Selected Title s i n Thi s Subserie s

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskff , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . N e t s v e t a e v , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . Polotovski i , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r (TRANS2/169 )

25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations (TRANS2/167 )

23 Yu . I lyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

(TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advance s i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4

20 R . L . Dobrush in , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4

19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4

18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4

17 D m i t r y Fuchs , Editor , Unconventiona l Li e algebras , 199 3

16 Serge i Gelfan d an d S i m o n Gindikin , Edi tors , I . M . Gelfan d seminar , Part s 1 an d 2 ,

1993

15 A . T . Fomenko , Editor , Minima l surfaces , 199 3

14 Yu . S . I l 'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2

13 V . P . Mas lo v an d S . N . Samborskit , Edi tors , Idempoten t analysis , 199 2

12 R . Z . Khasminski i , Edi tor , Topic s i n nonparametri c estimation , 199 2

11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2 10 A . V . B a b i n an d M . I . Vishik , Edi tors , Propertie s o f globa l at tractor s o f partia l

differential equations , 199 2

9 A . M . Vershik , Edi tor , Representatio n theor y an d dynamica l systems , 199 2

8 E . B . V inberg , Edi tor , Li e groups , thei r discret e subgroups , an d invarian t theory , 199 2

7 M . Sh . B i r m a n , Edi tor , Estimate s an d asymptotic s fo r discret e spectr a o f integra l an d

differential equations , 199 1

6 A . T . Fomenko , Edi tor , Topologica l classificatio n o f integrabl e systems , 199 1

5 R . A . Min los , Edi tor , Many-particl e Hamiltonians : spectr a an d scattering , 199 1

4 A . A . Susl in , Edi tor , Algebrai c K-theory , 199 1

3 Ya . G . Sinai , Edi tor , Dynamica l system s an d statistica l mechanics , 199 1

2 A . A . Kiri l lov , Edi tor , Topic s i n representatio n theory , 199 1

1 V . I . A r n o l d , Edi tor , Theor y o f singularitie s an d it s applications , 199 0

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Kirillov's Seminar o n Representation Theor y

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 18 1

Advances in the Mathematical Sciences — 35 {Formerly Advances in Soviet Mathematics)

Kirillov's Seminar o n Representation Theor y

G. I . Olshansk i Editor

American Mathematica l Societ y Providence, Rhod e Islan d

UVNDf5

http://dx.doi.org/10.1090/trans2/181

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

V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV

Transla t ion edi te d b y A . B . Sossinsk y

1991 Mathematics Subject Classification. P r i m a r y 05Exx , 17Bxx , 22E15 ; Secondary 53C35 .

ABSTRACT. Th e boo k i s a collection o f papers writte n b y students o f A. A. Kirillov an d participant s of hi s semina r o n Representatio n Theor y a t Mosco w University . Th e paper s dea l wit h variou s aspects o f representatio n theor y fo r Li e algebra s an d Li e group s an d it s relation s t o algebrai c combinatorics, theor y o f quantu m groups , an d geometry . Th e boo k i s usefu l fo r researcher s an d graduate student s workin g i n representatio n theor y an d it s applications .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-0669- 6

ISSN 0065-929 0

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Contents

Preface i x

Screenings an d a universa l Lie-d e Rha m cocycl e VICTOR GINZBUR G AN D VADI M SCHECHTMA N 1

Interlacing measure s SERGEI KERO V 3 5

Quasicommuting familie s o f quantu m Plucke r coordinate s BERNARD LECLER C AN D ANDRE I ZELEVINSK Y 8 5

Factorial supersymmetri c Schu r function s an d supe r Capell i identitie s ALEXANDER MOLE V 10 9

Yangians an d Capell i identitie s MAXIM NAZARO V 13 9

Hinges an d th e Study-Semple-Satake-Furstenberg-D e Concini-Procesi -Oshima Boundar y YURII A . NERETI N 16 5

Multiplicities an d Newto n polytope s ANDREI OKOUNKO V 23 1

Shifted Schu r function s II . Th e binomia l formul a fo r character s o f classica l groups an d it s application s ANDREI OKOUNKO V AN D GRIGOR I OLSHANKS I 24 5

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Preface

The present volume was prepared for publication by students and frienrds of Alexandr Alexandrovich Kirillov in connection with his 60th anniversary.

