Lie Groups an d Symmetric Space s In Memory o f F. I . Karpelevic h

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Friedrich Karpelevich , 1927-200 0

American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 0

Advances in the Mathematica l Sciences—5 4 {Formerly Advances in Soviet Mathematics)

Lie Groups an d Symmetric Space s In Memor y o f F . I . Karpelevic h

S. G . Gindiki n Editor

American Mathematica l Societ y !? Providence , Rhod e Islan d

http://dx.doi.org/10.1090/trans2/210

ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E

V. I . ARNOL D S. G . GINDIKI N V. P . MASLO V

2000 Mathematics Subject Classification. Primar y 00B30 , 22Exx, 53C35.

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Contents

Preface vi i

Friedrich Karpelevich : Hi s earl y year s i n mathematic s E. B . DYNKI N x i

Asymptotic distributio n o f eigenvalue s fo r certai n element s o f th e grou p rin g of a compac t Li e grou p D M I T R I N . A K H I E Z E R 1

Minimal homogeneou s submanifold s o f symmetri c space s D M I T R I V . A L E K S E E V S K Y AN D A N T O N I O J . D i S C A L A 1 1

The hea t kerne l o n noncompac t symmetri c space s

J E A N - P H I L I P P E A N K E R AN D P A T R I C K O S T E L L A R I 2 7

On th e uniquenes s o f Fourie r Jacob i model s fo r representation s o f [7(2,1 ) E H U D M O S H E B A R U C H , ILY A P I A T E T S K I - S H A P I R O , AN D S T E P H E N

R A L L I S 4 7

Notes o n integra l geometr y fo r manifold s o f curve s J O S E P H B E R N S T E I N AN D S I M O N G I N D I K I N 5 7

Quantization o f Alekseev-Meinrenke n dynamica l r-matrice s B E N J A M I N E N R I Q U E Z AN D P A V E L E T I N G O F 8 1

Analysis o n th e crow n o f a Riemannia n symmetri c spac e

J A C Q U E S F A R A U T 9 9

Quaternionic quasideterminant s an d determinant s I S R A E L G E L F A N D , V L A D I M I R R E T A K H , AN D R O B E R T L E E W I L S O N 11 1

Product formul a fo r c-functio n an d invers e horospherica l transfor m S I M O N G I N D I K I N 12 5

The dua l horospherica l Rado n transfor m a s a limi t o f spherica l Rado n transforms J . HlLGERT , A . PASQUALE , AN D E . B . VlNBER G 13 5

The Gindikin-Karpelevi c formul a an d intertwinin g operator s A. W . K N A P P 14 5

Multiplicity on e theore m i n th e orbi t metho d T O S H I Y U K I KOBAYASH I AN D S A L M A N A S R I N 16 1

vi C O N T E N T S

The c-functio n fo r non-compactl y causa l symmetri c space s an d it s relation s to harmoni c analysi s an d representatio n theor y BERNHARD KROT Z AN D GESTU R OLAFSSO N 17 1

A formal identit y fo r affin e roo t system s I. G . MACDONAL D 19 5

Canonical representation s an d overgroup s V. F . MOLCHANO V 21 3

Pencils o f geodesie s i n symmetri c spaces , Karpelevic h boundary , an d associahedron-like polyhedr a YURII A . NERETI N 22 5

Poisson formul a fo r a famil y o f non-commutative Lobachevsk y space s M. A . OLSHANETSK Y AN D V.-B . K . ROGO V 25 7

Real semisimpl e Li e algebras an d thei r representation s A. L . ONISHCHI K 27 3

A calculation o f c-functions fo r semisimpl e symmetri c space s TOSHIO OSHIM A 30 7

The Abe l transfor m o n symmetri c space s o f noncompac t typ e P. SAWYE R 33 1

Preface

This volume contains paper s which friends an d colleague s o f Friedrich Karpele -vich (1927-2000 ) dedicat e t o hi s memory .

