r t h a t we may t a k e A + t o be the image under exp:A-^A o f the u n i o n ' o f c e r t a i n Weyl chambers of J .R - 22 -c o n t a i n e d i n a f i x e d Weyl chamber o f RQ}« Then 1. may be r e p h r a s e d as 2. P r o p o s i t i o n . The map K / ( H r \ K ) A * A + G / H , (k(Hn K) A;as) v kaH, i s a d i f f e o m o r p h i s m onto an open dense s u b m a n i f o l d o f G/H. I s h a l l r e f e r t o t h i s map as " s p h e r i c a l c o o r d i n a t e s " on G/H. The r e l a t i o n between W' and WQ can be made more e x p l i c i t : 3.Lemma. W'/Wo = H / H Q , w h i c h i s o m o r p h i c w i t h <ar f i n i t e ' d i r e c t p r o d u c t o f c y c l i c g roups o f o r d e r two. P r o o f . /The p r o o f r e l i e s on a n o t h e r v e r s i o n o f t h e f i b r a t i o n -S ~T S -T theorem, w h i c h a s s e r t s t h a t t h e map K *G ' XG ' .—^ G, (k,X, Y) r-> kexp(X)exp.(Y) , i s a c t u a l l y a d i f f eomorphi sm ((s)r I', p.161). Thus G S = K S * G S ' " T , and H = (H P , K ) e x p ( G S ' ~ T ) . There-f o r e H / H Q = ( H n K ) / ( H n K ) ' o . To prove H / H Q = W'/WQ i t remains t o show t h a t e v e r y c o n n e c t e d component o f Rr\K c o n t a i n s an element n o r m a l i z i n g A. T h i s can be seen i n t h e same way a s 1.8 : F o r any X i n A t h e f u n c t i o n g t-»B(Ad(g)X,X) must have a c r i t i c a l p o i n t on e v e r y c o n n e c t e d component o f HAK. I f g i s such a c r i t i c a l p o i n t , t h e n |Ad.(g)X,x] = 0, as b e f o r e . X - S — T Thus Ad(g)X l i e s i n G ' '. -Moreover, i f r ( X ) j= 0 whenever i s i n R Q , t h e n Ad(g)X has t h e same p r o p e r t y , and t h e r e f o r e -23 -Ad(g)A = A d ( g ) G X ' " S ' " T = G A d ( 8 ) X ' " S ' " T =• A, i . e . g l i e s i n ( H A K ) A , F i n a l l y , t h e f a c t t h a t H / H Q . i s i s o m o r p h i c t o a f i n i t e d i r e c t p r o d u c t o f c y c l i c groups o f o r d e r two i s a g e n e r a l p r o p e r t y o f symmetric spaces. ( (8)1 p . 1 7 1 ) . 4. Lemma. W i t h a p p r o p r i a t e n o r m a l i z a t i o n o f t h e i n v a r i a n t measures: J f ( g ) d g = J . - f (kah) |©('a)| dk da dh G/H K x A x H where I^Y \ T r ( r - r N m + ( r ) , x , - r \ m " ( r ) 8 ( a ) = II (a - a L ) • (a + a ) ' r i n R m + ( r ) - dim m~(r) = dim G r ^ and R i s a s u b s e t of R c o n t a i n i n g e x a c t l y one element o f each p a i r +r. Proof.,. Denote: by (KX A X t i ) / ' ( H A K ) A : the q u o t i e n t ' o f K K A f H by the a c t i o n - ( k ' ; a , h ) \—T> ( k m T l ,a,mh) o f (H/"\K) . I f we i d e n t i f y (Hf- \K)~ w i t h the sub space [ ( - X , 0 , X ) : X i n ( H A . K ) - } o f K KA A H • A we'can i d e n t i f y t he t a n g e n t space, o f (K X A > H ) / ( H f l K ) a t (e,e,e) w i t h : ( K ^ A XH)/(Hf> K ) - . The ta n g e n t ' s p a c e a t ' a n a r b i t r a r y point- : o f : (Kx A X H ) / ( H can "then be mapped i s o m o r p h i c a l l y A • onto' ( K . X A K H ) / ( H A . K ) - by a l e f t t r a n s l a t i o n i n K, a ( l e f t - -r i g h t ) , t r a n s l a t i o n i n A , r ,and a r i g h t t r a n s l a t i o n i n H. A f t e r -24 such i d e n t i f i c a t i o n s t h e t a n g e n t map o f the s u r j e c t i o n ( K x A x H ) / ( H ^ K ) A - ^ > G a t (k,a,h) i s g i v e n by. t h e map (K x A x H)./ ( H O K ) -G 3 . ( X , Y , Z )R > A d ( h ~ 1 ) ( A d ( a " 1 ) X + Y + Z ) \ ( u p t o c o m p o s i t i o n w i t h en i n n e r automorphism from ( H r \ K ) ^ , w h i c h depends on the c h o i c e o f the r e p r e s e n t a t i v e ( k , a , H ) ) . The i n t e g r a l f o r m u l a i s e q u i -v a l e n t t o t h e a s s e r t i o n t h a t 0 ( a ) i s . a c o n s t a n t m u l t i p l e o f the " d e t e r m i n a n t " o f t h i s l i n e a r map ( t h e " d e t e r m i n a n t " b e i n g com-p u t e d w i t h r e s p e c t a r b i t r a r y - b u t f i x e d - b a s e s i n (K X A X H)/(HO | ) -and i n G). T h i s d e t e r m i n a n t i s e v i d e n t l y t h e same.as t h a t o f t h e map R / ( H r , K ) — > G / ( A + H ) , X + ( H A K ) ~ & A d ( a _ 1 )X + ( A + H ) . Composing w i t h t h e isomorphisms K / ( H A K ) ^ = K - ' " S + £ (G„ + G _ r ) T and G/(A + H) = K A ' " S + £ (£r•: + G _ r ) ~ S we g e t a map w h i c h sends K—3 i d e n t i c a l l y onto i t s e l f and w h i c h maps.'(G '+ G_ r) onto ( G r + G _ r ) ~ S by s e n d i n g X + T ( X ) ( X i n G ) i n t o A d ( a _ 1 ) ( X +. T ( X ) ) - S ( A d ( a - 1 ) ( X + T ( X ) ) ) .= ( a " r X - a r S T ( X ) ) - S ( a ~ r X a r S T ( X ) ) . •ST - CT N F r o m the d i r e c t d e c o m p o s i t i o n G = G + G one sees t h a t t h e r e q u i r e d d e t e r m i n a n t i s i n d e e d 9 ( a ) = T T ( a r - a ~ r ) m ^ r • I n v i e w o f 2. a d i f f e r e n t i a l o p e r a t o r on G / H c a n be p u l l e d back t o a d i f f e r e n t i a l o p e r a t o r on K / ( H n K ) A X A + , w h i c h I s h a l l c a l l i t s " e x p r e s s i o n i n s p h e r i e a D . c o o r d i n a t e s " . F o r a p p l i c a t i o n s t o a n a l y s i s one needs a f o r m u l a f o r the L a p a c i a n on G / H i n s'v s p h e r i c a l c o o r d i n a t e s . ( G / H i s t o be r e g a r d e d as a pseudo -- 25 -Riemannian m a n i f o l d under the unique G - i n v a r i a n t m e t r i c w h i c h c o i n c i d e s w i t h the K i l l i n g form o f G on the t a n g e n t space G a t the base p o i n t eH.) The d e r i v a t i o n o f such a f o r m u l a r e -q u i r e s some p r e l i m i n a r y remarks on i n v a r i a n t d i f f e r e n t i a l o p era-t o r s . I n t r o d u c e the n o t a t i o n " R Q " and " L Q " ( o r s i m p l y " R " and " L " i f c o n f u s i o n i s u n l i k e l y ) f o r the r i g h t and l e f t r e g u l a r r e p r e -s e n t a t i o n s o f G on complex v a l u e d f u n c t i o n s on G . ( Thus R ( g ] L ) f ( g 2 ) = f ( g 2 g 1 ) and L ( g 1 ) f ( g 2 ) = f ( g x X g 2 ) . ) Denote t h e c o r r e s p o n d i n g r e p r e s e n t a t i o n s o f the u n i v e r s a l e n v e l o p i n g a l g e b r a U ( G Q ) o f G Q on d i f f e r e n t i a b l e f u n c t i o n s by the same l e t t e r s . ( F o r X i n G, R ( X ) f ( g ) = f t f ( g e x p ( t X ) ) U = 0 , L ( X ) f ( g ) = | t f ( e x p ( - t X ) g ) I t = 0.) I d e n t i f y i n g : f u n c t i o n s on G / H w i t h r i g h t H - i n v a r i a n t f u n c t i o n s on G and w r i t i n g f * f o r the p u l l - b a c k o f a f u n c t i o n f on G / H by the map K K A —>G'/ H,. ( k , a ) H > kaH, we g e t iR A ( X ) f * ( k , a ) , i f X i s i n U ( A C ) R K ( A d ( a ) X ) f * ( k , a ) , i f X i s i n A d ( a _ 1 ) U ( K c ) (where R and Rv a c t on the f a c t o r s A and K o f K x A ) . A k F o r a i n A + , G = A d ( a _ 1 ) K + A + H (see p r o o f o f 1. )hence U(G_c) '= ( A d ( a " 1 ) U ( K c ) ) ' U ( A c ) * U ( H c ) . Thus any element i n U ( G C ) can be w r i t t e n as X = 1 Ad(a" 1)x£(a)-X A(a)Txj I(a) w i t h x£(a.) m U ( K C ) , X A ( a ) i n U ( A C ) , and x ^ ( a ) i n U ( H C ) . U s i n g the o b v i o u s f a c t t h a t R Q ( U ( G _ c ) H ) a n n i h i l a t e s the r i g h t H - i n v a r i a n t - 26 -f u n c t i o n s on G we see t h a t •(RgCX'JfXkja) = L R K ( X ^ ( a ) ) - R A ( X ^ ( a ) ) f * ( k , a ) , where the summation runs over those i n d i c e s i f o r w h i c h x j j ( a ) l i e s i n C. I f X i s an A d ( H ) - i n v a r i a n t element i n U(GQ), t h e n R f i s G r i g h t H - i n v a r i a n t whenever f i s (because R ( g ) R ( X ) f = R ( A d ( g ) X ) * R ( g ) f ) , so t h a t Rg(X) can be r e s t r i c t e d t o an o p e r a t o r R Q / H ( X ) on r i g h t H - i n v a r i a n t f u n c t i o n s . R ^ ^ H ( X ) can o f c o u r s e be r e -garded as a ( G - i n v a r i a n t ) . d i f f e r e n t i a l o p e r a t o r on G / H , and i t can be shown t h a t every G - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r on G / H can be o b t a i n e d by t h i s c o n s t r u c t i o n ( ( 4 ) p . 3 9 0 ) . Moreover, f o r an A d ( H ) - i n v a r i a n t element X i n U ( G Q ) the elements X X ( a ) i n U ( K Q ) can be chosen t o be Ad(Hr\K) - i n v a r i a n t (because t h e a c t i o n o f Ad(H/^K) i s c o m p a t i b l e w i t h the d e c o m p o s i t i o n G = Ad(a - i)K:'+ A + H) so t h a t the o p e r a t o r s R^.(X^(a)) can t h e m s e l v e s be r e s t i c t e d t o K - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on K / ( H A , K ) A . Combining t h e s e o b s e r v a t i o n s we a r r i v e a t the f o l l o w i n g r e c i p e : I f X i s an A d ( H ) - i n v a r i a n t element i n U X G ^ . ) , w r i t e X 5 % x£(a)°X^(a) mod u C G ^ H j W i t h Xfc(a) i n U ( K C ) and X^(a) i n U ( A Q ) . Then the e x p r e s s i o n o f Rgy-^(X) i n s p h e r i c a l c o o r d i n a t e s i s ^ R K / ( H ^ K ) ( x T i ( a ) ) , R A ( X A ^ a ) ) -To a p p l y t h i s p r e s c r i p t i o n t o the L a p l a c i a n A on G / H one has t o n o t e t h a t A = R Q / H ( Q ) where Q . i s the C a s i m i r element i n U ( G ) . To see t h i s , r e c a l l t h a t £1 = Z ± ( X j L ) 2 where { X ^ i s any - 27 -B -orthonormal b a s i s o f G (B(X^,X .) - +<^i 5j) a n d the s i g n on 2 1 J X i n the summation agree s w i t h B ( X . , X . ) . So i f f i s r i g h t i I I j H - i n v a r i a n t , then RrX)f(e) « X 1 ( 3 t ) 2 ' f ( e x P ( t X , ) ) | t = 0. where the summation can be r e s t r i c t e d t o a B-orthona'ormal b a s i s {X±{ o f G~ S. On the o t h e r hand, a c c o r d i n g t o a well-known f o r m u l a & f ( e H ) = I.^OZ.fCExpCtX^lt = -0 -S ° where Exp: G —^G/H i s t h e e x p o n e n t i a l map a t eH-of the pseudo-Riemannian ; m e t r i e . Because o f the r e l a t i o n Exp(X) = exp(X)H, v a l i d f o r a l l X i n G" S, w e . f i n d t h a t R (^0f (eH) A f (eH) f o r a l l G/H d i f f e r e n t i a l b l e f u n c t i o n s f on G/H. From the G - i n v a r i a n c e o f b o t h d i f f e r e n t i a l o p e r a t o r s we c o n c l u d e t h a t RQ/H(C0 ~ A. i d e n -t i c a l l y . . F o r the c o m p u t a t i o n o f RQ/H(£X) i t i s c o n v e n i e n t t o choose a b a s i s o f G c o m p a t i b l e w i t h the B - o r t h o g o n a l d e c o m p o s i t i o n G = G A + T (G_r + G _ r ) - S ' - T . So f o r r i n R and i , j = +1 p i c k a b a s i s : U k ^ / ^ r ) ] f o r ( G r + . G _ r ) l S ' ^ s a t i s f y i n g B ( X U ' ^Xv'*') .= - j i f u-r = v a n d = 0 o t h e r w i s e , •r r r r We can f u r t h e r a r r a n g e t h a t f o r a l l i n d i c e s k r the elements X^' J and X^1''"-3 a r e r e l a t e d as the X and Y i n 1.6. ( b ) ; To s i m p l i f y n o t a t i o n , f i x r and k r and omit th e s u b s c r i p t k r . i . j i-jST F o r i , j - 2.1: , t h e r e i s a unique element X J m G_r so t h a t x i , j = ^ x i ' J + i S ( X i * j ) ) ^ ( X 1 ^ + j T C X 1 ^ ) ) . - 28 -Thus i n p a r t i c u l a r ^ f o r j = +1: A d ( a 1 ) ( X i ' + ) = ^ ( a ^ X 1 + a ^ C X 1 ) ) = h(ar + a " r ) ( X i ' + ) - h(ar - . ' a " r ) ( X _ i ' " ) and ( c o n t i n u i n g the c o m p u t a t i o n i n U(G)) we g e t by s q u a r i n g : : A d ( a _ 1 ) ( X i , + ) 2 . = (i>(a r + a - ^ ) X i ' + ) 2 + ( % ( a r - a~^)X'l>') - % ( a 2 r - a " 2 r ) ( X i ' + . X " i ' - + X _ i ' " . X i y + ) . C o n s i d e r i n g t h e c a s e s • i = + l and i =-1 s e p a r a t e l y and u s i n g ; " " t o i n d i c a t e congruence mod U(G)H we f i n d ( i n v i e w o f f x 1 ' - ' j X - x ' J j = j Z r ) : A d ( a - 1 ) ( X + ^ + ) 2 = (%(a r' - a r ) ) 2 ( X ^ ' " ) 2 - % ( a 2 r - a " 2 r ) Z r AdCa^Xx"'4")2 - ( M a r •. a " r ) ) 2 ( x " ' + ) 2 + % ( a 2 r - a" 2 r)Z„. S o l v i n g f o r ( X ~ ' " ) 2 and f o r ( X ~ > + ) 2 : r ( X " ' " ) 2 •= ( 3 2 ( a r - a - r ) ) " 2 A d ( a " 1 ) ( X : ' ' ) 2 + a^-a"? Z r J r ~ a r - a ~ r (X~>T = (i;(a + a " J : ) ) " z A d ( a " J - ) ( x " ' " r ) ^ - a r + a ~ r z r . So i f j x ^ \ i s a B-orthonarmal b a s i s f o r G—, th e n :-- a r + a ~ r a r - a " r = 7 +(X k ) 2 +) [ m + ( r ) a r - ; . a - r +m~(r) a 1 + a~ r} Z r + A d ( a - l ) ^ (i>(a r - a- r))- 2(x£; +) 2 - AdCa" 1) £ O i ( a r + a - r ) ) - 2 ( X ^ 5 2 . I +ST +S T (As e a r l i e r , m-(r) = dim(G ) - = dim(G •+ G_)~ ' .) J- IT 1 3 j T h i s f o r m u l a a c t u a l l y h o l d s f o r any B-orthonormal b a s i s | x^ . ^ ,X^ .^ J c o m p a t i b l e w i t h the d e c o m p o s i t i o n G = G- + Z" (G + G_ r ) l S > J T * - 29 -I n d e e d , t h e r e a r e ( w e l l - d e f i n e d ) elements i n U(G) g i v e n by: STL A = X : X i n a B-o-.n. b a s i s f o r A A • • _ J _ Q A = { X 2 : X i n a B-o.n. b a s i s f o r K A ' " S ^ +S T 1 _n~r = [ x 2 : X i n a B-o.n. b a s i s f o r (G_r + G _ r ) ~ ' j . S i n c e a l l o f t h e s e elements a r e i n v a r i a n t s o f Ad(Hr*\K) A o u r •: a l g o r i t h m g i v e s : 5. P r o p o s i t i o n . The e x p r e s s i o n o f the L a p l a c i a n on G/H i n s p h e r i c a l c o o r d i n a t e s i s the f o l l o w i n g d i f f e r e n t i a l o p e r a t o r A 4-on K/(Hf\K) x A : " \ / ( H A K ) A ( i i K A ) - ? _ ( ^ ( a • a ) ) \ / ( H a K ) A ( r i r>-6. Remarks. The o p e r a t o r on A + d e f i n e d by t h e f i r s t l i n e o f the f o r m u l a i s the r a d i a l p a r t o f the L a p l a c i a n on G/H w i t h r e s p e c t t o the a c t i o n o f K. I t c a n a l s o be w r i t t e n as 6 ;R A(jn A )oe^ - e~'\ca A ) (e-0 (8 as i n 5.) i n agreement w i t h a r e s u l t o f S . Helgason ( ( 7 ) p . l 5 ) , F o r fixed-. ..a i n A the o p e r a t o r on K / ( H A K ) d e f i n e d by t h e r e -m a i n i n g t h r e e l i n e s can be i d e n t i f i e d w i t h the L a p l a c i a n on the K - o r b i t o f the p o i n t aH i n G/H, r e g a r d e d as a pseudo- Riemannian s u b m a n i f o l d o f G/H. - 30 -7. Example. The g l o b a l v e r s i o n o f 1.16.