A. A . Kirillov' s numerou s student s (an d no t onl y hi s students ) studie d a t hi s seminar o n representatio n theor y a t Mosco w Stat e University . Thi s semina r func -tioned fo r nearl y 3 0 years, beginning i n the early sixtie s when A. A. began teachin g at th e chai r o f function theor y an d functiona l analysi s o f the Mechanic s an d Math -ematics Departmen t o f MSU , an d continuin g unti l A . A . starte d workin g a t th e University o f Pennsylvani a i n Philadelphia . I first cam e t o A . A.' s semina r i n th e winter o f 1964-6 5 a s a freshman , s o I a m on e o f hi s firs t student s an d on e o f th e oldest participant s o f the seminar .

For man y year s Kirillov' s semina r wa s on e o f th e bes t know n an d popula r Moscow mathematica l seminars , an d fo r m e a s well as for Kirillov' s othe r students , the mos t customar y an d comfortin g one . I t too k plac e o n Mondays , tw o hour s before th e Gelfan d seminar . O n Thursdays , A . A . als o conducte d a semina r fo r beginners (firs t an d secon d yea r students) , whic h wa s especiall y wel l attended . Active student s o f th e latte r woul d eventuall y mov e o n t o th e Monda y seminar , intended fo r olde r undergraduates , graduat e students , an d professiona l researc h mathematicians.

The topic s discusse d a t th e semina r range d quit e widely , reflectin g Kirillov' s broad researc h interests. 1 I t include d finite-dimensional representatio n theory ; uni -tary representation s o f reductive , solvable , an d genera l Li e groups ; representation s of infinite-dimensiona l groups . O f course , th e orbit method] th e universa l formul a for characters ; symplecti c geometry . Th e fractiona l fields o f envelopin g algebra s and othe r noncommutativ e rings ; identitie s i n noncommutativ e rings . Infinite -dimensional Li e algebras . Superalgebras . C*-algebras . Combinatorics . Quantu m groups. Mathematica l physic s . . . ( I a m afrai d tha t I have misse d man y topics. )

The Kirillo v semina r wa s neithe r primaril y intende d t o infor m o n variou s top -ics, wit h expert s takin g turn s t o lectur e o n them , no r wa s i t a workin g grou p concentrating o n a specifi c cycl e o f papers, 2 althoug h t o som e exten t i t performe d

1When firs t talkin g t o a ne w student , A . A . woul d usuall y as k wha t th e latte r woul d lik e t o study unde r him—algebra , geometry , o r analysis .

2In general , th e organizatio n o f th e semina r di d no t involv e an y rigi d planning : lecturer s and title s wer e no t writte n ou t an d displaye d i n advance , an d everythin g seeme d t o tak e plac e spontaneously.

ix

x P R E F A C E

both functions . Abov e all , i t wa s a plac e wher e on e learne d t o d o mathematic s "according t o Kirillov" .

Participants o f the semina r woul d usuall y assembl e i n advance, an d whil e wait -ing fo r A . A . (wh o ofte n cam e a bi t late) , the y woul d conduc t animate d conver -sations.3 Whe n Kirillo v appeare d a t th e doorstep , al l th e participant s woul d rise . If n o tal k wa s planne d an d A . A . di d no t inten d t o lectur e himself , h e woul d con -duct a poll : wh o ha d don e somethin g new ? H e woul d the n cal l someon e t o th e blackboard an d as k t o stat e th e resul t "i n five minutes" . I n fact , fe w succeede d in complyin g wit h thi s sacramenta l tim e interval , bu t i f the topi c wa s interesting , A. A . would ofte n forge t thi s constraint , an d "fiv e minutes " coul d easil y becom e a detailed accoun t wit h a subsequen t discussion. 4

The atmospher e o f th e semina r wa s ver y free , relaxed , an d informal . Th e lec -turer wa s often interrupte d b y questions, an d wheneve r A . A. fel t tha t th e listener s were losin g track , h e woul d explai n th e difficul t part s i n hi s ow n wa y o r discus s improvised examples .