Friedrich Karpelevic h wa s bor n i n Moscow , o n Octobe r 2 , 1927 . Hi s teenag e years coincide d wit h th e difficul t wa r time . Fo r severa l year s hig h school s wer e closed. Friedric h worked on a factory; a s a result o f an accident he lost a part o f a fin-ger. H e was already 2 0 when h e entered th e university , havin g seriousl y considere d the optio n o f foregoin g universit y education . Fro m 194 7 to 195 2 h e wa s a studen t at Mosco w University . H e wa s on e o f mos t brillian t student s an d starte d researc h in hi s first undergraduat e year s unde r th e supervisio n o f E . Dynkin . Friedric h par -ticipated i n Dynkin' s semina r fo r hig h schoo l student s an d continue d t o participat e in Dynkin' s seminar s fo r a substantia l par t o f hi s mathematica l lif e (se e Dynkin' s contribution i n thi s volume) . H e establishe d himsel f a s a seriou s mathematicia n with hi s paper abou t th e characteristi c root s o f matrice s wit h nonnegativ e entries , published i n 1949 . Thi s pape r contain s a complet e solutio n o f a proble m pose d b y A. Kolmogorov . Befor e Karpelevich , thi s proble m ha d bee n considere d unde r som e restrictions b y N . Dmitrie v an d E . Dynkin .

In th e earl y fifties Karpelevic h studie d subalgebra s o f semisimple Li e algebras . He starte d b y givin g a descriptio n o f non-semisimpl e maxima l subalgebra s o f sim -ple complex Li e algebras . Th e classificatio n o f such subalgebra s ha d bee n obtaine d earlier b y V. Morosov. Fo r the nex t five years he studied semisimpl e subalgebra s o f real semisimpl e Li e algebras . Shortl y before tha t Dynki n ha d give n a descriptio n of th e semisimpl e subalgebra s o f comple x semisimpl e Li e algebras . Th e cas e o f real Li e algebra s i s muc h mor e difficult . Her e Karpelevich' s result s includ e a gen -eral statemen t abou t a canonica l embeddin g o f a rea l semisimpl e Li e subalgebra , which becam e widel y used . T o obtai n thi s result , Karpelevic h applie d th e theor y of symmetric spaces , which became hi s favorite mathematica l subject . T o complet e the stud y fo r th e cas e o f classica l Li e algebra s h e ha d t o wor k o n ver y comple x problems o f linear algebra . H e got a remarkable formul a fo r th e inerti a inde x o f a n invariant symmetri c o r Hermitia n for m o n th e spac e o f irreducibl e representatio n of th e rea l semisimpl e Li e algebra , whic h solve s th e proble m completely . Thes e results comprise d Friedrich' s Ph D thesis , an d the y wer e rewarde d i n 195 6 b y a very prestigiou s Mosco w Mathematica l Societ y Priz e fo r Youn g Mathematicians . A. Onishchi k prepare d fo r thi s volum e a n expositor y pape r abou t thi s work .

Due t o anti-Semitism , Karpelevic h wa s no t admitte d t o graduat e schoo l a t Moscow University . H e wa s preparin g hi s thesi s whil e teachin g i n a provincia l technical schoo l i n Novocherkassk and , startin g i n 1953 , in the Mosco w Institute o f

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Transport Engineering . Karpelevic h worke d i n thi s Institut e u p t o las t day s o f hi s life.

At th e en d o f th e fiftie s Friedrich' s interest s shifte d t o geometr y an d analysi s on homogeneou s manifolds . Togethe r wit h F . Berezi n i n 1958 , he compute d zona l spherical function s o n Grassmannian s i n terms o f specia l function s o f one variable . If th e spac e i s o f ran k one , zona l function s ar e function s o f on e variabl e an d ca n be expresse d i n term s o f th e Gaus s hypergeometri c function . Fo r comple x classi -cal group s the y wer e compute d b y Gelfan d an d Naimark . Numerou s attempt s t o do i t i n othe r case s wer e unsuccessful , s o th e computatio n mad e b y Berezi n an d Karpelevich remain s uniqu e u p t o thi s day .