(b) a r e t h e " r e a l hyper-b o l i c s p a c e s " S 0 ( p , q ) o / S 0 ( p , q - l ) Q . I n t h i s c a s e : G = S 0 ( p , q - l ) o , t h e c o n n e c t e d subgroup t o S0(p,q-1) . The i n v o l u t i o n s S and T a r e g i v e n by S(g) = s ( g - l ) t s T ( g ) = ( g - l ) t where s i s as i n 1.16.(b). H - = [g i n G K = { g i n G A =• [ g i n G S e n = en\ = S 0 ( p , q ~ l ) o (n = p+q) g(Re L+..,+Re ) C Re 1+...+le p\ = SO(p) SO(q) ge-^ = cosh(t)e.j- + s i n h ( t ) e n g.e„ = s i n h ( t ) e n v + i c o s h ( t ) e n l . n (some t i n E.) I i f i f l , n \ = S 0 ( l , l ) o (HA K ) A = { g i n (H K ) : g e 1 = e 1 } . Thus G/H may be i d e n t i f i e d w i t h the h y p e r b o l o i d X =|x i n R n: ( x , x ) — $ t ^ -x-, 2- ... -x 2+x v 2+...+x 2 = 1 and K / ( H r \ K ) A w i t h S p _ 1 X S q _ 1 i p p i 1 n e x c e p t when q = 1 (Riemannian c a s e ) t w h e r e one has t o add the r e s t r i c t i o n x n> 0 t o i n s u r e the connectedness o f X. The map K / ( H a K ) A K A — G / H , (k,X) \->kexp(X)H, i s t h e n i d e n t i f i e d w i t h the map s P " 1 ^ S q _ 1 x i — > X, ( x , y , t ) H > ( c o s h ( t ) y , s i n h ( t ) z ) . - 31 -Chapter I I I . P a r a b o l i c d e c o m p o s i t i o n s . By a " p a r a b o l i c d e c o m p o s i t i o n " I mean the d e c o m p o s i t i o n of a s e m i s i m p l e symmetric space G/H i n t o the o r b i t s o f a c e r -t a i n p a r a b o l i c subgroup of G. For Riemannian spaces t h i s i s p r o v i d e d by an Iwasawa. d e c o m p o s i t i o n , f o r L i e groups by a B r u h a t d e c o m p o s i t i o n . I n a d d i t i o n t o the n o t a t i o n s and c o n v e n t i o n s i n t r o d u c e d i n the. p r e v i o u s c h a p t e r s I^'shalL now-assume , t h a t G .has. f i n i t e z e n t e r . Choose a l i n e a r o r d e r i n A* and s e t N =. T_ G r. Then N ; i s a n i l -p o t e n t s u b a l g e b r a of G and A + N i s s o l v a b l e . 1. Lemma. G = H + (K--: + A -•• N), z ••• • . . The sum ( K ~ + A + N) i s d i r e c t and H C\ (K- + A + N) = ( H O K ) ~ . P r o o f . We know t h a t G = H + G _ S and t h a t G_"S = G_-'~S + T_ ( G r + G _ r ) ~ S S i n c e G^.'"S»T K- and dim ( G r +'G_ r)" S = dim(N) i t s u f f i c e s t o show t h a t (H + K— + k)(\ N = 0. Now i f X i s i n N, then.X - S.(.X) i s i n Z. G r. On the o t h e r hand i f X i s i n (H + K- + A ) , then •r e R X - S(X) i s i n G-. So i f X i s i n (H + + A) N 3 t h e n X - S(X) i s i n G- Q ]F G r = Q. Thus X = S(X) and c o n s e q u e n t l y X i s i n HryN = 0. L e t N be the a n a l y t i c subgroup of G w i t h L i e a l g e b r a N. - 32 -Then N i s a c t u a l l y c l o s e d i n G and the e x p o n e n t i a l map i s a d i f f e o m o r p h i s m of N onto N ( (4) p.225). S i n c e G A n o r m a l i z e s N i t i s c l e a r t h a t G^N i s a subgroup of G. I n f a c t : 2. Lemma. Pr G A N i s a p a r a b o l i c subgroup of G. 1 -T P r o o f . E x t e n d A t o a maximal a b e l i a n subspace A ' of G and l e t G = G~. + Z. G r be t h e r o o t space d e c o m p o s i t i o n of G under A ' . Then f o r any r i n R, G r = ^ [ G_r f • J r r i n R J so t h a t r T | A r j j so R c o n s i s t s p r e c i s e l y o f the n onzero r e s t r i c t i o n s of the r o o t s i n R T t o A . - Choose a n - o r d e r i n - A ' * c o m p a t i b l e w i t h the o r d e r i n A / V I h a t i s r ' > 0 ( r f i n R':) i m p l i e s r ' \ A ^ 0 . So i f - N ' =Y. G . , then N' = N' A + N. Thus G^ + N C G A' •• Nt , and -clearly., ..G, N"' C G N T , i . e . P i s a p a r a b o l i c subg roup o f G. A 3. Remark. G i s r e d u c t i v e , b e i n g i n v a r i a n t under the C a r t a n i n v o l u t i o n T of G. S i h c e N i s n i l p o t e n t we see t h a t G A i s a L e v i subgroup of P, w h i l e N i s i t s u n i p o t e n t r a d i c a l . A i s o b v i o u s l y a s p l i t subgroup of the c e n t e r of G A. But A need no t be a.maximal s p l i t subgroup of the c e n t e r . Indeed P i s the p a r a b o l i c subgroup a s s o c i a t e d w i t h the s e t 9 of s i m p l e r o o t s i n R' v a n i s h i n g on A ( Q-ty p.45 f f . . ) ; but A need not c o i n c i d e w i t h A Q = [ x . i n A T : r ' (X) = 0 whenever r ' J A = 0^ . The f a c t t h a t A r a t h e r then A Q i s r e l e v a n t t o the s t r u c t u r e of the symmetric space G/H can be e x p l a i n e d as f o l l o w s : S i n c e -33 s - s - s A Q i s s t a b l e under S, AQ = AQ + AQ . .Moreover, AQ = A and t h e r e f o r e A Q = A mod H (even mod H ^ ) . T h i s .seems .'to i m p l y . t h a t o n l y t h o s e r e p r e s e n t a t i o n s of the P - s e r i e s of G w h i c h A a r e i n d u c e d from r e p r e s e n t a t i o n s of P t r i v i a l on H (and hence t r i v i a l on A Q ) . e n t e r ! i n t o the harmonic a n a l y s i s on G/H. (Compare V. and V I . ) Set N = S(N) = T ( N ) . I n v i e w of the Br u h a t d e c o m p o s i t i o n NP i s open and dense i n G. T h i s f a c t i s the e s s e n t i a l i n g r e -d i e n t i n the p r o o f of the f o l l o w i n g g l o b a l v e r s i o n of 1. : 4. P r o p o s i t i o n . The -union UHWK'AAN, W r u n n i n g over a s e t of A double c o s e t r e p r e s e n t a t i v e s f o r ( H f\K) A\K^/K , i s d i s j o i n t , open, and dense i n G. More p r e c i s e l y , the m u l t i p l i c a t i o n map HxKAXAXN-5>G i s a submersion onto ah open, dense subset o f G w i t h f i b e r (hgTl,gm,a,n) : g i n ( H n K ) A \ above hman. 5. Remarks. ( a ) ( H r \ K )^\ K^/K A i s n a t u r a l l y i s o m o r p h i c w i t h t h e f i n i t e c o s e t space W/W (W = K A/K A, WT •= (H 0 K ) A / ( H n K ) A as b e f o r e ) . (b-)We' s h a l l see l a t e r t h a t G A = H AAK A. T h i s i m p l i e s t h a t .for m i n K A HrnP -..HmG^ = HGAmN = iHK^AmN ==.;. HmKAAN. So th e o r b i t o f mH i n G/H under P c o i n c i d e s w i t h the o r b i t of mH A under K AN. The p r o p o s i t i o n t h e r e f o r e shows t h a t the r e g u l a r - 3.4 -(=open) o r b i t s o f P ( o r , e q u i v a l e n t l y , of K . " A N ) i n G/H a r e p a r a m e t r i z e d by W/W and t h a t t h e i r u n i o n i s open and dense i n G/H, The i d e a of the p r o o f of 4. i s t o embed the symmetric space G/G d i f f e o m o r p h i c l l y i n t o G as the s u b m a n i f o l d Gg = £'S(g)g~l : g i n G ] by t h e map g G S H5>S.(g )g and t o de t e r m i n e the d e c o m p o s i t i o n of Gg o b t a i n e d by i n t e r s e c t i n g w i t h the • c open dense subset NP o f G. ( I t i s easy t o check t h a t gG° >—> g S(g)g"^- does i n d e e d d e f i n e a d i f f eomorphi sm of G/G onto Gg, t r a n s f o r m i n g t h e a c t i o n .of G on G / G ^ y l e f t t r a n s l a t i o n i n t o the a c t i o n g ( x ) = S:(g)xg~"1"( x i n G , g i n G) o f G on Gg.) s The d e t a i l s of the p r o o f a r e c o n t a i n e d i n a s e r i e s of lemmas. 6. Lemma. Ggr\NP i s dense i n Gg. P r o o f . For f i x e d g i n G c o n s i d e r the map NXP-^G, (n,p)v-}ngp. L e t g ! : NxP G be i t s t a n g e n t map a t ( e , e ) . S i n c e the com-plement G-NP c o n s i t s p r e c i s e l y of the u n i o n of the s i n g u l a r " I o r b i t s o f the actiongh>ngp: of NXP on G, we see t h a t G-NP = [ g i n G : d e t ( g ' ) = o},. the " d e t e r m i n a n t " of g 1 b e i n g d e f i n e d by i d e n t i f y i n g NXP w i t h N + P = G. - The p o i n t i s t h a t G-NP i s the s e t of z e r o s of an a n a l y t i c f u n c t i o n on G. T h e r e f o r e ? i f an open subset of the c o n n e c t e d , a n a l y t i c s u b m a n i f o l d Gg. - 35 l i e s i n G-NP, the n t h i s f u n c t i o n v a n i s h e s i d e n t i c a l l y on Gg^ and Gg l i e s c o m p l e t e l y i n G-NP. S i n c e t h i s n o t the case (e.g £ S ( n ) n n in'N] i s c o n t a i n e d i n GgO NP) i t f o l l o w s t h a t e v e r y open subset of Gg i n t e r s e c t s NP. 7. Lemma. For any x i n Gg NP t h e r e i s p i n P so t h a t S ( p ) x p A l i e s i n GgO G . P r o o f . Suppose n i s i n N, p i n P, and x = np i n Gg. Then S(x)••= x " 1 , i . e . S ( n ) S ( p ) = p" 1^" 1.. So pS(n) = n ~ 1 S ( p " 1 ) i s i n P G AN =.GA. Thus S ( p ) x p _ 1 = S ( p ) n i s i n G A, G A b e i n g i n v a r i a n t under S. 8.. Lemma. S ( g ) g ~ ^ i s i n G A i f , and o n l y i f , g i s i n K^AG^. P r o o f . S ( g ) g - ^ i s i n G A i f , and o n l y i f , f o r a l l X i n A: A d ( S ( g ) g " 1 ) X = X, i . e . A d ( g _ 1 ) X = A d ( S ( g T i ) ) X = S ( A d ( g _ 1 ) S ( X ) ) = - S ( A d ( g _ 1 ) X ) 5 i . e . i f and o n l y i f , A d ( g ~ 1 ) A CG S . S T h i s i s e v i d e n t l y the case i f g l i e s i n K ^ A C . On the o t h e r hand, i f Ad(g~ 1.)Ac: G~S and g = kah>• w i t h - 36, -k i n K, a i n A, and h i n G Sas i n I I . 1. ,then A d ( h " 1 a ~ 1 k 1 ) A c G _ S , !,-T ,-S, -T so A d ( a " 1 k " 1 ) A C G " ^ a n d Ad(k 1 ) A c" ( A d ( a ) G ~ S ) 0 G " T = G A ' ~ S ' ~ T (see 1.7. ). S i n c e any two maximal a b e l i a n subspaces of G are c o n j u g a t e by an element i n HP\ K (1.9.) t h e r e i s m i n H/l K so t h a t Ad(k )A = Ad(m)A, i . e . m^k ^ i s i n K . Moreover, Pi Ad(m)ACG means t h a t Ad(am)X = Ad(m)X f o r a l l X i n A, i . e . m "'"am i s i n G A. S i n c e m'J:am i s . a l s o c o n t a i n e d I n exp(Ad(m~^ ) A ) , ~S -T which i n t u r n i s c o n t a i n e d i n exp(G ' ), i t f o l l o w s t h a t m - 1am i s i n GAf\ exp t(_G~ S' " T ) = A. Thus g = kah = (km) (m _ 1am) ( m _ 1 h ) , g w h i c h i s i n K^AG . Combining 6., 7., 8 . , we see t h a t { S ( g ) g - 1 : g i n NAK A} S S i s dense i n .Gg, hence t h a t NAK^/G i s dense i n G/G . T h i s g proves t h a t G K-^ AN i s dense i n G. To show t h a t t h i s remains S . S. t r u e i f we r e p l a c e G by any subgroup H c o n t a i n i n g (G ) i t S S S S s u f f i c e s t o show t h a t G - (G ) o ^ ^ , i . e . t h a t i n t e r s e c t s S ev e r y c o n n e c t e d component o f G . T h i s was done d u r i n g t he pro o f of I I . 3. A A Wow suppose t h a t Hm]_K AN = Hm2K Al1! f o r some m]_,m2 i n K^. ' Then m^  i s i i i ^ t a ^ K ^ A N = HK AA(m2Nm 2 ^ )'^ 2 > a n d t h e r e f o r e m^mg"^ i s i n K A HK^AXir^Nm-} "*"). Thus t h e r e i s m i n so t h a t m-^ m2 . ^m Is i n HAvir^Nn^" 1}. S i n c e n^Nm^ "^Vis j u s t the " i s " a s s o c i a t e d w i t h a d i f f e r e n t o r d e r i n A*,.:the n e x t lemma shows t h a t the f i b e r s of the m u l t i p l i c a t i o n map H * X A *N — > G a r e as a s s e r -t e d i n the p r o p o s i t i o n . 9. Lemma. f\ HAN = (H f)K)A. P r o o f . HAN = HNA, and i f han i s i n (h i n H, a i n A, n i n N) t h e n Ad(hn)A = A i . e . Ad(n)A = A d ( h _ 1 ) A . S i n c e Ad(n)A C -1 -S A + N, w h i l e Ad(h )A CG , i f f o l l o w s t h a t n and h n o r m a l i z e -1 ^ A s e p a r a t e l y . So f o r any a i n A, a nan ^ i s i n N o A = {e i . On t h e o t h e r hand, i f a =/ 0 f o r a l l r i n R, t h e n the map n > a ~ l n a n - I i s a d i f f e o m o r p h i s m o f N onto i t s e l f ( (4 ) p.234). T h e r e f o r e n = e and h i s i n ( H H K ^ . vi. I t remains t o show t h a t our m u l t i p l i c a t i o n map i s a sub-m e r s i o n . For t h i s i t s u f f i c e s t o compute i t s t a n g e n t map A H x K , . X A x N — ^ G a t the p o i n t s (e,m,e,e) w i t h m i n K ^ . T h i s t a n g e n t map sends (X,Y,Z,W) i n t o A d ( m _ 1 ) X + Y + Z + W. So i t s image i s Ad(m _ 1)H"+ K~ +•A + N = A d ( m _ 1 ) ( H + K - + A + Ad(m -- G by 1.. T h i s completes the p r o o f o f 4. . „ The next lemma g i v e s the. i n t e g r a l f o r m u l a c o r r e s p o n d i n g to.-'.. 4 . . The..measure' dgH o c c u r i n g i n the statement i s the unique G i n y a r i a n t - measure,;6n;,G,/H . s a t i s f y i n g .... r.;f(g> dg- = j . dgH j^ ; f ( P h ) dh •J • G/H .^ H f o r - c o m p a c t l y • s u p p o r t e d f u n c t i o n s f on G. ' ' ' • - 3 8 -10. Lemma. For a p p r o p r i a t e n o r m a l i z a t i o n s o f the l e f t i n -v a r i a n t measures (a) J " G f ( g ) = i H ^ A x N f(hman) a 2P dh dm da dn (b) J , f ( g H ) dgH = J f(namH) a" 2P-dn.da dm G/H • .. 6 NAAXK A/(H^K) A • where p = h^2— m ( r ) r ; 'm(r) = dim G r. P r o o f . S i n c e the i n v e r s i o n g^g"-'- maps H K A A N d i f f e o m o r p h i c a l l y onto NAK^H t r a n s f o r m i n g a 2P>dh*dm-da-dn i n t o a dn•da-dm-dh (a) i s e q u i v a l e n t t o ( b ) . To prove (b) I have t o compute the " d e t e r m i n a n t " o f the tangent maps of N A A X K A/(Hr\ K ) A ~ ^ G by i d e n t i f y i n g t h e tangent spaces w i t h NX A X K A / ( H 0 ^ ) — and G/H by l e f t trams-A ' A ~ l a t i o n s and NxAxK.-/(H r \ K ) _ w i t h G/H by an a r b i t r a r y , . i s o -morphism. At the p o i n t (n,a,m(H^K)^) the t a n g e n t map sends (X,Y,Z.