I a m convince d tha t fo r us , just beginnin g t o d o mathematics , th e mai n profi t from participatin g i n the semina r ha d t o do with the impac t o f A. A. Kirillov's per -sonality, hi s manner o f explaining things simply, his light iron y concerning an overly "scientific" styl e of exposition, hi s sharp remarks , and strong dislike of artificial con -structions. Al l thi s contribute d t o for m a prope r tast e i n mathematica l style , suc h an importan t componen t o f one' s mathematica l education . And , o f course , a cru -cial rol e wa s playe d b y th e problem s tha t Kirillo v systematicall y produce d durin g the seminar . Som e wer e prepare d i n advance , other s aros e spontaneousl y durin g discussions. A goo d resul t leadin g t o ne w problem s wa s particularl y praise d b y Kirillov.5

One of the specifi c trait s o f A. A. Kirillov's styl e a s a teacher wa s that h e neve r liked to impos e researc h topic s fo r kursovye (ter m papers) , diplomnye (M S theses) , or kandidatskie (Ph D dissertations) . I t wa s assumed tha t eac h studen t mus t find a topic himsel f o n th e basi s o f problem s se t a t th e seminar . O f course , thi s wa s no t an absolut e rule , bu t t o th e student s tha t h e rated amon g the best , Kirillo v alway s gave complet e freedo m i n th e choic e o f a researc h topic .

Now, when Kirillov's seminar i n Moscow no longer functions , whil e his student s have dispersed al l around the world and mostly communicate by e-mail,6 I piercingly realize ho w muc h I ow e t o th e seminar . I hav e n o doubt s tha t simila r feeling s ar e experienced b y Kirillov' s othe r students .

* * *

I shal l briefl y revie w th e content s o f the contributions . 1. I n the pape r "Screening s an d a universa l Lie-d e Rha m cocycle " b y V. Ginz -

burg and V. Schechtman, a generalization o f the classica l Feigin-Fuchs constructio n

3Many Mosco w seminar s were , t o som e extent , somethin g lik e club s (thi s wa s especiall y tru e of I . M . Gelfand' s famou s seminar ) an d th e discussion s befor e the y formall y began , a s wel l a s th e positive influenc e o f th e lat e arriva l o f th e seminar' s head , deserv e specia l analysis .

4Such a poll would invariabl y take place at th e firs t sessio n after vacations . I vividly remembe r the feelin g o f frustratio n tha t aros e i f m y tur n t o b e polle d wa s no t reached .

5 In assessin g mathematica l achievements , A . A . half-jokingl y use d "economic " terminology , distinguishing result s tha t destro y "workplaces " fo r mathematician s fro m thos e tha t creat e them .

6Recently A . A . tol d m e tha t h e ca n invit e peopl e t o hi s ne w semina r i n Philadelphi a fro m within a radiu s o f $30 0 (tha t i s th e amoun t tha t ca n b e allocate d fo r trave l expenses) . Unfortu -nately, typica l distance s ar e no w measure d b y large r amounts .

P R E F A C E x i

is presented. I t provide s canonica l mappings fro m th e homology of one-dimensiona l local system s o n th e configuratio n space s appearin g i n conforma l fiel d theor y t o the Ext-spa,ces betwee n module s o f semi-infinit e form s ove r th e Virasor o algebr a or Wakimot o module s ove r affin e Li e algebras . A n analo g o f thi s constructio n fo r finite-dimensional semisimpl e Li e algebras i s given .

2. Th e pape r "Interlacin g measures " b y S . Kero v deal s wit h th e asymptoti c behavior o f pair s o f interlacing sequences ,

xi < 2/1 < x 2 < • • • < x n-i < yn-i < x n.

A typica l exampl e o f suc h pair s i s provide d b y root s o f polynomial s o f adjacen t degrees i n a famil y o f orthogona l polynomials . Th e autho r introduce s an d studie s a mor e genera l object , a pai r o f interlacin g measures . A s a matte r o f fact , t o eac h pair of interlacing measures with difference r ther e corresponds a unique probabilit y distribution \x such tha t

e x p [ \n-L-T(du)= / ^ l , Tmz^O. J z-u J z-u

This equatio n ha s a numbe r o f interesting applications , includin g (1) th e connectio n betwee n additiv e an d multiplicativ e integra l representation s

of analyti c function s o f negative imaginar y type ; (2) th e Marko v momen t problem ; (3) distribution s o f mean value s o f Dirichle t rando m measures ; (4) th e theor y o f spectra l shif t functio n i n scatterin g theory ; (5) th e Planchere l measur e o f the infinit e symmetri c group .