The natura l developmen t o f Friedrich' s interes t i n spherica l function s wa s ou r collaboration o n the computatio n o f the c-functio n o f Harish-Chandra i n 1962 , an d later o n th e invers e horospherica l transform . Yo u ca n fin d mor e detail s abou t thi s work i n m y reminiscence s i n thi s volume .

One o f the mos t importan t o f Karpelevich' s result s i n the theor y o f symmetri c spaces i s hi s constructio n o f th e boundar y o f symmetri c space s o f non-positiv e curvature i n 1965 . I t i s based o n a detailed stud y o f the asymptoti c behavio r o f th e geodesies. Karpelevich' s boundar y ha s numerou s application s i n th e theor y o f th e eigenfunctions o f th e Laplace-Beltram i operator , whic h h e studie d fo r som e time . Soon h e change d hi s fiel d an d starte d t o wor k i n probabilit y theory . Yu . Nereti n wrote a n expositor y pape r o n boundarie s o f symmetri c space s fo r thi s collection . Karpelevich's work s i n probabilit y ar e reflecte d i n th e memoria l volum e "Analyti c methods i n applie d probability" , Yu . M . Suho v (ed.) , America n Mathematica l Society, Providence , RI , 2002 .

Friedrich Karpelevic h wa s one of the deepes t an d mos t origina l mathematician s working i n Lie groups an d symmetri c space s i n the secon d hal f o f the 20t h century . The contributor s o f thi s volum e hav e differen t relationship s wit h him . Som e o f them wer e happ y t o kno w Friedric h personall y an d t o collaborat e wit h him , som e know hi m onl y throug h hi s works , bu t the y al l shar e th e highes t opinio n abou t Karpelevich's mathematica l merits .

As we already mentioned, th e volume includes Dynkin's recollection on Karpele-vich's firs t step s i n mathematic s an d tw o expositor y paper s o n Karpelevich' s re -sults an d thei r development : Onishchik' s pape r o n subalgebra s o f rea l semisimpl e Lie algebra s an d Neretin' s pape r o n boundarie s o f symmetri c spaces . Severa l pa -pers ar e connecte d wit h th e produc t formul a fo r th e Harish-Chandr a c-function . A. Knap p discusse s it s application s t o intertwinin g operators . Th e paper s b y Os -hima an d b y Krot z an d Olafsso n conside r computation s o f c-function s fo r som e pseudo-Riemannian symmetri c spaces . I recal l tha t Friedric h ha d a stron g interes t in geometr y an d analysi s o n pseudo-Riemannia n symmetri c space s an d undertoo k several attempt s t o wor k i n thi s direction . I wrote dow n m y reminiscence s o n ou r joint wor k o n th e c-functio n an d th e horospherica l transform .

The pape r o f Hilgert , Pasqual e an d Vinber g consider s a n algebrai c versio n o f the inversio n o f th e horospherica l transfor m whic h i s als o connecte d wit h th e c-function.

Several paper s ar e dedicate d t o Karpelevich' s favorit e area , symmetri c spaces . Sawyer gav e a surve y o f th e Abe l transfor m o n noncompac t Rieman n symmetri c spaces whic h ha s stron g connection s wit h th e horospherica l transfor m an d it s in -version. I t als o ha s connection s wit h estimate s o f th e hea t kerne l o n suc h spaces . Anker an d Ostellar i prepare d th e surve y o n thi s area . I n Faraut' s contribution ,

PREFACE i x

some analyti c problem s o f complex crown s o f Riemann symmetri c space s ar e stud -ied.

The volum e contain s severa l contribution s o n theor y o f representation s an d other aspect s o f Li e groups : Kobayash i an d Nasrin' s pape r o n a ne w multiplicit y one theorem , Molchanov' s pape r o n canonica l representation s o n Hermitia n sym -metric spaces , an d Baruch , Piatetski-Shapir o an d Rallis ' pape r o n representation s of C/2, 1 f° r loca l fields. Akhieze r investigate s asymptoti c problem s fo r grou p ring s of compac t Li e groups .