+ (H_. K ) ~ ) i n t o A d(m~ 1a" 1)X .+ A d ( m _ 1 ) Y + Z + H. Mapping A A G/H i s o m o r p h i c a l l y onto NXAyK -/(HnK)— by the r u l e ( X ' , 0 . 0 ) i f X i s i n G r w i t h r >0 f S ( X ) , 0 , 0 ) i f X i s i n G r w i t h r < 0 X + H H > ( (0,X,0) i f X i s i n A ( 0 , 0,X+(H K ) — i f X i s i n K-A A our t a n g e n t map c o r r e s p o n d s t o the endomorphism o f ' N*A*K—/(HoK)— - 3.9 -(X,Y,Z) <• d e f i n e d by (a" rAd(m" 1)X,Ad(m" 1)Y,Z) i f X i s i n G_r w i t h r > 0 and Ad(m ) r > 0, (-a" rS(Ad(m- 1)X),Ad(m- 1)Y,Z) i f X i s i n G r w i t h sr^O and A d ( m _ 1 )r'< 0. S i n c e A d ( m _ i ) i s a B T - o r t h o g o n a l map the d e t e r m i n a n t o f t h i s endomorphism i s +7T a ~ m ( r ) r ' = + .a~ 2?. P r o p o s i t i o n 4. g i v e s a p a r a m e t r i z a t i o n of t h e r e g u l a r o r b i t s o f P i n G/H o n l y . So f a r I have not been a b l e t o o b t a i n a complete d e t e r m i n a t i o n o f the o r b i t s o f P, i n c l u -d i n g the s i n g u l a r ones. 11. Examples. (a) Riemannian spaces. I n t h i s case S = T and H = K. The Iwasawa d e c o m p o s i t i o n G = KAN shows t h a t t h e r e a r e no s i n g u l a r o r b i t s o f P = K AAN. P a c t s t r a n s i t i v e l y on G/K. T h i s agrees w i t h 4.,.i w h i c h says t h a t t h e r e i s o n l y one r e -g u l a r o r b i t of P (because W - W ). (b) L i e groups. I n a n a l o g y w i t h . I . 1 6 . ( a ) a con n e c t e d semi-s i m p l e r e a l L i e group G r may be c o n s i d e r e d as a s e m i s i m p l e symmetric space G/H,' where G = G U G' and H i s the d i a g o n a l i n G. Thei_ i n v o l u t i o n s ' S. and T. a r e defimed by- S(g-^, p^') ^'^g.2'81^ and .T(g]_,g 2) = (T ' ( g L ) , T' ( g 2 ) ) where T' i s any C a r t a n i n -- .40 -v o l u t i o n of G' . L e t G' = K'A'N' be an Iwasawa d e c o m p o s i t i o n o f G'. One e a s i l y v e r i f i e s t h a t K = K' X K' A = | ( a , a _ 1 ) : a i n A ' ] K A = [-(m]_,m2.) : m] ,m2 i n K' so t h a t riling ^ i s i n K1' N = N'X N 1. Thus 4. says t h a t £ (grn-^an, gn^a" ^ -n ) : g i n G,,.ni1,m2 in;.K T -1 A t — _V so t h a t m^n^ i s i n K' . , n. i n N } n i n N j i s open and dense i n G. Mapping G/H d i f f e o m o r p h i c a l l y onto G' by ( g i sg2 ) l"^'g]_g2 ^ t h i s says t h a t N » K ' A ' A I N . ' i s open and dense i n G' . N'K' A' A'N' i s o f c o u r s e p r e c i s e l y t he unique open o r b i t of the Br u h a t d e c o m p o s i t i o n o f G'. T h i s was a f t e r a l l the c r u c i a l s t e p i n t he p r o o f o f 4., whi c h we j u s t r e c o v e r i n t h i s manner. The Bru h a t d e c o m p o s i t i o n shows f u r t h e r t h a t t h e o r b i t s o f P i n G/H a r e i n one-to-one c o r r e s p o n d e n c e w i t h the e l e -ments o f the Weyl group o f the r o o t system of the p a i r G',A'. - I do n o t know how t o i n t e r p r e t t h i s i n terms o f the s t r u c t u r e of t he symmetric space G/H. ( c ) R e a l h y p e r b o l i c s p a c e s . Here G =' S0(p,q) , H ¥ S 0 ( p , q - l ) as i n 1.16. (b) and 11.10. As we found e a r l i e r (p. 13) W =i-,W' ( = WQ =./Z2) u n l e s s p — - l , - i n which case W' i s t r i v i a l . Moreover N = |_g i n G : g ( e L + e n ) = e x + e n , g e ^ e ^ - B L C e ^ e n ) ) . • .P = J g.'.in G : gE . ( e i + e h-)-= E ( e 1 + e n ) j - 41 -I t f o l l o w s from 4. t h a t t h e r e i s o n l y one o r b i t o f P on G/H, except when p = 1, i n which case t h e r e a r e two o r b i t s . •This can a l s o be seen d i r e c t l y . I d e n t i f y G/H w i t h the h y p e r s u r f a c e [ x i n K. : ( x 3 x ) = • 1 £ as i n 11.10. Suppose x = p e n f o r some p i n P. S i n c e p ( e ^ + e n ) = c ( e ^ + e n ) f o r some c 4 0, we g e t pe-^ = -x + c C e - ^ + e n ) . T h e r e f o r e -1 = ' (pe 1,pe ]_) = 1 - 2c(ei + e n , x ) : , i . e . (e-j^ + e n , x ) = 1/c 4 0. Conversely/, i f c H: (e-^ + e n , x ) =/ 0 f o r some x i n G/H, we can f i n d g i n 0(p,q) so t h a t g e n = x, and ge-^ = -x + c.(e^- e n ) , ( T h i s i s c l e a r s i n c e ( x , x ) = 1 = ( e n , e n ) and (-x + c X e ^ ++ * e n ) , -x + c(e-^ + e n ) ) = -1 = (ejL,e - j _ ) . ) Moreover, o n l y the f i r s t and the l a s t coloum of g a r e de t e r m i n e d by x so t h a t we may even choose g i n SO (p, q ) Q , , p r o v i d e d p> l„and q > l . I f q = 1 we a r e i n the Riemannian c a s e . I f p = 1, we cannot choose g i n S 0 ( l , q ) o f o r a l l x i n G/H: f o r example, i f x = - e n , the n c = -1 and (ge-pe-^) = (-x + .c(e]_ + e n ) , e 1 ) : ••= -1 i s n e g a t i v e , hence cannot l i e i n S 0 ( l , q ) o (because o f the r e s t r i c t i o n s on the s u b d e t e r m i n a n t s of g ) . Thus i f p = 1 t h e r e a r e two r e g u l a r o r b i t s o f P, namely P e n and P ( - e ), i n agreement w i t h the t h e o r y . The argument shows f u r t h e r t h a t the u n i o n o f t h e s i n g u l a r o r b i t s o f P i s the i n t e r s e c t i o n of G/H w i t h the h y p e r p l a n e ( x i n R n : . ( x , e i + -e n) = o ] . - 42 -(b) SL(p+q)/SO(p,q). T h i s i s t h e symmetric space c o r -r e s p o n d i n g t o 1 . 1 6 . ( c ) . I t may a l s o be r e a l i z e d as t h e space of symmetric n * n m a t r i c e s (n = p + q) o f d e t e r m i n a n t one and s i g n a t u r e ( p , q ) . . From I.16^.(c) we know t h a t W = Sp+^ and Wr = SpX S . For N we may t a k e the upper t r i a n g u l a r m a t r i c e s i n G = SL(n) w i t h u n i t d i a g o n a l , and f o r P (- AN i n t h i s c a s e ) the upper t r i a n g u l a r m a t r i c e s w i t h p o s i t i v e d i a g o n a l . The p a r a b o l i c d e c o m p o s i t i o n o f G/H reduces e s s e n t i a l l y t o the Gram - Schmidt process.: f o r any g i n GL(n) s e t x l - & e l and i n d u c t i v e l y x r•= g e r - T.(ger,xi)/(xi,xi)iei p r o v i d e d (x^,x^_) 4 0 f o r i = l , . . . n . Set y^ = J ( x ^ ,x^ )1 x^-and d e f i n e a m a t r i x p by pe^ = g'^-y^. p i s upper t r i a n g u l a r w i t h d i a g o n a l e n t r i e s | ( x ^ , x ^ ) | S i n c e ( y ^ , y j ) = +1 i f i = j , and =0 o t h e r w i s e , t h e r e i s a p e r m u t a t i o n i H > i T o f l , . . , n so t h a t ( y - [ i , y j t ) = ( e ^ e j ) . L e t m be a m a t r i x i n so t h a t me^ = +e.j_i. Then h : e^v->y^ I s i n 0(p,q) and hm~1e;i, = y^ =: gpe.^. Thus g = hnT^p , and by t a k i n g d e t e r m i n a n t s we see t h a t d e t ( h ) = +],.. whenever det (g) = ..1. . I t f o l l o w s t h a t every g i n G = SL(n) f o r w h i c h ( x ^ j X ^ ) 4 0 f o r i = l , . . . n l i e s i n HwP f o r some w i n W. - 43 -Ch a p t e r IV. The c o n i c a l d u a l o f a s e m i s i m p l e symmetric space. The " c o n i c a l d u a l " X* of a s e m i s i m p l e symmetric space X i a a g e n e r a l i z a t i o n o f the space o f h o r o c y c l e s of a Riemannian symmetric space ( 5 ) . I n t h e Riemannian case X* i s " d u a l " t o X i n a d o u b l e sense: ( a ) t h e s t r u c t u r e o f X* c l o s e l y resembles t h a t o f X; (b) harmonic a n a l y s i s on X* i s i n t i m a t e l y r e l a t e d t o harmonic a n a l y s i s on X. Both of t h e s e phenomena c a r r y o v e r t o a r b i t r a r y s e m i s i m p l e symmetric spaces, but w i t h ., e s s e n t i a l m o d i f i c a t i o n s (compare 7. below and V I . )• The e x p l a n a t i o n o f the t e r m " c o n i c a l " comes from the example X = S O ( p , q ) o / S 0 ( p , q - 1 ) Q i n wh i c h case X* can be i d e n t i f i e d w i t h a ( d o u b l e ) cone w i t h v e r t e x d e l e t e d (compare 9. b e l o w ) . T h i s example i s t y p i c a l i n t h a t X* i s always d i f f e o m o r p h i c t o a f i n i t e d i s j o i n t u n i o n o f p r o d u c t m a n i f o l d s o f a-com-p a c t space and a E u c l i d e a n space (compare 7. b e l o w ) . By a " c y c l e " ( s h o r t f o r " h o r o c y c l e " ) i n a s e m i s i m p l e symmetric space X .= G/H I mean the o r b i t o f a subgroup of G c o n j u g a t e t o N ( n o t a t i o n as b e f o r e ) . T h i s d e f i n i t i o n de-pends o n l y on the symmetric space G/H, not on the p a r t i c u l a r -S -T c h o i c e o f t h e C a r t a n i n v o l u t i o n T, the subspace A o f G , or on t h e o r d e r i n A* d e t e r m i n i n g N. A c y c l e i s " r e g u l a r " if- f t he r e s t r i c t i o n o f the pseudo - Riemannian m e t r i c on X - 44 -t o t h i s s u b m a n i f o l d i s non-degenerate. I f a c y c l e c o n t a i n s a p a r t i c u l a r p o i n t x i n X, I a l s o say t h a t the c y c l e "passes t h r o u g h x". The base p o i n t eH o f X = G/H w i l l always be de-not e d X Q . 1. Lemma. A c y c l e t h r o u g h Xq i s r e g u l a r i f , and o n l y i f , i t i s the o r b i t o f a subgroup o f G c o n j u g a t e t o N by an element i n HK^AN. P r o o f . The c y c l e gNg~l«x 0 i s r e g u l a r i f . and o n l y i f , the r e s t r i c t i o n o f Bg to' Ad(g)N i s non-degenerate, i . e . B s(Ad(X),Ad(g')N) = 0 w i t h X i n N i m p l i e s X = 0 i . e . Bg.CAdCSCg"1 )g)X,N) = 0 w i t h X i n N i m p l i e s X = 0 i . e . A d ( S ( g _ 1 ) g ) X i n G—H-N w i t h X in:N,. i m p l i e s X = 0 i . e . A d < S ( g _ 1 ) g ) N (G^+N) = 0 i . e . Nf\Ad(g.S(g _ 1))(G-+N) = 0. Now N r\xPx~l- i s n a t u r a l l y i s o m o r p h i c t o t h e s t a b i l i z e r o f x under the the a c t i o n xt->nxp-"'- of N X"'P on G. So Nn A d ( x ) P = 0 i f , a n d o n l y i f , x l i e s i n the r e g u l a t o r b i t NP o f t h i s a c t i o n . For x = gS(g ) t h i s i s e q u v a l e n t t o g b e i n g an element o f HK AAN, as was shown d u r i n g . t h e p r o o f o f I I I . 4 . . The a c t i o n o f G on X permutes the c y c l e s i n X: g^gc-Ng^-*) -1 ( g g 0 ) N ( g g 0 ) ' *g*x; thus g maps the c y c l e s t h r o u g h x onto t h e c y c l e s t h r o u g h g*x. The r e g u l a r c y c l e s a r e permuted among - ,45 -themselves so t h a t we g e t an a c t i o n o f G on X * . T h i s a c t i o n i s i n g e n e r a l not t r a n s i t i v e : X * decomposes i n t o iW'\w| o r b i t s o f G. More p r e c i s e l y , , f o r any w i n W l e t n^.be a r e p r e s e n -t a t i v e o f w i n K A. Set w(N) = mwNmw''" and denote the ( r e g u l a r ) c y c l e w ( N ) * x Q by x w . Then 2 . P r o p o s i t i o n . X * = • G'x w* ; G*x w* = G'xw;) , i f , a n d w i n W -L o n l y i f , Ww-j_ = W!W2» P r o o f . I t s u f f i c e s t o c o n s i d e r c y c l e s t h r o u g h x 0 . I f g N g ~ l # x 0 i s a r e g u l a r c y c l e t h r o u g h x 0 , the n g = hman f o r a p p r o p r i a t e h i n H, m i n K^, a i n A, and n i n N. So g N g - i * x 0 = h*mNm~l*x 0 A = h»x w* where w =-mK . T h i s proves t h a t X * = ( j G * x w * . w • 1:1 ow suppose ex, * = x •* f o r some g i n G. S i n c e b o t h x * r r ° w]_ w2 w l and x w * a r e r e g u l a r c y c l e s t h r o u g h x G we may assume t h a t g = h l i e s i n H. From h*w-^(N)«x0 = h«W2(N)«x o we g e t t h a t f o r / a l l n^ i n W]_ (N) t h e r e i s n2 in'w 2(N)- so t h a t S(hn^)(hn-^) ^ = S( n 2 - ) n 2 ., i . e . hS(n-|_)n-[_~1h = S ( n 2 ) n 2 ~ . D i f f e r e n t i a t i n g w i t h r e s p e c t t o n^ we f i n d t h a t f o r a l l X-j_ i n w^(N) t h e r e i s ' X 2 i n w 2 ( K ) so t h a t '(*) A d ( h ) ( S ( X 1 ) - X L ) = S ( X 2 ) - X 2 . Now •JS(X 1) - X i : X x i n w]_(N)i - £s (X 2 ) - X 2 : X 2 i n w 2 ( N ) ( 5~ - S = r > 0 ^ - r + G_ r) . :By; (*) Ad(h) l e a v e s t h i s subspace of G i n -v a r i a n t , hence ; a l s o ; < i t s B g - o r t h o g o n a l complement,: w h i c h i s ;-G^>~S = K - 5 _ S + A. We t h e r e f o r e f i n d t h a t A d ( h ) G - ' " S = G-'~S - 4.6 -and s i n c e A c o n s i s t s p r e c i s e l y o f t h o s e e l e m e n t s X o f K—' ^ + A f o r w h i c h ad(X) has r e a l e i g e n v a l u e s , we even g e t Ad(h)A = A. i . e . h i s i n HA. = ( H r ^ K ) A H A (see the p r o o f o f I I . 3 ) . Thus w = hH^ may be c o n s i d e r e d as an element o f Wr. From (*) i t f o l l o w s t h a t Ad(m wm w-^)G r = Ad(m W 2)G_ r f o r a l l r i n R, i . e . ww-^(r) = w 2 ( r ) f o r a l l r i n R, i . e . . ww-^  = w 2. T h e r e f o r e W' w-| - W' w 2. C o n v e r s e l y , i f w i s i n W , t h e n G'x*^ = G*ww.^(N)*x0 =. Gm w'w 1(N)-m w" 1«x 0 '= G.w 1(N)-x 0 =0-G'xw*. The p r o o f a l s o g i v e s 3. C o r o l l a r y . The s t a b i l i z e r o f x * i n G i s H Aw(N). Make X* i n t o an a n a l y t i c m a n i f o l d by r e q u i r i n g t h a t the maps G/H Aw(N )-} G«x w* , gH Aw(N) H> g x w * , be a n a l y t i c d i f f e o -morphisms. Equipped w i t h t h i s a n a l y t i c s t r u c t u r e X* w i l l be c a l l e d the " c o n i c a l d u a l " o f X. F o r more i n f o r m a t i o n on the s t r u c t u r e o f X* we need: 4. Lemma. G A = K A A H A . ' P r o o f . A p p l y i n g I I . 1. t o the q u o t i e n t group of ( G A ) Q by i t s c e n t e r g i v e s ( G A ) Q C K A A H A . I t i s a l s o known t h a t K A i n t e r s e c t s A A ev e r y c o n n e c t e d component o f G ( .,(10) p. 75). T h e r e f o r e G = H AAK A. - 47 -5. P r o p o s i t i o n . The map K/(HA K ) A X A ~ > G/HAN, (k(Hr\ K ) A , a ) H> kaH AN, i s an a n a l y t i c d i f f e o m o r p h i s m . A A • P r o o f . S i n c e P = G N i s p a r a b o l i c , G = KG N by the Iwasawa de-c o m p o s i t i o n . So G / H \ - K G ^ / H ^ = KAH^N/H^ by 4. T h i s shows t h a t our map i s s u r j e c t i v e . S i n c e KA AH^N = (H/^K.) A i t i s a l s o i n f e c t i v e . F i n a l l y , i t s t a n g e n t map a t (e(H/^ K ) A , e ) maps K / ( H ^ K ) - X A onto G/(H- + N) by se n d i n g ( X + ( H O K ) A , Y ) i n t o X + Y + (H^ + N), hence i s an isomorphism. 6. C o r o l l a r y . The map K / ( H r \ K ) A \ A x W - * X * , ( k ( H O K ) A , a ,w) ka*x w' r, i s a c o v e r i n g w i t h f i b e r { (kwQ~-'- ,w Q(a) ,wQw) : w Q i n W T ] above ka«x w*. 1 7-.. Remarks. ( a ) Choosing a s e t o f c o s e t r e p r e s e n t a t i v e s f o r W ' \ W g i v e s an a n a l y t i c d i f f eomorphism K/(Hr\ K ) A X A x W ' \ W —>X*. But t h i s d i f f e o m o r p h i s m i s not " n a t u r a l " i n t h a t i t depends on the c h o i c e o f the c o s e t r e p r e s e n t a t i v e s . A n a t u r a l i d e n t i -A f i c a t i o n o f X* w i t h t h e q u o t i e n t of K / ( H A K ) X A X W by t h e obvious a c t i o n of W i s immediate from 6.. T h i s i d e n t i f i c a t i o n ;;..) X* - K / ( H q K ) A XA XW V/' s h o u l d be compared w i t h X = K/(HA K ) A X A . W . . . . (see I I . 2 . ). 48 (b) The r e g u l a r c y c l e s a r e c l o s e d ' s u b m a n i f o l d s o f X. To see t h i s i t s u f f i c e s t o show t h a t the c y c l e s x , w i n W a r e c l o s e d i n X, and we may even assume w = e , the o t h e r c a s e s b e i n g a n a l o g o u s . So l e t ^ n ^ H ^ b e a sequence i n x^ = N'H w h i c h converges i n G/H. Then the,sequence(S(n.)n~^( converges i n G. S i n c e NN i s ( w e l l known t o be) c l o s e d i n G and d i f f e o m o r p h i c w i t h N A N , i t f o l l o w s t h a t ln^\converges i n N and hence [n_^H^ i n G/H. The. d u a l i t y between X and X* c a n be c a r r i e d one s t e p f u r t h e r i n t h a t p o i n t s i n X can a l s o be c o n s i d e r e d as s u b m a n i f o l d s o f X*. I n f a c t , w i t h a p o i n t x i n X we can a s s o c i a t e the s e t C*(x) of a l l c y c l e s p a s s i n g t h r o u g h X. The a c t i o n o f G on X* permutes t h e s e " c o c y c l e s " t r a n s i t i v e l y so t h a t g«C"(x) = C * ( g . x ) . More-over, the a s s o c i a t i o n xH> C"(x) i s a b i s e c t i o n of.X. onto the c o l l e c t i o n £C*(x) : x i n X] . To prove t h i s i t s u f f i c e s t o show t h a t g . C * ( x Q ) = C * ( x Q ) o n l y i f . g l i e s i n H. Now C * ( x Q ) = U H-x* so i f g * C " ( x 0 ) = C * ( x Q ) then i n p a r t i c u l a r g*x* ~ h - x * for'-some w i n W„and .some h i n H. By 2. and 3. t h i s can happen o n l y i f w i s i n W and ( a f t e r m u l t i p y i n g h on the r i g h t by an element i n ( H n K ) A ) j h; ^g i s i n H AN. By m u l t i p l y i n g w i t h an element i n H A on the l e f t we may as w e l l assume t h a t h _ i g l i e s , i n N, say h _ 1 g = n. From gH«x* - H'x* i t f o l l o w s t h a t g i v e n h, i n H t h e r e i s h ^ i n H s o . t h a t n h ^ ' x * e = h^'x* i , e * h ^ 1 1 ^ i s i n H AN. - 49 -T h e r e f o r e .nH C HN , whic h a l s o g i v e s nHn~^ C HN and by d i f f e -r e n t i a t i o n , A d(n)H C. H + N = FA + Y~ G^ .. T a k i n g B - o r t h o g o n a l complements shows t h a t A d ( n ) G ~ S D G^'"S, o r A d ( n _ 1 ) G ^ ' " S G G~ S. On the o t h e r -handr , Ad(n" 1)G-,-S C G^ + N, so A d ( n _ 1 ) G ^ ' " S C G~ S r i G— + N A - S -1 A = G—' I n p a r t i c u l a r , i f a i s i n A, t h e n n an l i e s i n ; G , 1 - 1 r hence so does n ~ l a n a . Choosing a so t h a t a 4 e f o r a l l r i n R we g e t t h a t n = e as i n the p r o o f o f I I I . 9 . I n v i e w o f the symmetric r e l a t i o n between X and X* i t seems more n a t u r a l t o r e g a r d X* as an " a b s t r a c t " m a n i f o l d on i t s own r i g h t and the i d e n t i f i c a t i o n o f i t s . p o i n t w i t h submani-f o l d s o f X as a p a r t i c u l a r " r e a l i z a t i o n " . I n anal o g y w i t h the n o t a t i o n i n t r o d u c e d above I s h a l l t h e r e f o r e w r i t e C ( x * ) f o r the s u b m a n i f o l d o f X c o r r e s p o n d i n g t o a p o i n t x * i n X*. 50 8. Lemma. W i t h a p p r o p r i a t e n o r m a l i z a t i o n s o f t h e l e f t i n v a -r i a n t measures ( . fCgHAwCN)') dgH Aw(N) - j . f C k a H ^ C N ) ) a 2 w ( P ) • • G/FTw(N) •' K/(H/"IK) A X A ' - dk(H f) K ) - da. P r o o f . I d e n t i f y t h e t a n g e n t spaces o f G / P A ^ N ) and o f K/(H K ) A A w i t h G/H^wCN) and w i t h K / ( H O K ) - X A by l e f t t r a n s l a t i o n s . Then t h e t a n g e n t map. o f K/(H«s K ) A X A - ^ G/lA/CN) a t ( k ( H r ] K ) A , a ) maps (X + ( H n K ) - , . Y ) i n t o A d ( a _ 1 ) X + Y + (H- + w(N)). 'Map G/HA+W(N) i s o m o r p h i c a l l y back onto K / ( H O K ) - x A by t h e r u l e f ( 0 , X ) i f X i s i n A X + (H^+wCN)) <(X + ( H ^ K ) ^ , 0 ) i f X i s i n _(X + T(X) + (HnK)-,0) i f X i s i n G w ( r ) w i t h r < 0 . Then our t a n g e n t map c o r r e s p o n d s t o t h e endomorphism o f K / ( H o K ) - X A d e f i n e d by. (X + T(X) + ( H Q K ) - , Y ) | - > ( a w ( r \ x + T(X) + ' ' A ( H-nK ) - , Y ) i f X i s i n Gr w i t h r > 0. I t s d e t e r m i n a n t t h e r e f o r e i s U a M ( * M ' R » ^ a 2 w ^ . . • 9. Example. The c o n i c a l d u a l o f the r e a l h y p e r b o l i c space X = S 0 ( p , q ) o / S 0 ( p , q - l ) o has a s i m p l e g e o m e t r i c i n t e r p r e t a t i o n : I d e n t i f y X w i t h {x i n I n : ( x , x ) = 1\ as b e f o r e . Then the c y c l e s t h r o u g h a p o i n t x i n X a r e p r e c i s e l y the c o n n e c t e d .com-ponents of x i n the i n t e r e s c t i o n s of X w i t h the h y p e r p l a n e s t a n g e n t . t o the cone |y i n I - : (y - x,y - x ) = o] , i . e . hyperplanes/bi; the form £y i n & n : "(x*,y)'= ( x * , x ) f o r b o r n e x* 4 0 such t h a t ( x * , x * ) = o} . - 5L -A c y c l e t h r o u g h x i s r e g u l a r i f , and o n l y i f , t he c o r r e s p o n d i n g x , v , s s a t i s f y ( x * , x ) 4 0. I n t h a t case x* may be u n i q u e l y de-t e r m i n e d by the r e q u i r e m e n t ( x V r , x ) - 1. So X* may be i d e n t i -f i e d w i t h the cone £x* i n l n :. ( x * , x * ) = 0,.x* 4 0 ] , To see a l l of t h i s i t s u f f i c e s t o c o n s i d e r c y c l e s t h r o u g h x o = en« I f g N g - i * x o i s a n y c y c l e t h r o u g h x Q , s e t x* - g ( e ^ + e ) ( t h i s i s w e l l d e f i n e d because N. = [ g i n G : g(e]_ + e ) = e-^  .+ e n , g e i •= ei+E-(e1+"en')ifl,nOl.If y = g n g _ 1 * x 0 l i e s i n g N g " 1 * x o 3 t h e n ( x * , y ) = ( g ( e L + e n ) , g n g _ 1 e n ) = ( g n ~ 1 g " 1 g ( e i + e n ) , e n ) = (g(ei-.+ e n ) , e n ) • = ( x * , x 0 ) . • ' • • Thus gNg-l^Xo i s c o n t a i n e d i n t h e i n t e r s e c t i o n o f X w i t h the h y p e r p l a n e ^y i n f n : ( x , v , y ) = ( x * , x ) ^ . Comparing d i m e n s i o n s ( dim(N) = n - 2).one f i n d s t h a t g N g " I * x 0 i s a c t u a l l y e q u a l t o the c o n n e c t e d component of x Q i n [ y i n BLn : ( y , y ) = l , ( x * , y ) = • ( x * , x 0 ) ] . (One may even show t h a t t h i s s e t i s c o n n e c t e d i f ( x * , x 0 ) •/- 0.) As f o r the a s s e r t i o n about the r e g u l a r i t y o f the c y c l e c o r r e s p o n d i n g t o x*, we o n l y have to observe t h a t i t s t a n g e n t space a t x Q may be i d e n t i f i e d w i t h {z i n E.n : ( x Q , z ) = ( x * , z ) = 05 . - 52 -P a r t Two . A p p l i c a t i o n s t o A n a l y s i s . In t h i s second p a r t I s h a l l a p p l y the s t r u c t u r e t h e o r y d e v e l o p e d so f a r t o some problems o f a n a l y s i s r e l a t e d t o semi-s i m p l e symmetric spaces. I s h a l l c o n t i n u e t o employ the n o t a t i o n s and c o n v e n t i o n s o f t h e p r e v i o u s c h a p t e r s . I n a d d i t i o n I make the f o l l o w i n g d e f i n i t i o n s , which w i l l remain i n f o r c e throughcn.it: ,. f o r a C - m a n i f o l d X l e t D(X) =. the space o f compac t l y s u p p o r t e d , c o m p l e x - v a l u e d C - f u n c t i o n s on X E(X) = the space o f a l l c o m p l e x - v a l u e d C - f u n c t i o h s on X D'(X) =• the space o f d i s t i b u t i o n s on X E T ( X ) = the space o f compactly s u p p o r t e d d i s t r i b u t u i o n s on X. When a p p r o p r i a t e , t h e s e spaces w i l l be c o n s i d e r e d as t o p o l o g i c a l v e c t o r spaces i n the u s u a l manner. F o r f i n D(X) find F i n D.'-(X) i t i s sometimes c o n v e n i e n t t o w r i t e F ( f ) as J f ( x ) F ( x ) d x i n X o r d e r t o keep t r a c k o f d i f f e r e n t v a r i a b l e s . I f a L i e group G a c t s on X on the l e f t ( r e s p . on the r i g h t ) d e f i n e a " l e f t ( r e s p 0 r i g h t ) r e g u l a r r e p r e s e n t i o n " o f G on complex-v a l u e d f u n c t i o n s on X by L x ( g ) f ( x ) = f ( g - l ' ; x ) r e s p . R ( g ) f ( x ) = f ( x g ) . The s u b s c r i p t "X" w i l l be o m i t t e d ;when c o n f u s i o n i s u n l i k e l y . The r e s t r i c t i o n s o f t h e s e r e p r e s e n t a t i o n s t o D(X) and E ( X ) , w i l l be denoted by t h e same l e t t e r s as w i l l be the c o r r e s p o n d i n g r e p r e -- 53 -sen t a t i o n s o f U ( G < - ; ) . For the c o n t r a g e d i e n t r e p r e s e n t a t i o n s on D ' ( X ) and on E ' ( X ) I use the symbol " L T " ( r e s p . "R'">. X X I f X = G / H i s a homogeneous space .1 s h a l l f r e q u e n t l y i d e n t i f y f u n c t i o n s on X w i t h r i g h t H - i n v a r i a n t f u n c t i o n s on G . I f G / H admits an i n v a r i a n t measure, I u s u a l l y denote i t by "dg" and assume i t n o r m a l i z e d so t h a t x J f ( g ) dg = j j f ( g h ) dh. dg ( f i n D ( G ) ) G , G / H \ n f o r l e f t i n v a r i a n t measures dg on G and dh on H . - 54 -C h apter V. A n a l y s i s on the c o n i c a l d u a l . I n t h i s c h a p t e r I s h a l l a n a l y s e the r e g u l a r r e p r e s e n t a t i o n s 2 o f G on E(X-') and on L ( X * ) . To d i s c u s s the common p r o p e r t i e s o f t h e s e r e p r e s e n t a t i o n s i t i s c o n v e n i e n t t o i n t r o d u c e t h e no-t a t i o n " F ( )" t o s t a n d f o r e i t h e r "E(. )" o r " L 2 ( • ) » . We have seen t h a t the c o n i c a l d u a l X* o f X decomposes i n t o [ WT\W|open o r b i t s o f G : X* = yG*x&, where w runs o v e r a system of c o s e t r e p r e s e n t a t i v e s f o r W'\W. As a homogeneous space the o r b i t G.x* i s i s o m o r p h i c w i t h G / H A N ; the r e g u l a r r e p r e s e n t a t i o n o f G oh F ( X * ) i s t h e r e f o r e i s o m o r p h i c w i t h the d i r e c t sum ^ F ( G / l F w ( N , ) ) w & w'\W So i t s u f f i c e s t o c o n s i d e r the r e g u l a r r e p r e s e n t a t i o n s o f G on F C G / H ^ W C N ) ) , W dri-W.We may even r e s t r i c t o u r s e l v e s t o the case w = e as w ( N ) i s j u s t "the N " c o r r e s p o n d i n g t o a n o t h e r o r d e r i n A*. . The r e p r e s e n t a t i o n o f G on F ( G / H \ ) i s e a s i l y d e s c r i b e d i n terms o f " P - s e r i e s r e p r e s e n t a t i o n s " o f G ( w i t h P = G ^ N ' ) , i . e . r e p r e s e n t a t i o n s o f G i n d u c e d from f i n i t e - d i m e n s i o n a l 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 o f P. I t i s a well-known f a c t t h a t r e s t r i c t i o n A t o G e s t a b l i s h e s a one-to-one c o r r e s p o n d e n c e between the f i n i t e -d i m e n s i o n a l 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 o f P and the f i n i t e -A A d i m e n s i o n a l . 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 o f G . ( S i n c e G n o r -A m a l i z e s N i t i s c l e a r t h a t every r e p r e s e n t a t i o n o f G extends u n i q u e l y t o a r e p r e s e n t a t i o n o f P w h i c h i s t r i v i a l on N . ) The - 55 -P - s e r i e s r e p r e s e n t a t i o n s r e l e v a n t t o the a n a l y s i s on G/H^N a r e p r e c i s e l y those' c o r r e s p o n d i n g t o i r r e d u c i b l e , s u b r e p r e s e n t a t i o n s o f t h e l e f t r e g u l a r r e p r e s e n t a t i o n o f G A on f u n c t i o n s on G A / H A . To d e s c r i b e such P - s e r i e s r e p r e s e n t a t i o n s i n a form appro-p r i a t e f o r comparison w i t h F ( G / H A N ) , s t a r t w i t h t h e o b s e r v a t i o n A t h a t G a c t s on G/N on the r i g h t : (gN)«m = gmN f o r gN i n G/N and m i n G A . We t h e r e f o r e have a r i g h t r e g u l a r r e p r e s e n t a t i o n R o f • G A on G / N , and f o r a space V o f f u n c t i o n s on G A / H A i n v a r i a n t A under the l e f t r e g u l a r r e p r e s e n t a t i o n L o f G we can d e f i n e a space . : F G A ( G / N , V ) o f f u n c t i o n s v :G/N—?