Alekseevsky an d D i Scal a generaliz e Karpelevich' s 195 3 resul t o n totall y geo -desic orbit s o f reductiv e isometr y group s o n Rieman n symmetri c spaces . Th e vol -ume also contains a paper by Macdonald on affine roo t systems , a paper by Gelfand , Retakh, an d Wilson o n quaternionic determinant s an d quasideterminants , a contri -bution b y Olshanetsk y an d Rogo v o n non-commutativ e hyperboli c spaces , a pape r by Enrique z an d Etingo f o n quantizatio n o f dynamica l r-matrices , an d a pape r b y Bernstein an d Gindiki n o n integra l geometr y fo r curves .

We can see that man y of these contributions hav e a close relation with Karpele -vich's heritag e an d al l o f them , withou t a singl e exception , represen t th e area s o f mathematics whic h h e love d al l hi s life .

Simon Gindiki n Marc h 200 3

Papers o f F . Karpelevic h o n Li e Group s an d Symmetri c Space s

1. F . I . Karpelevich , On nonsemisimple maximal subalgebras of semisimple Lie Algebras, Dok -lady Akad . Nau k SSS R 7 6 (1951) , 775-778 . (Russian )

2. , Classification of simple subgroups of the real forms of the group of complex unimodular matrices, Doklad y Akad . Nau k SSS R 8 5 (1952) , 1205-1208 . (Russian )

3. , Subgroups of real Lie groups, Uspekh i Mat . Nau k 7 (1952) , no . 5 , 203-204 . (Russian ) 4. , Surfaces of transitivity of a semisimple subgroup of the group of shifts of a symmettric

space, Doklad y Akad . Nau k SSS R 9 3 (1953) , 401-404 . (Russian ) 5. , The classification of the simple subalgebras of the real forms of classical Lie algebras,

Doklady Akad . Nau k SSS R 9 3 (1953) , 613-616 . (Russian ) 6. , Simple subalgebras of the real Lie algebras, Trud y Moskov . Mat . Obshch . 4 (1955) ,

3-112. (Russian ) 7. , On semisimple subgroups of semisimple Lie groups, Uspekh i Mat . Nau k 1 0 (1955) ,

no. 1 , 196 . (Russian ) 8. , Hermitian and bilinear invariants of subalgebras of the matrix algebra, Uspekh i Mat .

Nauk 1 0 (1955) , no . 4 , 190-191 . (Russian ) 9. F . I . Karpelevic h an d A . L . Onishchik , Homology algebra of the path space, Doklad y Akad .

Nauk SSS R 10 6 (1956) , 967-969 . (Russian ) 10. F . I . Karpelevich , On the fibering of homogeneous spaces, Uspekh i Mat . Nau k 1 1 (1956) ,

no. 3 , 131-138 . (Russian ) 11. F . A . Berezi n an d F . I . Karpelevich , Zonal spherical functions and Laplace operators on some

symmetric spaces, Doklad y Akad . Nau k SSS R 11 8 (1958) , 9-12 . (Russian ) 12. F . I . Karpelevich , V . N . Tutubalin , an d M . G . Shur , Limit theorems for compositions of

distributions on the Lobachevski plane and space, Teor . Veroyatnost . i Primenen . 4 (1959) , 432-436; Englis h transl . i n Theor y Probab . Appl , 4 (1959) .

13. F . I . Karpelevich , Geodesies and harmonic functions on symmetric spaces, Doklad y Akad . Nauk SSS R 12 4 (1959) , 1199-1202 . (Russian )

14. , Horospherical radial parts of the Laplace operator on symmetric spaces, Doklad y Akad. Nau k SSS R 14 3 (1962) , 1034-1037 ; Englis h transl. , Sovie t Math . Dokl . 3 (1962) , 528-531.

15. S . G . Gindiki n an d F . I . Karpelevich , Plancherel measure for Riemannian symmetric spaces of nonpositive curvature, Doklad y Akad . Nau k SSS R 14 5 (1962) , 252-255 ; Englis h transl. , Soviet Math . Dokl . 3 (1962) , 962-965 .

x PREFAC E

16. F . I . Karpelevich , Nonnegative eigenf unctions of the Beltrami-Laplace operator on symmetric spaces of nonpositive curvature, Doklad y Akad . Nau k SSS R 15 1 (1963) , 1274-1276 ; Englis h transl., Sovie t Math . Dokl . 4 (1963) , 1180-1182 .