> V by t h e c o n d i t i o n s (1) (R(m)v)(gN) = a ( m ) ^ L ( m - l ) ( v ( g N ) ) f o r a l l m i n , G A and a l l gN i n G / N . A ( F o r m i n G the element a(m) i n A i s d e f i n e d by m H A K A a ( m ) ; thus a(m) P= ( detAd(m)| N | ^  = . jdetAd(m)( P|'2. ) (2) (a)I.f F = E, then V C E I G W ) and G/N * G A / H A — > £:, (g,m) I—;> v ( g ) ( m ) i s C'*°. ( b ) l f F = L , then V i s a H i l b e r t space and K/(HAK) A-5>V, k H>v(k) i s an L 2 - f u n c t i o n . G a c t s on F A ( G / N , V ) t h r o u g h i t s a c t i o n on G / N , and i t i s c l e a r t h a t t h i s r e p r e s e n t a t i o n o f G i s j u s t the r e p r e s e n t a t i o n i n d u c e d from the r e p r e s e n t a t i o n o f P wh i c h r e s t r i c t s t o the r e g u l a r r e p r e s e n t a t i o n o f G A on V and wh i c h i s t r i v i a l on N. (The m u l t i ^ p l i e r a ( m ) P i n s u r e s t h a t L Q A ( G / N , V ) i s a u n i t a r y r e p r e s e n t a t i o n - 56 -of G whenever V i s a u n i t a r y r e p r e s e n t a t i o n o f G A . See IV.8.) A I n p a r t i c u l a r - , i f Vis i r r e d u c i b l e under G (and n e c e s s a r i l y f i n i t e -d i m e n s i o n a l as we s h a l l see b e l o w ) , t h e n F Q A ( G / N , V ) i s a P - s e r i e s r e p r e s e n t a t i o n . On the o t h e r hand, i f V = F ( G A / H A ) , t h e n F G A ( G / N , V ) i s i s o m o r p h i c w i t h F ( G / H A N ) by the map F ( G / H A N ) —>F A ( G / N , V ) , G f ^ > v f , d e f i n e d by v f ( g N ) ( m H A ) = a ( m / f (gmH AN). (As a p a r e n t h e n t i -c a l remark, t h i s i somorphism i s a case o f i n d u c i n g i n s t a g e s : from H AN t o G A N t o G . ) 2 I t i s a well-known f a c t t h a t the a n a l y s i s o f L A ( G / N , V ) ; -. and hence t h a t o f L 2 ( G / H A N ) - can be r e d u c e d t o the a n a l y s i s o f the r e p r e s e n t a t i o n o f P on V from which i t was i n d u c e d . F o r t h i s r e a s o n I s h a l l f i r s t c o n s i d e r the r e g u l a r r e p r e s e n t a t i o n o f G A on L 2 ( G A / H A ) . , A A To b e g i n w i t h , n o te t h a t G /H i s a r e d u c t i v e symmetric space ( a l t h o u g h n o t n e c e s s a r i l y c o n n e c t e d ) . Moreover, the d i f f e o m o r -phism G A / H A = K A / ( H r , K ) A A ( I V . 4 . ) i s a c t u a l l y an isomorphism of symmetric spaces ( i n the sense o f ( 8 ) ) . I t f o l l o w s from g e n e r a l f a c t s about r e d u c t i v e symmetric spaces t h a t t h i s - ' i s o -morphism can be i n d u c e d by an isomorphism o f the c o r r e s p o n d i n g t r a n s f o r m a t i o n groups: i f ( G A ) ° i s the normal subgroup o f G A w h i c h l e a v e s G A / H A p o i n t w i s e f i x e d , and (K A)° the normal sub-A A A group o f K which l e a v e s K /(Hr\K) r p o i n t w i s e f i x e d , then t h e r e i s a L i e group isomorphism G A / ( G A ) ° - ^ - = > K A / ( K A ) ° XA, m H>(k(m) ,a(m) ) , so t h a t mkaH A= k(m)ka(m)a;H A f o r a l l m i n G A , k i n K A, and a ?n A. (The element a(m) i n A i s the same as the one d e f i n e d above.- For - 57 a p r o o f o f th e s e f a c t s / s e e the appendix t o t h i s c h a p t e r . ) As a consequence , the r e g u l a r r e p r e s e n t a t i o n o f G A on G^/H^ i s - i n an ob v i o u s sense - " i s o m o r p h i c " w i t h the r e g u l a r r e -p r e s e n t a t i o n o f . K A X A on K A/(Hf\ K ) A K A. The d i r e c t i n t e g r a l d e c o m p o s i t i o n o f L (A) i s o f c o u r s e j u s t c l a s s i c a l F o u r i e r a n a l y s i s on the v e c t o r group A. The d e c o m p o s i t i o n o f L 2 (K A/(H/"\ K') A) f o l l o w s from F r o b e n i u s R e c i -p r o c i t y and the Pe t e r - W e y l Theorem: by F r o b e n i u s R e c i p r o c i t y , 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 V , o f K A i s i s o m o r p h i c w i t h a 7 A A s u b r e p r e s e n t a t i o n o f L (K / ( H A K ) ) i f , and o n l y i f , ^ c o n t a i n s A a non-zero v e c t o r f i x e d byTICHnK) , t h e m u l t i p l i c i t y . o f V. i n L 2 ( K A / ( H A K ) a ) b e i n g equaLto the d i m e n s i o n o f the subspace o f V-^-f i x e d by ( H P | K ) A . A c c o r d i n g t o Pe t e r - w e y l , the L - o r t h o g o n a l p r o j e c t i o n E^ o f L 2 ( K A / ( H 0 K ) A ) onto the Tf - p r i m a r y s u b r e -p r e s e n t a t i o n !/• ( K A / ( H r t K ) A ) i s g i v e n by E ^ v C m ) •= dim ( I f ) i^A^m^ 1)v(m^m) dm^ - dim(ir) i R A X ( n i ^ 1 ) v ( m m 1 ) d m 1 , where % i s the c h a r a c t e r o f If • S i n c e v i s r i g h t - i n v a r i a n t XA under (HA.K) the c o n v o l u t i o n w i t h X ^ c a n be r e p l a c e d by a r i g h t c o n v o l u t i o n w i t h the c o r r e s p o n d i n g " s p h e r i c a l f u n c t i o n " o^. d e f i n e d by qj(m) = i ( H n K ) A r X 1 T ( m m 1 ) d m 1 < Thus E ^ v C - m ) = dim(Ti) $ K A / ( H n K ) A Sr ^ I " 1 ) v ( m m 1 )dm 1, - 58 -E v e r y v i n L 2 ( K A / ( H f \ K) ) can be r e p r e s e n t e d i n an L - c o n v e r -gent s e r i e s v = 2T E_v = 2"dim('Tr)v* CT. Moreover, the s e r i e s converges i n E ( K A / ( H A K ) A ) f o r v i n E(KA/(HC] K ) A ) . To r e p h r a s e t h e s e f a c t s i n terms o f p r o p e r t i e s o f G d e f i n e f o r a s p h e r i c a l f u n c t i o n Q~ on K A / ( H A K ) A and f o r i n A g a space Vff./^= . ( G A / H A ) o f f u n c t i o n s v: G /H A—> C by the c o n d i t i o n s (1) v(ma) = a ^ v ( m ) f o r a l l m i n G A and a i n A (2) dim(qr) ^ A / ( H D K ) A v ^ i r a n l ) ( 3^ I T 11 1 ) d j n 1 ~ v ( m ) f o r a l l m i n G A. ( dim(<y) = dim( 1$ i f ' (7= Gjj.). (3) (a) I f F = E, then v i s i n E ( G A / H A ) . (b) I f . F = L 2 , then k H>v(k) i s i n L 2(K A/(Ho K ) A ) 9 A A A F o r r e a l ^ L ^ ^(G /H ) i s a u n i t a r y r e p r e s e n t a t i o n o f G , i n J 2 A A f a c t a q u o t i e n t r e p r e s e n t a t i o n o f L (G / r i A ) : the p r o j e c t i o n L 2 ( G A / H A ) — ^ L25- ^ ( G A / H A ) , v - ^ V g . ^ , i s d e f i n e d , by v _(m) = j .., v(mm" 1a~ 1 )a v , / c-(m )dm da •^•"A K X A ' 1 1 1 and g i v e s the d i r e c t . i n t e g r a l d e c o m p o s i t i o n L 2 ( G A / H A ) =X • $ , L 2 ( G A / H A ) dim(a-) d A cr A * ( a p p r o p r i a t e n o r m a l i z a t i o n s o f the measures p r e s u p p o s e d ) . From g e n e r a l f a c t s about i n d u c e d r e p r e s e n t a i o n s we . can c o n c l u d e t h a t F ^ G / N j V ^ ^ ) i s i s o m o r p h i c w i t h a d i r e c t p r o d u c t o f a f i n i t e number;, o f c o p i e s •(= the m u l t i p l i c i t y o f "TT i n F ( K A / ( H / ) K ) A ) ) o f P - s e r i e s r e p r e s e n t a t i o n s . The same t h e r e f o r e h o l d s f o r the space F ^ . ^ . ( G / H A N ) o f f u n c t i o n s on G / H A N which c o r r e s p o n d s t o F Q A ( G / N , V G ^ ) under the isomorphism f"->v d e f i n e d above.• I t i s c l e a r t h a t F ^ ^ ( G / H A N ) c o n s i s t s p r e c i s e l y o f the f u n c t i o n s f : G / H A N - 7 > (C which s a t i s f y (1) f ( g a H A N ) = a?*^ f ( g H A N ) f o r a l l a i n A and g i n G ( 2 ) .dim(«)j* A f(gmHAN){r(m" 1)dm = f ( g H A N ) f o r a l l g i n G (3) ( a ) l f E = E , t h e n f i s i n E ( G / H \ ) . ( b ) l f F = L 2 , t h e n k >—>f ( k H A N ) i s i n L 2 ( K A / ( H H K ) A ) . Moreover, L ^ ^ C G / H ^ ) i s u n i t a r y f o r r e a l */\, i n f a c t a q u o t i e n t r e p r e s e n t a t i o n o f L 2 ( G / H A N ) : the p r o j e c t i o n L 2 ( G / H A N ) -> L 2 - ^ ( G / H A N ) f H > f c ^ , i s d e f i n e d by f 0 ^ ( g H A N ) = I . ^ f ( g m - 1 a - 1 ) a ' f 4 ' " V ( m - 1 ) d m da and g i v e s t h e d i r e c t i n t e g r a l d e c o m p o s i t i o n L 2 ( G / H A N ) S v L 2, ( G / H A N ) dim(cr) dA . ° " " ^ A ' 2 T h i s c ompletes the a n a l y s i s on G / H A N i n the u n i t a r y c a s e F . = L . T u r n i n g now to the d i f f e r e n t i a b l e c a s e , the o b j e c t i s t o d e t e r m i n e the common e i g e n f u n c t i o n s o f a l l G - i n v a r i a n t d i f f e r -e n t i a l o p e r a t o r s on G / H A N . But t h i s t a k e s some p r e p a r a t i o n s . To b e g i n w i t h , we need a managable d e s c r i p t i o n o f the a l g e -b r a I D ( G / H A N ) o f i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on G / H A N . T h i s can o b t a i n e d as • f o l l o w s : the r i g h t r e g u l a r r e p r e s e n t a t i o n o f G A on E ( G / N ) c o r r e s p o n d i n g to the r i g h t a c t i o n o f G A on G / H A N a l s o g i v e s a r e p r e s e n t a t i o n o f the c o m p l e x i f i e d L i e a l g e b r a U C G ^ ) ^ The s u b a l g e b r a o f A d ( H A ) - i n v a r i a n t s , denoted U ( G ^ ) ^ , l e a v e s the "subspace" E ( G / H A N ) o f r i g h t H ^ - i n v a r i a n t f u n c t i o n s i n E ( G / N ) - 60 -A H A A I n v a r i a n t . T h i s g i v e s an a c t i o n o f U ( G £ ) on E(G/H AN) by , A d i f f e r e n t i a l o p e r a t o r s , hence an a l g e b r a homomorphism o f U(G ) i n t o ]D(G/H%), which I denote by R . I n f a c t : » j^A . 1. P r o p o s i t i o n . R : U(G_-C) —~^ID(G/H AN) i s s u r j e c t i v e . P r o o f . F o r the p r o o f we have t o r e c a l l some g e n e r a l f a c t s about i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on homogeneous spac e s . Suspen-d i n g p r e v i o u s n o t a t i o n a l c o n v e n t i o n s f o r the moment, l e t G/H be any homogeneous space. The r i g h t r e g u l a r representation?RQ o f G g i v e s the u s u a l isomorphism o f U(G^) onto the a l g e b r a o f l e f t -i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on G. For X i n U(G_£), R g ( X ) f = 0 f o r a l l r i g h t r H - i n v a r i a n t , f u n c t i o n s f on G i f , and o n l y i f , X l i e s i n U(G_C)H. The subspace U(G C)H o f U(G_C) i s A d ( H ) - s t a b l e , so H a c t s on the q u o t i e n t , space U(G_c)/U(G )H by the a d j o i n t r e p r e -s e n t a t i o n . Denote the i n v a r i a n t s o f t h i s a c t i o n by (U(G^)/U(G^)H) I f X + U(G_C)H i s such an i n v a r i a n t , then A d ( h ) X s X mod U(G_C)H f o r a l l h i n H; ' . . . . i f f i s r i g h t H - i n v a r i a n t , so i s R ( ; ( X ) f : R G ( h / P . G ( X ) f = R G ( A d ( h ) X ) R G ( h ) f = R G ( X ) f . Thus RQ(X) can be " r e s t r i c t e d " t o a d i f f e r e n t i a l o p e r a t o r RQ/H^ ) on G/H, n e c e s s a r i l y G - i n v a r i a n t . I t can be shown t h a t t h i s map ^G/K: (U(G C)/U(G C)H ) I L^ID(G/H) i s a v e c t o r space isomorphism. (The p r o o f i s e s s e n t i a l l y t h e same as t h a t of Lemma 2 . 2 p.390 i n ( 4 ) . ) There i s a n o t h e r d e s c r i p t i o n o f ffi(G/H) i n terms o f the - 61 -symmetric a l g e b r a S ( G ) : t h e c a n o n i c a l v e c t o r space i s o m o r - ; phism S ( G Q ) — > U ( G Q ) commutes.*.:with the a d j o i n t r e p r e s e n t a t i o n o f G , hence passes t o the q u o t i e n t t o a v e c t o r space isomor-phism o f ( S ( G C / S ( G C ) H ) H w i t h ( U ( G _ C ) / U ( G _ C ) H ) H and t h e r e f o r e w i t h m(G/H). A p p l y i n g t h e s e remarks t o the case a t hand,we f i n d t he f o l l o w i n g isomorphisms: S ( G | ) H A ~ U C G J ) ^ ( S ( G C ) / S ( G C ) ( H A + N ) ) H A N = ( U ( G C ) / U ( G C ) ( H A + N ) ) H A N ( U ( G C ) / U ( G C ) . ( H ^ + N ) ) H A N ~ 3 D ( G / H A N ) . So t o prove the p r o p o s i t i o n ' i t s u f f i c e s t o show t h a t the com-p o s i t e map S ( G ^ ) H ^ - > U ( G ^ ) H A — J D C G / P A N O - ^ - ^ ( S ( G C ) / ( H ^ + N ) ) H A N i s s u r j e c t i v e . D i s e n t a n g l i n g d e f i n i t i o n s , one f i n d s t h a t t h i s c o mposite i s j u s t the r e s t r i c t i o n o f the n a t u r a l map P h ^ P mod S ( G C ) ( H ^ + N ) . To prove s u r j e c t i v i t y ^ enumerate the p o s i t i v e r o o t s o f the p a i r G , A so t h a t 0 < r < r, <. .. < r n ( w i t h r e s p e c t t o the o r d e r 1 ^ d e f i n i n g N ) . F o r e a c h l ^ . i ^ n choose a b a s i s { X i 3 k ^ f o r G _ R I so t h a t S T ( X i j k ) - + X i j k , | x 0 k \ f o r c f ' ^ . W r i t i n g X.-(X. ,X. ) and u s i n g m u l t i p l e i n d e x n o t a t i o n , e v e r y element i n . S " ( G Q ) / S ( G Q ) ( H A + N ) has a unique r e p r e s e n t a t i v e o f the f o i )rm -: 62 -P = 7 c ( m 0 , m i ; . . . ) X^°.X^....X^n where =. ( m ^ , ^ j 2,... ), X™*- = X ^ - X ^ and c ( . . . ) I f P i s A d ( H A N ) - i n v a r i a n t mod S ^ K H ^ N ) , then a d ( X ) P = 0 mod S(G r)(H-+N) f o r a l l X i n H^+N. I n p a r t i c u l a r , c h o o s i n g X i n G v ( r the maximal r o o t ) we g e t — r n n 0 = a d ( X ) P =. c(m 0..)X 0°-.'ad(X)X J...Xgn J , (.mi ) J }- s c ( m 0 . . ) x " ; 0 . . . X m n - l . a d ( X ) X ^ (mi) u 0 n-1 1 1 because a d ( X ) G r C G C N f o r 0 i ^  n - 1 . Furthermore m 111 . J-a d ( X ) X ^ = 2- X ^ 1 l . . . { X , X n j j ] . m n 3 J - X n ^ J and s i n c e r X , X n j j * ] i s i n G— we must a c t u a l l y have \ x , X n ? j " ] i n f o r a l l j such t h a t m . 4 0 . I f t h e r e i s such a j , p i c k X = S ( X n ) j ) ; s o [ s ( X n j j ) , X n ? i s i n 0 n t h e o t h e r hand,by I . 6 . ( a ) :B(X n i J;T(X n j ) ) - Z r n - 63 -A whi c h l i e s i n A. S i n c e K~C\ A = 0, B ( X n j j , T ( X n j ,) )' = 0, and t h e r e f o r e X_ • = 0 (B T,-being p o s i t i v e d e f i n i t e ) . T h i s c o n t r a -d i c t i o n shows t h a t j = 0 f o r a l l . ; j . R e p e a t i n g the argument, r e p l a c i n g r n s u c c e s s i v e l y by r n _ - j _ , . . mo A r ^ , we c o n c l u d e t h a t P ='. cCm^Xg l i e s i n S ( G ^ ) , n e c e s s a r i l y ' A . A d ( H ) - i n v a r i a n t . 2. C o r o l l a r y . 1 D ( G / H A N ) , i D ( G A / H A ) , and U ( K A / ( H O K) A)®fl)(A) a r e i s o m o r p h i c , commutative a l g e b r a s . A H A A P r o o f . The k e r n e l o f t h e epimorphism R: U ( G _ C > — > I D ( G / H ^ N ) A H A . A . i s c l e a r l y the same as t h a t o f the homomorphism U ( G Q ) . — - > I D ( G A / H ^ ) , T h i s l a t t e r homomorphism i s a l s o s u r j e c t i v e , because H — has the A d ( H A ) - s t a b l e complement (A*~S i n G - so t h a t ( S C G ^ / S C G ^ H - ) 1 ^ H A = . S ( G - ' " S ) . I t f o l l o w s t h a t ffi(G/HAN) i s i s o m o r p h i c w i t h D ( G A / H A ) . Moreover, 1 D ( G A / H A ) i s commutative: i f G A / H A i s c o n n e c t e d , t h i s a well-known r e s u l t on Riemannian symmetric spaces; i t a l s o h o l d s f o r d i s c o n n e c t e d s p a c e s , because r e -s t r i c t i o n o f i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s t o the' c o n n e c t e d component o f t h e base p o i n t i s an i n f e c t i v e homomorphism. F i n a l l y , the isomorphism ] D ( G A / H A ) = ffl ( K A / ( H A K ) A ) $ I D ( A ) i s a consequence o f the isomorphism o f symmetric spaces G A / H A = K A / ( H A K ) A A A . . - 64 -/A I t f o l l o w s from L a n d 2., t h a t a f u n c t i o n f on, G / H N i s an e i g en f u n c t i o n of ID (G/H AN) i f , and o n l y i f , the map v f : G / N - > E ( G A d e f i n e d above tak e s on v a l u e s i n an ei g e n s p a c e o f I D ( G A / H A ) . A HA In t h a t case t h e r e a r e a l g e b r a homomorphisms yUJU(K-£-) ——=5> (E and •V:U(A C) = S ( A C ) — s o t h a t R(X1 + X 2 ) f = CJU(X1) +V(X.2))f f o r a l l X x i n U ( K § ) H A and X 2 i n _ U ( A ). The c o n d i t i o n R.(X)f = 'i/(X)f f o r a l l X i n U ( A C ) i s e q u i v a l e n t to f ( g a H A N ) = a V f ( g H A N ) f o r a l l a i n A and g i n G where "V i n Agois. t h e ' r e s t r i c t i o n o f ."V t o AQ. I f the subspaces E ^ ( K A / ( H A K ) A ) .of E(K A/(Hr> K ) A ) a r e n o t o n l y p r i m a r y , b u t a c t u a l l y i r r e d u c i b l e (as i s always t h e case i f . K . A / ( H r \ K ) A i s c o n n e c t e d ) , then the o t h e r c o n d i t i o n uA R-(X)f = Jx(X)f f o r a l l X i n U ( G _ | ) h o l d s f o r a l l f i n E C J . ^ ( G / H A N ) by Schur^s Lemma. I n g e n e r a l , however, we can o n l y say t h a t E Q- ^ ( G / H A N ) decomposes i n t o the sum o f a f i n i t e number o f ei g e n s p a c e s o f ffl(G/HAN). We can a l s o a s s e r t t h a t E ( j ^ ^ ( G / H A N ) always c o n s i s t s o f e i g e n f u n c t i o n s o f the c e n t e r o f U ( G _ C ) (which, a c t s as a s u b a l g e b r a o f ID (G/H AN) by the r e g u l a r r e p r e s e n t a t i o n ) . Indeed, i f X i s an element i n the c e n t e r o f U(GQ) t b e r e must be an element X i n t h e c e n t e r o U ( G ^ ) so t h a t RQ^P^(X) = R ( X ) . As an element o f the c e n t e r X does a c t by s c a l a r m u t i p l i c a t i o n „on the p r i m a r y subspace E < j ^ ( G A / H A ) o f E ( G A / H A ) . - 65 -For the C a s i m i r element SD. i n U(G) thes e remarks can be made more e x p l i c i t : 3. Lemma. D e f i n e .Q, A and X ' i ^ as i n 1.5. L e t Z p be t h e element i n A s a t i s f y i n g B.(Zp,Z) = p(Z) f o r a l l Z i n A . Then D. = - - Q K A + n A ;+ 2z p . A A I f ^ U ^ . i s t h e e i g e n v a l u e o f XI K A on E f f(K / ( H A . K ) A ) , then /*<r " B(p , f») - B (AA) i s t he e i g e n v a l u e o f T^(jQ)on E <^(G/H AN). P r o o f . The f o r m u l a f o r -TL i s e q u i v a l e n t t o the a s s e r t i o n t h a t SX= - X 2 K A + -CZA mod UCGXH^ + N). To prove t h i s , choose a b a s i s \ X, I o f G f o r each r i n R, s a t i s f y i n g ( I ) S(X. ) = + T(X, ) and (2) B(X.. ,Xi ) = -t-§ . ". . Then fx. ,X, 1 = + Z r (1.6. ( a ) ) , i r j r . — 1 r , j r *- x r - L - X •> — L so Ci= - Q . A + D + I + (x X. + x. x± ) k A sr>r> L r i - r 1 r r , = - Z r z K A + o +. z ^ ( r ) z r w h i c h proves the f o r m u l a f o r X 2 . • The a s s e r t i o n about i t s e i g e n -v a l u e i s the n c l e a r . - 66 -R e t u r n i n g t o X*, l e t nu be a system o f double c o s e t r e p r e s e n t a t i v e s f o r ( H A K ) A K T / K A , s e t w.„= m-KA, N. = m.NmT1, x* = x* , X* = G ' x t . A A i i ' i l i ' i W i ' i i For a U s p h e r i c a l f u n c t i o n " cr on KA/(Hr\ K ) A ( a s d e f i n e d a b o v e ) , and f o r i n A g , l e t F a ^ y(X£) be the space o f f u n c t i o n s f:X^—>J!E s a t i s f y i n g : (1) f ( g a . x * ) = a w i ( - f + i ^ f ( g . x t ) (2) dim(cr) $ A f ( g m - x ^ M n T 1 ) dm = f(m) K (3) ( a ) I f F = E , then f i s i n E ( X | ) . (b) I f F = L 2 , then k K > f ( k * x f ) i s i n L 2 ( K ) . As a r e p r e s e n t a t i o n o f : - : G ; F ^ ^ X t ) i s e v i d e n t l y i s o m o r p h i c w i t h FCT N ( G / H A N . ). Moreover, i n the u n i t a r y case we have a d i r e c t I n t e g r a l d e c o m p o s i t i o n L 2 ( X * ) = , ^ - j L L ^ A C X * ) dim(cr-) d ^ . In th e d i f f e r e n t i a b l e case we can a s s e r t t h a t E Q - ^ C X * ) con-s i s t s o f e i g e n f u n c t i o n s o f the c e n t e r o f U ( G Q ) under the r e g u l a r r e p r e s e n t a t i o n and i s the d i r e c t sum o f a f i n i t e number o f e i g e n s p a c e s of the (commutative) a l g e b r a o f G - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on X*. One f i n i a l remark: the spaces F c ^ ( X ? ) depend o f c o u r s e on t h e c h o i c e o f the d o u b l e c o s e t r e p r e s e n t a t i v e s . But d i f f e r e n t c h o i c e s l e a d t o spaces c a r r y i n g i s o m o r p h i c r e p r e s e n -t a t i o n s o f G . Moreover, the c o n d i t i o n (1) above i s i n d e p e n -dent o f such a c h o i c e . I f a c t , , t h e r e i s a ( w e l l - d e f i n e d ) r i g h t a c t i o n o f A on X * g i v e n by ( g ' x / O a ^ gw ( a O*x* T h i s e x p l a i n s ; the w. i n ( 1 ) . - 67 -4. Example. To i l l u s t r a t e the r e s u l t s o f t h i s c h a p t e r I r e t u r n t o the c o n i c a l d u a l o f the r e a l h y p e r b o l i c space S 0 ( p , q ) o / S 0 ( p , q - l ) o . As i n IV.9. i d e n t i f y S 0 ( p , q ) o / S 0 ( p , q - l ) The r i g h t a c t i o n o f A on- X* mentioned above i s j u s t s c a l a r m u l t i p l i c a t i o n : x * * a t .= e^Sc*. K A c o n s i s t s o f m a t r i c e s i n K = S O ( p ) x S O ( q ) which l e a v e the subspace spanned by e-j. and e n i n v a r i a n t and r e s t r i c t t o +1 on t h i s subspace. ( H A K ) ^ c o n s i s t s o f m a t r i c e s i n K A which a r e t r i v i a l on He + l e . T h e r e f o r e ••' o f m a t r i c e s a t i n BL, d e f i n e d by a t * e ^ = c o s h ( t ) e i + s i n h ( t ) e . at ' e n = s i n h ( t ) e ^ + c o s h ( t ) e a *e. = e. f o r i ^ l . n . t i I n n 1 i f p = 1 o r q = 1. We a l s o know t h a t 1 i f p. ~i I' ' or q = 1 l Z 2 i f p = L and q 4 1. So i f p,q 4:,1, then X* i s connected and the " s p h e r i c a l f u n c t i o n s it - 68 -on K / ( H r ) K ) A a r e the c h a r a c t e r s 0" = +" 1 o f Z 2. The space F ^ ^ C X * ) (A i n Ag,tf~= + 1) c o n s i s t s o f homogeneous f u n c t i o n s o f d e g r e e — p + C^ , even o r odd a c c o r d i n g ascr=.+ 1 or(T= - 1. Tf; b u t : q ; / l : , then X * has two connected components: X * =r X|UX* and F<^(X*.) ( A i n A*, <5~ = +1>.) c o n s i s t s o f homogeneous f u n c t i o n s o f d e g r e e ~ v-'A s u p p o r t e d on X * . I n the r e m a i n i n g c a s e , when q - 1, the symmetric space X - i s R i e m a n n i a n , and we a r e on f a m i l i a r ground. / - 69 -5. Appendix. L e t G-^  and G 2 be the t r a n s f o r m a t i o n groups on the symmetric space G A/H A c o r r e s p o n d i n g t o the a c t i o n s o f G A and o f K A A r e s p e c t i v e l y . ( I n the n o t a t i o n i n t r d u c e d e a r l i e r G1 ^ G A / ( G A ) " and G 2 - K A/(K A)°X A. ) The a s s e r t i o n i s t h a t G]_ = G 2. A A Si n c e K i n t e r s e c t s e v e r y c o n n e c t e d component o f G (as remarked d u r i n g the p r o o f o f .IV.4.) i t s u f f i c e s t o show • t h a t G]_ = G_2« D e n o t i n g the i n v o l u t i o n s o f G^ and G^ i n d u c e d - S by the i n v o l u t i o n S o f (Thy the same l e t t e r , we have G-^  = G_2^  (•= G—/H^), because b o t h G^ and G 2 a c t t r a n s i t i v e l y on G A/H A. S i n c e G-^  and G 2 a l s o a c t e f f e c t i v e l y (by d e f i n i t i o n ) ^  S S n e i t h e r G^ . n o r G _ 2 c o n t a i n s : a ^ . n o n - . t r i v i a l i d e a l o f o r ^  . : F i n a l l y , i t i s c l e a r t h a t bothi:. G^ and G_2 a r e r e d u c t i v e . The a s s e r t i o n G^ = G_2 now f o l l o w s from: • Lemma. L e t G be a r e d u c t i v e L i e a l g e b r a , S an i n v o l u t i v e S automorphism o f G. I f G does n o t c o n t a i n any n o n - t r i v i a l i d e a l o f G, then G = rG~ S,G~ S] + G~ S ( i . e . G S = U G~S,G~.SQ)v P r o o f . S i n c e t h e c e n t e r o f G i s S - s t a b l e and cannot i n t e r s e c t S - S G , i t must l i e i n G . By s p l i t t i n g o f f the c e n t e r we may t h e r e f o r e assume t h a t G i s s e m i s i m p l e i n the f i r s t p l a c e . .In t h a t case the K i l l i n g form o r t h o g o n a l complement o f - S - S • • - S • ' £.G- jG 3+G (which i s an i d e a l o f G ,.as one e a s i l y checks u s i n g S -q S G = G . + G ) i s an i d e a l o f G and l i e s i r i G.-^  .hence i s t r i v i a l - 70 -C h apter V I . The Radon t r a n s f o r m and i t s d u a l . The r e s u l t s o f the a n a l y s i s on the c o n i c a l d u a l X* can be-.used as a t o o l f o r t h e a n a l y s i s on t h e symmetric space X i t s e l f . The v e h i c l e f o r the t r a n s i t i o n from X t o X* and v.v. i s the "Radon t r a n s f o r m " . T h i s t r a n s f o r m has been i n t r o d u c e d f o r Riemannian symmetric spaces by S. Helgason (5 ) from a g e o m e t r i c p o i n t of v i e w . I t can a l s o be m o t i v a t e d from a p u r e l y a l g e b r a i c - a n a l y t i c s t a n d p o i n t by a p p l y i n g F r o b e n i u s R e c i p r o c i t y t o the problem o f r e l a t i n g t h e r e g u l a r r e p r e s e n t a t i o n o f G on G/H t o P - s e r i e s - > r e p r e s e n t a t i o n s . Here I. s h a l l adopt the " g e o m e t r i c " d e f i n i -t i o n , m a i n l y because i t i s m a n i f e s t l y i n t r i n s i c , i . e . depends o n l y on the s t r u c t u r e o f the symmetric space G/H. R e c a l l t h a t t h e r e i s a one-to-one cor r e s p o n d e n c e between p o i n t s i n X and c e r t a i n submanif olds: o f X>" a n d v . v . : w i t h a p o i n t x i n X a s s o c i a t e the s u b m a n i f o l d C*(x) ~>f X* c o n s i s t i n g . o f a l l c y c l e s p a s s i n g t h r o u g h x; w i t h a p o i n t x* i n X* a s s o c i a t e the s u b m a n i f o l d C ( x * ) of X which -as a s e t - c o i n c i d e s w i t h x*. Thus C * ( g e x 0 ) = U gH-xf and C(g»x*) = gw(N)-x measures on C*(g»x Q) and on C(g«x&) by o* D e f i n e ( dh an H - i n v a r i a n t measure on H/H A), and - 71 -j f = j f(gn w»x 0) dn •C*(g-x*) w(N) where dn i s an i n v a r i a n t measure on w(N) n o r m a l i z e d so t h a t w 5 • , x f ( n w ) d n w = i f.Cmwnm^1) dn. w(M) N The measure dn on N can be f u r t h e r n o r m a l i z e d so t h a t the measure on C ( x * ) c o i n c i d e s w i t h t h e measure d e r i v e d from the pseudo-Riemannian m e t r i c o n . t h i s s u b m a n i f o l d o f X ( f o r a l l x* i n X * ) . To p rove t h i s a s s e r t i o n i t s u f f i c e s (by G - i n v a r i a n c e ) t o show the volume forms on C ( x * ) (w i n W) d e r i v e d - from the m e t r i c s a t i s f y A d ( m w ) * e w ( x 0 ) - 0 e ( x o ) i f we i d e n t i f y the t a n g e n t space o f . C ( x * ) = w ( N ) * x 0 w i t h w(N) i n the u s u a l manner. To see t h i s , s t a r t w i t h the d e f i n i t i o n of 9 w ( x Q ) : f o r X L , X 2 . . . X n i n w(N) (n = d i m ( N ) ) , 0 w ( x o ) ( X L , . . . X n ) = ( d e t ( B g ( X i , X j ) ) ^ j 1^ * I n P a r t i c u l a r , i f X-^,...Xn i s a b a s i s o f N • so t h a t Bg.(X i,Xj) = ±S.±},j and-ST(X^) .= ±'X±, t h e n A d ( m w ) * e w ( x 0 ) ( X 1 , . . . . X n ) _ = & w ( x 0 ) ( A d ( m w ) X 1 , . . . A d ( m w ) X n ) = [ d e t ( B s ( A d ( m w ) X i , A d ( m w ) X i ) l ' 2 = l d e t ( B s ( A d ( S ( n C 7 1 ) m w ) X i , X j ) ) l = |det Ad(S(m^ 1)m w)|N \ w = 1 , ( f o r the f o u r t h equality;,' one has t o remember t h a t S(m - i)m i s i n K A f o r a l l m i n K A - see I I I . 8 ) . 72 -The "Radon t r a n s f o r m " i s t h e map D ( X ) — ? E ( X - ) , fR>f*, d e f i n e d by f * ( x * ) = J ' f C ( x * ) o r e q u i v a l e n t l y •f*(g»x*) = I f ( g m w n m ^ 1 r x 0 ) dn . (The i n t e g r a l converges f o r a l l f i n D ( X ) , because C ( x " ) i s c l o s e d i n X- see I V . 