17. M . I . Graev , F . I . Karpelevich , an d A . A . Kirillov , Represenration theory of the Lie groups, Proc. Fourt h All-Unio n Math . Congres s (Leningrad , 1961) , vol . II , Nauka , Leningrad , 1964 , pp. 275-281 . (Russian )

18. S . G . Gindiki n an d F . I . Karpelevich , A problem of integral geometry, I n Memoriam : N . G . Chebotarev, Izdat . Kazan . Univ , Kazan , 1964 , pp . 30-43 ; Englis h transl. , Select a Math . Sovietica 1 (1981) , 169-184 .

19. F . I . Karpelevich , Geometry of geodesies and eigenf unctions of the Beltrami-Laplace operator on symmetric spaces, Trud y Moskov . Mat . Obshch . 1 4 (1965) , 48-185; Englis h transl. , Trans . Moscow Math . Soc . 196 5 (1967) , 51-199 .

20. S . G . Gindiki n an d F . I . Karpelevich , On certain special functions of several variables related with Lie groups, Studie s Contemporar y Problem s Constructiv e Theor y o f Functions , Izd . Akad. Nau k Azerb . SSR , Baku , 1965 , pp . 545-554 . (Russian )

21. , An integral associated with Riemannian symmetric spaces of nonpositive curvature, Izv. Akad . Nau k SSS R Ser . Mat . 3 0 (1966) , 1148-1156 ; Englis h transl. , Amer . Math . Soc . Transl. (2 ) 8 5 (1969) , 249-257 .

22. F . A . Berezi n an d F . I . Karpelevich , On associative algebras of functions, Vestni k Moskov . Univ. Ser . I Mat . Mekh . 1976 , no . 1 , 33-38 ; Englis h transl. , Mosco w Univ . Math . Bull . 3 1 (1976), no . 1 , 29-34 .

Amer. Math . Soc . Transl . (2) Vol . 210 , 200 3

Friedrich Karpelevich : His Earl y Year s i n Mathemat ic s

E. B. Dynkin

We me t firs t i n 194 6 whe n Karpelevic h wa s a senio r i n hig h school , an d I was a graduat e studen t a t Mosco w Universit y (MGU) . H e cam e t o a mathematic s circle a t MGU , t o a sectio n whic h I wa s runnin g fo r th e secon d year . Amon g about 2 0 participant s o f th e sectio n wer e Agranovich , Berezin , Chentsov , Minlos , Uspenskii, an d Yushkevich , al l o f who m late r contribute d significantl y t o variou s areas of mathematics. Ver y soon Karpelevich distinguishe d himsel f b y solving some challenging problems .

In 198 9 he remembers : "I solved a nic e specia l cas e o f th e fou r colo r problem : i f th e numbe r o f ever y

country's borders is a multiple of three, then a regular coloring is possible. I used th e familiar fac t tha t a coloring of countries can be reduced t o a coloring of boundaries , and I prove d a crucia l lemm a b y inductio n o n th e numbe r o f countrie s insid e a contour. I remember tha t yo u claimed al l attempts t o us e induction i n this contex t failed, an d I was proud whe n I demonstrated tha t sometime s i t works. "

[Here and late r I cit e a tap e recordin g o f ou r conversation s i s made durin g m y visit t o Mosco w i n 198 9 on a n exchange progra m betwee n th e Academ y o f Science s of the USS R an d th e Nationa l Academ y o f Science s o f the USA. ]

I aske d him : "When you graduated fro m th e high school , you told m e that yo u are not goin g

to apply fo r th e universit y bu t rathe r g o to a work i n a factory. I was disappointed , and aske d why. "

"You know, " h e responded , " I grew u p i n a ver y poo r family . M y mothe r ha d three childre n an d als o he r mothe r t o car e for . M y parent s divorce d whe n I wa s 7 an d m y siste r wa s 1 . Practically , th e fathe r ha s no t supporte d us . M y mothe r worked a s a sorte r a t a factory , earnin g 7 0 ruble s pe r month . M y brothe r cam e back from th e arm y disabled afte r th e war , an d h e studied a t a medical school . Th e sister wa s i n school .