7 . ( b ) . ) A n a l o g o u s l y , - the " d u a l Radon t r a n s f o r m " i s t h e map D(X*)—>.E(X), " a l s o w r i t t e n as f v* f * , d e f i n e d by f - ( x ) = 5 f o r e q u i v a l e n t l y f*(g«x ) = j f(gh«x&) dh. . H/H A ( A g a i n , t h e i n t e g r a l converges f o r a l l x i n X because C*(x) i s c l o s e d i n X* - t h i s f o l l o w s f r om t h e f a c t t h a t HN i s c l o s e d i n G, w h i c h wasshown i n I V . 7 . ( b ) . ) The f o l l o w i n g lemma shows t h a t t h e d u a l Radon, t r a n s f o r m i s i n d e e d t h e d u a l map o f t h e Radon t r a n s f o r m : 1. Lemma. F o r f i n D(X) and g i n D(X*) J f * ( x * ) g ( x * ) dx* = \ f ( x ) g * ( x ) dx. X* X P r o o f . By l i n e a r i t y we may assume t h a t t h e s u p p o r t o f g i s c o n t a i n e d i n G'x^ f o r some w i n W. I n t h a t c a s e t h e lemma f o l l o w s from ( ( 6 ) p.16). 73 I t f o l l o w s from t h e lemma t h a t we can -and s h a l l - e x t e n d the d u a l Radon t r a n s f o r m t o a map E'(X*).-7>D'(X), Fh->F*, by - s e t t i n g F * ( f ) - F ( f * ) f o r F i n E'(X*> and f i n D ( X ) . (The Radon t r a n s f o r m i t s e l f can o f c o u r s e be extended a n a l o g o u s l y , but such an e x t e n s i o n w i l l n o t be used h e r e . ) More g e n e r a l l y , f o r F i n D'»(X*) d e f i n e F* i n D'(X) by F * ( f ) = j" f*(x*) F ( x * ) dx*' (1) X* p r o v i d e d t h i s " i n t e g r a l " can be g i v e n a s u i t a b l e i n t e r p r e t a t i o n f o r a l l f i n D ( X ) , e.g. i f F i s a f u n c t i o n f o r which i t conver-ges. S p e c i f i c a l l y , suppose F i s a f u n c t i o n i n E q ^ ( X t ) ( n o t a t i o n as i n V.); i f (1) converges a b s o l u t l y f o r a l l f i n D(X) (and i t w i l l be shown below t h a t i t does, a t l e a s t f o r 'A i n an open s u b s e t o f A " ) , we can compute F* i n terms o f the r e s t i c t i o n o f F t o the K - o r b i t K»xJ: F * ( f ) - F ( f * ) J A f ( g n i ) F(g.x*) d n ± dg l * l =T fCkam.nmT1) F ( k a - x * ) a w i ( 2 p ) dk da dn KXAXN 1 1 = j f(kam inmT 1) F ( k - x * ) a W i ( P H ^ } dk da dn KKAXN 1 1 = f f(km.anm: 1) F(k'x*) a ^ C^dk da.dn KAAXN 1 1 1 =5 , A ^ A f(kmm anm" 1) F:(km-x*) a f + t ^ d m da dnl K/K K AXAXN I I i - 74 -•' = ] HA- f (kn-ia-V-imT 1) F C k m " 1 ^ L ' x * ) a " ^ " * ^ dm da dn] dk . K /K K A A N 1 To s i m p l i f y t h i s e x p r e s s i o n , d e f i n e f o r - g i n t h e open dense sub-s e t UHm .KAAN o f G: j J a ( g ) : the u n i q u e element i n A so t h a t g ^ Hm.K Aa(g)N f o r a p p r o p r i a t e j . m(g): any element i n K A so t h a t g e Hm m(g)AN f o r a p p r o p r i a t e j . (Of c o u r s e m(g) i s a c t u a l l y u n i q u e up t o l e f t m u l t i --1 A. p l i c a t i o n by elements, from m^ ( H f ) K ) my) "0 i f g e HmjK AAN ( 1 i f g & HmjK AAN. S e t t i n g g = n ~ l a _ l m " l m ^ and u s i n g I I I . 1 0 . t h e above i n t e g r a l becomes: ' - . ' { • • C f ( k g ) > ( k m ( g - l ) - l m i 1 . x * ) ^ ( g - l ) a ( g - 1 ) * ( ^ ' 4 ) dg dk J K / K A ;G/H 1 - =J f ( g ) J" A F ( k m ( g - l k ) - 1 m ; 1 - x * ) & ( g _ 1 k ) a ( g - 1 k ) " ( f ' i ' A ) dk dg. G/H . K/K i l l So F , v i s the f u n c t i o n on X g i v e n by , . •F*(g.x 0) = j . F ( k m ( g - l k ) - l m T 1 . x ^ ) 5 , ( g _ 1 k ) a ( g - 1 k ) ~ ( p H l ^ dk. ( 2 ) K / K 1 So f a r we o n l y used the r e l a t i o n F ( k a - x * ) = a w i ( ~ f + c ^ ) F ( k - x * ) . 1 i But f u n c t i o n s i n E ^ ^ X ? ) a l s o s a t i s f y F ( g . x * ) = dim(tr) j" F(gm.x*) tf-(m_1) din.' 1 K A/(HAK) 1 R e p l a c i n g g by k m ( g ~ i k ) ~ 1 m 7 1 i n t h i s f o r m u l a we g e t - 75 -F C k m C g ^ k ) - 1 ™ : 1 ^ . ) - dim(cx) j F(km(g- 1k)" 1m~ 1m'xf) cr(m? 1) dm 1 1 K A / ( H C N K ) A 1 1 = dim(o) S A A FCkmmT^-.x'. ) C7(m1-m~1m(g~1k) 1m:;" K / ( H n K ) 1 . dm S u b s t i t u t i n g i n t o (2) g i v e s : F * ( g . x Q ) = J A{dim ( c r ) S F(kmm.-x^) c ^ m " ^ ^ ^ ) " 1 ! ! ^ 1 ) dm ] K / K K A / ( H K ) A 1 _ ( f H ^ ) _ ^ ( g ^ k ) aCg-^k) X dk = dim(<r).f A FCkm^.x*) cr(m.m(g 1k)~ 1m-: 1) &. ( g - 1 k ) I n t r o d u c i n g t h e n o t a t i o n F ^ k ) • = FCkm- 1^*) cr\(m) = cy(m jmn ) . , a i ( g ) = S i ( g ) a ( g ) ( H r i K ) A = ^ ^HQKA. w e , f i n a l l y ; a r r i v e a t the f o r m u l a F*(g-.x 0) = dim(a)J A F . ( k ) ct. ( m C g^k) a. ( g ^ k ) " ^ d k . ( 3 ) K / ( H A K ) . Some remarks c o n c e r n i n g t h i s f o r m u l a a r e i n o r d e r . F i r s t , ( 3 ) was d e r i v e d under the as s u m p t i o n o f a b s o l u t e convergence o f (l)„for a l l f i n D ( X ) . But r e a d i n g the argument backwards one sees c o n v e r s e l y t h a t ( 1 ) h o l d s p r o v i d e d ( 3 ) i s a b s o l u t e l y c o n v e r g e n t . Second, ( 3 ) has a g e n e r a l i z a t i o n t o d i s t r i b u t i o n s : l e t D / . ^ ( X £ ) be the space o f d i s t r i b u t i o n s on X * w h i c h s a t i s f y - 76 -(1)5 .... f ( g a - l - x f ) F(g:.-x*). dg G/HANi x 1 = aWi(p-KA)_f f(g«x*)- dg G/H N i -1 ( 2 ) J .. : J dim(a) f f (gm-x*) C(m) dm 1 F(g'x*) dg G/H N . . J K A / ( H K ) A 1 J 1 i = J A f(g«x*) F(g.x*) d l . G/H N. 1 1 l Then E < r ^ ( X ^ ) C D'0 ^ ( X ? ) , and.(3) remains v a l i d i n the d i s -t r i b u t i o n sense f o r a l l F i n D ' t f ^ ( X | ) p r o v i d e d the d e n s e l y d e f i n e d f u n c t i o n . ' g«x 0h> C^Cm'CgT1 ) _ 1 ) a i ( g " 1 ) " ( P * ^ ) extends t o a d i s t r i b u t i o n on X. Of co u r s e i n t h a t case F^ w i l l be a d i s t r i b u t i o n on K/(HaK) A and (3) r e p r e s e n t s the d i s t r i b u t i o n on X d e f i n e d by F*(f> =T AS' f(kg)C- i(m(g-l)-l) a . t g - l y ^ ) d g J F ^ k ) d k . ( 4 ) K/(HftK) A G/H 1 (The f a c t t h a t a d i s t r i b u t i o n i n D T f f A (X*) can be " r e s t r i c t e d " t o t he K - o r b i t K«xt t o produce the d i s t r i b u t i o n F^ can be shown by t h e argument used i n (6)p.80.) T h i r d , I mention p a r e n t h e t i c a l l y t h a t the r e l a t i o n o f the d u a l Radon t r a n s f o r m and F r o b e n i u s r e c i p r o c i t y r e f e r r e d t o e a r l i e r i s a p p a r e n t from ( 4 ) . Indeed, ( 4) shows t h a t the b i l i n e a r form E e ^ ( X * ) XD(X)-S> <C, (F,'f )H> F * ( f ) , c o i n c i d e s w i t h t h e " i n t e r -t w i n i n g form" c o r r e s p o n d i n g t o the E < _ ( K A / ( H 0 K ) A ) - v a l u e d d i s -t r i b u t i o n g b> dim(<j-)a^ ( g ~ ^ )~ ^ ^ L ^ A ( m ( g ~ ^ ) )^ Tj_ under F r o b e n i u s r e c i p r o c i t y ( c f . ( l O ) v . I , 5.3.2.1. p.404.).' - 77 -I now t u r n t o the problem o f e x t e n d i n g the d e n s e l y d e f i n e d f u n c t i o n s ' gH \—> a i ( g ~ 1 ) " ^ ' i t ^ 0 " i ( m ( g - l ) " : 1 - ) t o d i s t r i b u t i o n s on G/H. To b e g i n w i t h , i t i s c l e a r t h a t such a f u n c t i o n i s l o c a l l y i n t e g r a b l e i f , and o n l y i f , gH H) a . ( g " 1 ) " ^ 4 1 ^ i s ; and t h i s i s the case f o r a l l i (and f o r f i x e d ^ ) i f , and o n l y i f , gHv-^a(g"f) ^ i s l o c a l l y i n t e g r a b l e . Moreover, s i n c e g h v > a ( g _ l ) i s a c t u a l l y r i g h t i n v a r i a n t under the f u l l f i x e d p o i n t s e t G o f S (because i n t e r -s e c t s e v e r y c o n n e c t e d component o f G , see I I . 3 . ) we may assume S H!.':.= G f o r d e t e r m i n i n g l o c a l i n t e g r a b i l i t y . As i n I I I . we embed G/G S i n t o G as t h e s u b m a n i f o l d G =js(g)g _ 1 : g I n G ) (by t h e map gG l-> S ( g ) g ~ l ) . Our e x t e n s i o n problem w i l l be t r a n s f e r r e d from the symmetric space G/G^ - Gg t o the L i e group G w i t h the h e l p o f t h e f o l l o w i n g ; lemma, wh i c h shows t h a t the s u b m a n i f o l d Gg o f G i s t r a n s v e r s a l t o the o r b i t s o f the a c t i o n o f H^w^) (w i n W) on G by l e f t t r a n s l a t i o n . 1. Lemma. (a) F o r any w i n W, the m u l t i p l i c a t i o n map F: H Aw(N) X Gg-> G, F ( q , x ) = qx, i s a l o c a l d i f feomorphisnv onto an open s u b s e t (H\7(N)3g) o f -G. (b) W i t h a p p r o p r i a t e n o r m a l i z a t i o n s o f the i n v a r i a n t measures dq, dx, dg on H Aw(N), Gg, G: F*(dg) = dqxdx. (The i n v a r i a n c e o f dx means of c o u r s e i n v a r i a n c e under the a c t i o n g: x rtS(g)xg"l of G on Gg. ) - 78 -P r o o f . I t s u f f i c e s t o prove ( b ) , which i m p l i e s i n p a r t i c u l a r t h a t the tangent mapsoof F a r e l i n e a r isomorphisms. L e t c be the f u n c t i o n on H^wCtOxGg d e f i n e d by F*(dg) = c^dq dx. S i n c e F ( q , x ) = q F ( e , x ) , c ( q , x ) = c ( e , x ) ; s i n c e F ( e , S ( g ) g " l ) = S ( g ) " F ( e , e ) g - 1 - , c ( e , S ( g ) g _ i ) - c ( e , e ) . So c s. c ( e , e ) i s c o n s t a n t . F i n a l l y , from the d i r e c t d e c o m p o s i t i o n G = H— + W ( N ) + G i t f o l l o w s t h a t t h e tangent map o f F a t (e,e) i s an isomor-phism, so t h a t c =f- 0. 2. C o r o l l a r y . F o r a l e f t H A w ( N ) - i n v a r i a n t d i s t r i b u t i o n F on the open s u b s e t H^wCN )GC o f G t h e r e i s a unique d i s t r i -b u t i o n F on Gg so t h a t i I j A f ( q , x ) dq[ F ( x ) dx = ^ ^ H f ( q , y ) ] F ( x ) dx Gg lAvCN) H w(N) " ; qy=x f o r a l l f i n D ^ N j X G g ) . Moreover, i f F i s a l o c a l l y i n t e g r a b l e f u n c t i o n on ) G O , then F:: i s a l o c a l l y i n t e g r a b l e f u n c t i o n on Gg. P r o o f . I f F i s a l o c a l l y i n t e g r a b l e f u n c t i o n , F i s j u s t the r e s t r i c t i o n o f F t o Gg, which i s l o c a l l y i n t e g r a b l e by the above lemma a n d F u b i n i ' s Theorem and c l e a r l y v e r i f i e s the a s s e r t i o n o f the c o r o l l a r y . I n the g e n e r a l case t h e c o r o l l a r y f o l l o w s from a s i m i l a r a p p l i c a t i o n o f F u b i n i ' s Theorem f o r d i s -t r i b u t i o n s (see (3) p.11 f o r d e t a i l s ) . - 79 -I t f o l l o w s from the c o r o l l a r y t h a t i n o r d e r t o s o l v e our l o c a l i n t e g r a b i l i t y p r o b lem i t s u f f i c e s t o produce a l e f t H^w^) i n v a r i a n t f u n c t i o n on G ( o r even o n l y on H A w ( N ) ^ f o r some w i n W) which i s l o c a l l y i n t e g r a b l e and whose r e s t r i c t i o n t o Gg co-i n c i d e s w i t h the d e n s e l y d e f i n e d f u n c t i o n S(g ) g ~ l W>>•a(g~-'- ) To c o n s t r u c t such a f u n c t i o n I have t o i n t r o d u c e some more n o t a t i o n : s e t M = K ^ H ^ - G^/A , and f o r g i n the open dense s u b s e t NMAN of G, l e t a G ( g ) be the unique element o f A so t h a t g l i e s i n NMa^(g)N. Then S ( g ) g _ 1 v — > a X g " 1 ) c o i n c i d e s w i t h the r e s t r i c t i o n o f gj~->.a G(g) 2 t o Gg, i . e . a ( g - 1 ) 2 =' a G ( S ( g ) g - l ) : . i f g"-1- = hman ( w i t h h i n G S, m i n K ^ , a = a ( g ) i n A, n i n N ) , t h e n S ( g ) g - _ 1 = S(n" 1)aS(rn" 1)man = S ( n _ 1 ) S ( m _ 1 )ma 2n, which l i e s i n NMa 2N (because S(m~-*-)m i s i n K A f o r m i n K ^ ) . To d e t e r m i n e the l o c a l i n t e g r a b i l i t y o f g H - > a G ( g ) ( V i n A G ) I need so^e i n t e g r a l f o r m u l a s : 3. Lemma. W i t h a p p r o p r i a t e n o r m a l i z a t i o n o f the i n v a r i a n t measures: (a) 3 f ( g ) dg = $ A § f(kman) a^drn da dn dk G K / K MxAxN . (b) J f ( g ) dg =[_ f(nman) a 2 ^ dn dm da dn G NxMxAxN ( c ) j f ( k ) dk = $_ f.(k(n")) a F ( n ) f dn" K 7 K a N where f o r g i n G k ( g ) i n K and a ^ ( g ) i n A a r e elements so t h a t g l i e s i n k ( g ) M a K ( g ) N . - 8G -P r o o f . (a) and (b) f o l l o w from the Iwasawa d e c o m p o s i t i o n and the Bruhat d e c o m p o s i t i o n r e s p e c t i v e l y (by f a m i l i a r a r g u ments), ( c ) i s o b t a i n e d . b y comparing ( a ) and ( b ) . From 3.(a) and t h e r e l a t i o n a^(gman) = a G(g)«a i t f o l l o w s t h a t g \-5>ac(g) i s l o c a l l y i n t e g r a b l e on G i f , and o n l y i f , the '• • r • "V i n t e g r a l J A a r ( k ) dk converges a b s o l u t e l y . N o t i n g t h a t K / K _ y _ _ _j_ a g ( k ( n ) ) = a^(n) ^ (because n = kman i m p l i e s k = nn x a lm = nm-la'-^-n' w i t h n T i n N) we see from 3.(c) t h a t t h i s i s e q u i v a -C - -(if-tv) .-• l e n t t o the a b s o l u t e convergence o f J _ a^.(n) dn. Thus 2"V N (by 2.) g H l - ? a ( g - i ) i s . l o c a l l y i n t e g r a b l e on G/H whenever J _ a^(n) ^ 2"f > +" v^ dn converges a b s o l u t e l y . ( T h i s c r i t e r i o n i s N -v s u f f i c i e n t , b ut i n g e n e r a l n o t n e c e s s a r y , because gi—=>aQ.(g) need n o t be l o c a l l y i n t e g r a b l e on a l l o f G, o n l y on t h e open s u b s e t H ANGg.) Convergence c i t e r i a f o r t h i s i n t e g r a l a r e known, i f i n a s l i g h t l y d i f f e r e n t form ( (j.0) v . I I , Theorem 9.1.6.4 p.322). To f a c i l i t a t e c o m p a r i s o n ( e x t e n d : A t o an S - s t a b l e maximal a b e l i a n subspace A T as i n I I I . Choose a l i n e a r o r d e r i n A , V c c o m p a t i b l e w i t h the o r d e r i n A* d e f i n i n g N, i . e . A i n A T*,ft>0, i m p l i e s A[A ^ 0, a n d ^ ( A > 0 i m p l i e s ^ A 7 0. The a c t i o n o f S on A T * c l e a r l y permutes the s e t o f r o o t s R T o f the p a i r G,A', hence a l s o the Weyl chambers o f R' in . A'. I n p a r t i c u l a r , t h e r e i s a u n i q u e element wg i n t h e Weyl group o f R' so t h a t Sow c l e a v e s - 81 -the p o s i t i v e Weyl chamber i n A' i n v a r i a n t , hence a l s o t h e s e t Rj^ o f p o s i t i v e r o o t s i n R'. Moreover: 4. Lemma. The' f o l l o w i n g c o n d i t i o n s on a p o s i t i v e r o o t r T i n R' a r e e q u i v a l e n t : . ( a ) r ! \ A = 0. • (b) S(r») = r i ( c ) S(r») > 0. (d) w s ( r ' ) > 0. P r o o f . (a)<=>(b) f o l l o w s from A' = A' S + A'" s w i t h A'" S = A. • (b) = ) ( c ) t r i v i a l l y . (c)<=>(d) i s a r e s t a t e m e n t o f t h e d e f i n i t i o n o f W g : Sowg(Rj) = R_j, so W g(R|) - S ( R | ) . ( c ) =>(a): I f r ' 6 R ' , r» > 0, and r T \ A 4 0, then r ' | A > 0 (by c o m p a t i b i l i t y o f o r d e r s ) , so S ( r ' ) | A = - r T | A <0, and conse-q u e n t l y S ( r ' ) < 0 ( a g a i n by c o m p a t i b i l i t y ) . T h i s p r o v e s ( c ) =i> ( a ) by c o n t r a p o s i t i o n . Set N T =Y { G r i : r ' i n R | ( ' and N ' =f. { G_r, : r ' i n R]. j Then E = i G_r: r i n R + j = { G_ r T : r ' i n R^ J , r ' A 4 0 } = { G _ r i : r ' i n R|, w g ( r ' ) i n -R^\ = NT p| Ws-1(N' ) . - 82 -D e n o t i n g by a'(g) the "A 1-component" o f g a c c o r d i n g t o the Iwasawa d e c o m p o s i t i o n G = KA'N', ( 10) l o c . c i t . . a s s e r t s t h a t the i n t e g r a l $_ a'(H)" (f' + v ) dn ( V i n A'*) N T converges a b s o l u t e l y i f , and o n l y i f , Re B ( V , r ' ) > 0 f o r a l l r ' i n R' w i t h r ' IA 4 0. (Here f 1 = \ 2. m(r» ) r T , m(r' ) = dim G_r, . ) Regard A*as a. subspace o f A T * by e x t e n d i n g i n A* t o A T* by s e t t i n g i t e q u a l t o z e r o on A' . A l s o , i f i s i n A 1*, w r i t e *Al+ -v V_ a c c o r d i n g to.?A?'*= A'*S.+ A*. Thus ?V_- 'A'lA w h i l e ^ \ \ f \ - 0 . For ^  i n A* we o b v i o u s l y have a ' ( g ) ' ~ a ^ ( g ) ^ - . I t f o l l o w s t h a t L a K(H)-(K+ V) dn N i s a b s o l u t e l y c o n v e r g e n t i f , and o n l y i f , Re B( p + V , r ' ) ? 0 f o r a l l r ' i n R' w i t h r'JA 4 0. S i n c e p - f + t f_ w i t h f_= f> t h i s i s e q u i v a l e n t t o Re B ( p * V , r ' ) > B ( p ' , r ' ) f o r a l l r ! I n R| w i t h r'|A ^/ 0, o r Re B(f+V,r) > max |^B(^',r') : r 1 U = r j f o r a l l r i n R. Summerizing: 5. P r o p o s i t i o n . The d e n s e l y d e f i n e d f u n c t i o n s gH P> (J". (m(g ) ~ 1 ) . a i ( g ~ ) a r e l o c a l l y i n t e g r a b l e on-.. G/H p r o v i d e d Re B(-C/\,r) i s s u f f i c i e n t l y l a r g e f o r a l l r i n R. -.83 -I t s h o u l d be p o i n t e d out t h a t the b e h a v i o u r o f t h e s e f u n c t i o n s can be much b e t t e r t h a t one might e x p e c t from t h i s p r o p o s i t i o n . F o r example, i f G/H i s a Riemannian symmetric space ( i . e . H = K ) , then g w > a ( g ) = a^(g) i s a c t u a l l y an a n a l y t i c f u n c t i o n on G, hence so i s g H i ^ a ( g 1 ) f o r any V i n A g . I f G/H'" i s " a s e m i s i m p l e .Lie group r e g a r d e d as a symmetric space (as i n I I I . 1 1 . ( b ) ) , then the f u n c t i o n s gH H ? a ( g ) can be i d e n t i f i e d w i t h " c o n i c a l d i s t r i b u t i o n s " i n the sense o f ( 6 ) . In f a c t , w r i t i n g G = G'^G', as b e f o r e , we. g e t a ( g _ i ) = a^T (g^g2~"1" i f g = ( g ] _ , g 2 ) i n G'XG'* The r e s u l t s o f =(6). show that i.thej. d e n s e l y d e f i n e d f u n c t i o n s g't-^a^, (g) extend t o d i s t r i b u t i o n s on G l de-pending m e r o m o r p h i c a l l y on~V . The same t h e r e f o r e h o l d s f o r the d e n s e l y d e f i n e d f u n c t i o n s gH i-> a ( g " i ) on G/H c o r r e s p o n -d i n g t o CT s ' l i n 3. ( F o r a L i e group W = W and a ( g ) = a i ( g ) . ) F o r t h e r e a l h y p e r b o l i c spaces S O ( p , q ) Q / S O ( p , q - l ) Q i t i s a l s o p o s s i b l e t o d e f i n e an e x t e n s i o n by a n a l y t i c c o n t i n u a t i o n i n "V (see the example b e l o w ) . - 84 -6. Example. The Radon t r a n s f o r m on r e a l h y p e r b o l i c , s p a c e s . The r e a l h y p e r b o l i c spaces X •= S 0 ( p , q ) o / S 0 ( p , q - l ) o p r o -v i d e a c o n c r e t e example f o r the use o f the ( d u a l ) Radon t r a n s -form i n the a n a l y s i s on X. To b e g i n w i t h , we need a f o r m u l a f o r the f u n c t i o n g«x 0 ^ a ( g " l ) on X. So suppose g = hman, g i n G, h i n H, a = a ( g ) i n A, and n i n N ( n o t a t i o n as i n . 1 . 1 6 . ( b ) , 11.10., I I I . 1 1 . ( c ) , I V . 9 . ) . We know t h a t a a t i s a m a t r i x o f the form a«e^ = c o s h ( t ) e ] _ + s i n h ( t ) e n a«e n = s i n h ( t ) e - ^ + c o s h ( t ) e n a* e^ = e^ i f i . V 1,n f o r a p p r o p r i a t e t i n I , Moreover h , e n = e n m ' e i = i e i n . ( e i + e n ) . = eL + e R . T h e r e f o r e ( g _ 1 * e n ' e l + e n ) = ( n " 1 a " 1 m " 1 h - l . e n , e i + e n ) - + ( a _ 1 . e n , e i + e n ) = + ( - s i n h ( t ) e 1 + c o s h ( t ) e n , e i +e n) •= + ( s i n h ( t ) + c o s h ( t ) ) . So i f we i d e n t i f y A w i t h I. by l e t t i n g t h e element a = a - 85 -c o r r e s p o n d t o e , th e n a ( g _ 1 ) - l ( g _ 1 ' . ' e n , e i + e n ) \ . (4) The f u n c t i o n XA X" ~>A, g]_« xo»g 2' xe "^"^  a ( g ~ ^ g 2 ) , which i s d e f i n e d f o r any s e m i s i m p l e symmetric space, becomes pa r -t i c u l a r l y s i m p l e f o r the case a t hand: w i t h the i d e n t i f i c a t i o n X ={x i n R n : ( x , x ) = l j and X* = {x* i n JRn : ( x*,x*) = 0, x* 401 t h i s f u n c t i o n sends the p a i r x , x * i n X x X* i n t o l.(x,x*) 1 i n E.x = A. ( T h i s i s c l e a r from (4) and G - i n v a r i a n c e . ) I t i s now easy t o see what f o r m u l a (3) amounts t o : - the K - o r b i t K.x* i s the s u b s e t Y o f X* d e f i n e d by Y = |y i n l n : y ± 2 +y 2 2 +.-. +y p 2.-= i = - y p + 1 2 • + . . - y n 2 ] = s P " 1 ^ - 1 = K / ( H / I K A Assuming f o r s i m p l i c i t y t h a t p 4 L a n d q / 1 we have. X* = G«xg, K A/'(HnK) A - Z2, and the map G/HXR'/( Hj-\R ) A K A / ( H r ) K ) A , g , k i - > m ( g _ 1 k ) _ 1 becomes XXY- :5>Z2, x,y V-^3> s i g n ( x , y ) ; t he " s p h e r i c a l f u n c t i o n s " on K A / ( H o K ) A " a r e " the c h a r a c t e r s <7 =+ 1 o f Z 2; w r i t i n g C ( s i g n ( x , y ) ) = c T ( x , y ) a n d F Q f o r the f u n c t i o n ( o r d i s r t i b u t i o n ) on Y c o r r e s p o n d i n g t o the f u n c t i o n ( o r d i s t r i -b u t i o n ) . ^ on K/(Hr\K) A i n f o r m u l a (3) we f i n d ' t h a t (3) reads:. • F*(x) = j -F 0(y)cr(x,y) | ( x , y ) | " ( f ^ dy (5) Y w i t h P = J$(p + q - 2) and A i n = E. T h i s f o r m u l a admits a n o t h e r i n t e r p r e t a t i o n w h i c h c l a r i f i e s i t s meromorphic dependance on ^  . The c o n s t r u c t i o n i s as f o l l o w s ( c f . (9) ); A-- 86 -L e t F be i n E C ^ ( X * ) , a homogeneous f u n c t i o n o f degree - f>*V *\ on X", even. o r odd a c c o r d i n g as G" = +1 o r cT= -1 (see V . 4 . ) , F t h e r e s t r i c t i o n o f F t o Y as above. Because o f the i n t e -o g r a l f o r m u l a \ f ( x * ) dx* = \ ( f ( y a ) a1? dy a _ 1 d . . X* E + Y (wh i c h i s a s p e c i a l case o f IV.8.) we have a j f ( x ' * ) . F ( x * ) cix*.= f 'J f ( y a ) F G ( y ) a ^ ' ^ ' d y da X--' I * Y = % j J f ( y a ) F o ( y > U l P + V ' A _ l d y da (6) U Y (where Q~(a) = G*(sign(a».) I f cT(a) \ a ^"'extends t o a d i s t r i b u t i o n on R, the n (6) makes sense f o r a l l f i n D ( l . n ) and t h e r e f o r e d e f i n e s a d i s -t r i b u t i o n F^,c on fi.n s u p p o r t e d on X*. Now i t i s w e l l known t h a t cr(a) \ a\ " i s " i n f a c t a meromorphic, D T (E.)-valued f u n c t i o n o f V ; moreover, T(t<-A"))\'^\ ? + ^  ^  1 and F ( ^ ( f ^ - 0 ) ^ " w M ^ f ^ 1 a r e e n t i r e f u n c t i o n s o f * A ( 1 ). So i f f>-K'A i 1,0,-1,-2,... the n we can t a k e the " F o u r i e r t r a n s f o r m " o f t h i s d i s t r i b u t i o n r n ( w i t h r e s p e c t t o t h e b i l i n e a r form (,) on I. ) and compute ( i n the d i s t r i b u t i o n s e n s e ) : . ^ ( x ) = j F ( x * ) e 1 ^ * * ) dx* X X* * Y ° F ( y ) e i ( x ' y ) a (7(a) l a f ^ ' d y da - 87 -| r(^cA')cos(TT/2(^i-A)) J F Q ( y ) l ( x , y ) | i f (T = +1 (ir ( e u - A)sin(ir/2(^vA)) \ F ( y ) s i g n ( x , y ) l ( x , y ) l dy Y i f (T- -1 (where I have used t h e f o r m u l a f o r the F o u r i e r t r a n s f o r m o f <T(a) \ a \ f 4 c A " 1 - see ( 1 ) ) . B e i n g a homogeneous d i s t r i b u t i o n on HL1^ F ^ * c a n be " r e -s t r i c t e d " t o a d i s t r i b u t i o n Fx*\X on X (because X i s a sub-m a n i f o l d o f BLn t r a n s v e r s a l t o t h e m u l t i p l i c a t i v e a c t i o n o f R x. - T h i s can be shown by an argument s i m i l a r t o t h a t used i n ( 7 ) p . l l , and i s o b v i o u s whenever (6) c o i n c i d e s w i t h a l o c a l l y i n t e g r a b l e f u n c t i o n . ) Comparing (5) and (6) we f i n d t h a t (V(^X)cos(w / 2 ( ^ l - A ) ) ) " 1 Fx*\X i f cr = + 1 F* =' ( i r ( ^ c - A ) s i n ( ( T / 2 ( P l o \ ) ) ) " 1 F^ c\X i f ( f = - 1. We see .that as. long...as ^ c y \ i s n o t an i n t e g e r the d u a l Radon t r a n s f o r m extends t o a map o f ^ ( X * ) i n t o D ' ( X ) . The r e s u l t s o f ( 9 ) show t h a t t h e s e maps s u f f i c e f o r t h e d i r e c t i n t e g r a l d e c o m p o s i t i o n o f L ( X ) : as l o n g as tr*«-v\ i s n o t an . i n t e g e r L^-^(X*) (see V . 4 . ) i s i r r e d u c i b l e and the d u a l Radon t r a n s f o r m i s i n j e c t i v e on L^^(X<').; i f p-\,-'>A i s an i n t e g e r l e s s - 8 8 -than p the d u a l Radon t r a n s f o r m s t i l l extends t o a map o f D ' ^ l X " ) i n t o D'(X) (because i n t h a t case the i n t e g r a l i n (5) c o n v e r g e s ) , but i s no l o n g e r i n f e c t i v e . However, th e k e r n e l 2 o f i t s r e s t r i c t i o n t o L g + ^ ( X * ) has a G - i n v a r i a n t complement L ^ ^ X * ) 0 w h i c h is_ i r r e d u c i b l e . Moreover, we g e t a G-isomorphism: L 2 ( X ) =1 \+ L 2 ^ ( X * ) \c(«r,*)r2 d ^ where e(<r,$0 i s a c e r t a i n merpmorphic f u n c t i o n on Z2*(C w i t h o u t r e a l z e r o s , and p-^cN runs over the i n t e g e r s l e s s than(° . E x p l i c i t l y , e v e r y f i n L 2 ( X ) can be r e p r e s e n t e d i n the form f ( x ) = Z. J j v ( y ; ^ ) c r(x,y) j ( x , y ) | P + \\ (<r, A) \~2 dy c • JY^(y;<r^°) 0*(x,y.) I ( x , y ) | f"* - oj .^ 2 w i t h v ( ;s,>0 and v°( ;<r5J?) i n L ( J . ( Y ) . For A € BL*~the f u n c t i o n s v ( ; a,K) a r e u n i q u e l y d e t e r m i n e d by f , i n f a c t t hey a r e g i v e n by the " F o u r i e r t r a n s f o r m " r - - f - f A .v.(y;<y,X) = J f ( x ) o-(x,y) \ ( x , y ) \ dx. X - 89 -For '-(^^-Cp^A^Jeffi the f u n c t i o n v°( ;<r,?f) a r e unique o n l y i f they a r e r e q u i r e d t o l i e i n the subspace o f L 5 - (Y) c o n s i s t i n g 9 O o f the r e s t r i c t i o n s t o Y o f f u n c t i o n s i n L^_^(X*) . F i n a l l y , one has the " P l a n c h e r e l f o r m u l a " : J l f ( x ) | 2 dx = ^ 5 S \ v ( y ; c r , ^ ) \ 2 \ c ( c - ^ ) \ " 2 dy X o- E Y ^ Z $ |v°(y;<r,^)\2 dy. One w i l l n a t u r a l l y wonder t o what e x t e n t the r e s u l t s o f t h i s example can be g e n e r a l i z e d t o o t h e r s e m i s i m p l e symmetric spa c e s . As i s w e l l known, f o r Riemannian symmetric spaces such a g e n e r a l i z a t i o n i s i n d e e d p o s s i b l e . F o r L i e groups the s i t u a t i o n i s a l r e a d y much.more complex, and i t seems t h a t e s s e n t i a l e x t e n -s i o n s o f the t h e o r y a r e n e c e s s a r y i f i t i s t o a p p l y i n t h i s ge-n e r a l i t y . - But t h e s e s p e c u l a t i o n s l e a d t o problems f a r beyond the scope o f t h i s t h e s i s . - 90 -B i b l i o g r a p h y . 1. G e l ' f a n d , I , M . and S h i l o v , G . E . : G e n e r a l i z e d f u n c t i o n s , v o l . I , Academic P r e s s , New York, 19 64 2. H a r i s h - C h a n d r a : S p h e r i c a l f u n c t i o n s on a s e m i - s i m p l e L i e group I , I I , Amer. J . Math., v o l . 8 0 (1958), pp. 241-310, 553-613. 3. - Harmonic a n a l y s i s on s e m i - s i m p l e L i e g r o u p s , B u l l . Amer. Math. Soc., v o l . 7 6 (1970), pp. 529-551. 4. He l g a s o n , S.: D i f f e r e n t i a l geometry and symmetric s p a c e s , Academic P r e s s , New Yo r k , 19 62. 5. - D u a l i t y and Radon t r a n s f o r m f o r symmetric s p a c e s , Amer. J . Math., v o l . 8 5 (1963), pp. 667-692. 6. - A d u a l i t y f o r symmetric spaces w i t h a p p l i c a t i o n s t o group r e p r e s e n t a t i o n s , Advances i n Math., v o l . 5 ( 1 9 7 0 ) , pp. 1-154. 7. - A n a l y s i s on L i e groups and homogeneous sp a c e s , Amer. Math. Soc., R e g i o n a l Conference S e r i e s , number 14, 1972. 8. Loos, 0., Symmetric s p a c e s , I , I I , W. A. Benjamin, New Y o r k , 1969. 9. S t r i c h a r t z , R.S., Harmonic a n a l y s i s on h y p e r b o l o i d s , J . o f F u n c t i o n a l A n a l y s i s , v o l . 1 2 . (1973), pp.341-383. 10. Warner, G., Harmonic a n a l y s i s on s e m i - s i m p l e L i e groups I , I I , S p r i n g e r - V e r l a g , New York, 1972. 

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