"During th e wa r m y famil y staye d i n Moscow . Fo r tw o academi c years , 1941 -42 an d 1942-43 , school s i n Mosco w di d no t function . I wa s 1 4 an d I worke d a s a milling-machine operato r fo r tw o years . Whe n regula r schoo l resumed , I returne d there."

In 194 7 several activ e participant s i n the circl e (includin g Fred ) wer e admitte d to the Departmen t o f Mechanics an d Mathematic s a t MG U an d th e circl e itself wa s

©2003 America n Mathematica l Societ y

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transformed int o the seminar "Selecte d problems of contemporary mathematics " fo r freshmen. Th e semina r wa s continue d unde r variou s name s fo r a numbe r o f years . In 195 5 i t wa s divide d int o tw o daughte r seminars : fo r algebr a an d fo r probabilit y theory. Participant s o f these seminar s wer e encourage d no t onl y t o solv e problem s but als o t o prepar e thei r solution s fo r publication . Ove r a perio d o f time , th e firs t publications o f Dobrushin , Karpelevich , Kirillov , Margulis , Sinai , Uspenskii , an d others appeare d thi s way . Karpelevic h recalls :

"My firs t publicatio n wa s o n pseudo-norm s o f integers . I wa s requeste d t o review paper s o f Mahle r an d t o improv e hi s presentation . M y not e wa s rewritte n at leas t fou r time s an d eac h tim e yo u requeste d a ne w revision . I believe tha t thi s drill wa s a n excellen t training. "

As a sophomore , Friedric h mad e hi s firs t seriou s contributio n t o mathematics . It i s relate d t o a proble m pose d b y Kolmogoro v i n 1945 : t o describ e th e se t o f all eigenvalue s o f n x n stochasti c matrices . Dmitrie v an d Dynki n gav e a partia l solution t o thi s problem . A fina l solutio n wa s give n b y Karpelevich . Hi s pape r was published i n Izvestiya Akademi i Nauk SSS R and translate d int o English b y th e American Mathematica l Society .

"When I wa s a fourt h yea r student , yo u aske d m e t o revie w a pape r o f Gant -makher devote d t o Cartan' s classificatio n o f rea l form s o f comple x semisimpl e Li e algebras. I simplified th e presentation b y using simple roots. I n particular , I proved a theore m o n canonica l embeddin g whic h i s referre d t o i n recen t textbooks . Thi s was th e content s o f my diplom a work. "

Fred wa s on e o f the brightes t student s i n hi s class . H e was regarded highl y b y Kolmogorov an d b y Petrovski i (Karpelevic h worke d successfull y a t hi s seminar) . However i n 195 2 — th e pea k o f Stalin' s anti-Semitis m — i t wa s impossibl e fo r Karpelevich, a Jew , t o b e admitte d t o a graduat e school . H e was sen t t o teac h a t a provincia l technica l schoo l i n Novocherkassk . A s a resul t o f a seriou s illnes s h e was permitte d t o retur n t o Mosco w i n 1953 .

He continued t o work on subalgebras o f semisimple Li e algebras. H e obtained a number o f remarkable results on this subject (se e Onishchik's survey in this volume) which wer e th e cor e o f hi s Ph.D . thesis , publishe d i n 1955 . H e wa s awarde d a prestigious priz e b y th e Mosco w Mathematica l Societ y i n 1956 .

Karpelevich wa s on e o f th e activ e participant s i n th e semina r o n Li e group s which I ra n i n 1957-196 2 a t Mosco w University . Hi s well-know n wor k o n a ge -odesic approac h t o th e boundar y proble m starte d a t thi s seminar . [Mor e o n th e seminar ca n b e foun d i n "Li e Group s an d Li e Algebras : E . B . Dynki n Seminar" , S. G . Gindikin , E . B . Vinberg (eds.) , America n Mathematica l Society , Providence , RI, 1995. ]