Apparently, th e pape r gives th e firs t unifie d surve y o f al l thes e topics . A specia l emphasis i s on the combinatoria l connection s between the moments o f the measure s r an d \x in the abov e formula . On e o f the ne w result s i s an explici t formul a fo r th e multiplicative integra l representatio n o f the Gaussia n measur e o n th e rea l line .

3. Th e pape r "Quasicommutin g familie s o f quantu m Plucke r coordinates " b y B. Lecler c an d A . Zelevinsk y i s devoted t o th e stud y o f g-deformation s o f Plucke r coordinates on the flag variety. Th e authors give a criterion for quasi-commutativit y of tw o suc h coordinate s an d stud y thei r maxima l quasi-commutin g familie s (her e "quasi-commutativity" mean s "commutativit y u p t o a powe r o f g") . Th e result s have applications t o the description o f canonical bases for the quantum grou p GL n , the geometr y o f Bott-Samelso n desingularization s o f Schuber t varieties , an d com -binatorics o f the "secon d Bruha t order " du e t o Mani n an d Schechtman .

4. Th e pape r "Factoria l supersymmetri c Schu r function s an d supe r Capell i identities" b y A . Mole v i s devoted t o supe r generalizatio n o f a remarkabl e clas s of combinatorial functions—th e so-calle d factoria l Schu r polynomials . Thes e polyno -mials, introduced b y the mathematica l physicist s L. C. Biedenharn an d J . D . Louck and furthe r studie d b y I . G . Macdonal d an d othe r authors , ar e certain multidimen -sional inhomogeneou s polynomial s whos e highes t degre e term s ar e ordinar y Schu r polynomials. The y hav e numerous application s i n algebraic combinatoric s an d rep -resentation theory . Th e autho r develop s th e supe r counterpar t o f the theory . Th e main application s o f hi s result s ar e "factorial " analog s o f th e Jacobi-Trud i an d Sergeev-Pragacz formulas ; constructio n o f a distinguishe d linea r basi s i n th e cen -ter o f th e universa l envelopin g algebr a o f Ql(m\n); a supe r analo g o f th e highe r Capelli identities . Relate d topic s ar e discusse d i n th e paper s b y M . Nazaro v an d by A . Okounko v an d G . Olshansk i (se e below) .

xii P R E F A C E

5. I n the paper "Yangian s and Capell i identities " b y M. Nazarov, the i^-matri x formalism i s applie d t o highe r Capell i identities . Recal l tha t th e classica l Capell i identity (whic h i s discussed i n H . Weyl's famous boo k o n classica l groups ) provide s remarkable determinanta l expression s fo r canonica l generator s o f the cente r o f th e universal envelopin g algebr a U(g{(n)). Th e highe r Capell i identitie s ar e state d fo r a muc h wide r famil y o f central elements , whic h for m a distinguishe d linea r basi s i n the cente r o f U(gi(n)). Not e tha t unde r th e Harish-Chandr a isomorphism , thes e basis element s tur n int o th e factoria l Schu r polynomial s mentione d above . Th e methods o f th e pape r ar e inspire d b y quantu m grou p theory . Th e autho r studie s the imag e o f th e universa l it!-matri x fo r th e Yangia n Y(g{(ri)) wit h respec t t o th e evaluation homomorphism o f Y(gi(n)) t o U(gi(n)). Th e fusion procedur e as define d by I . Cherednik i s used. Th e highe r Capell i identitie s ar e obtained a s a corollary of this machinery . Althoug h th e Yangian technique s use d i n the pape r ma y first see m rather sophisticated , th e Yangians ar e actually a very natura l an d powerfu l too l fo r handling man y problem s concernin g classica l Li e algebras . Not e tha t i n anothe r paper by the same author, th e same approach i s carried over the "true " super analo g of gl(n), the quee r Li e superalgebra q(n) , and i n the recen t pape r b y A . Molev an d M. Nazarov , th e Yangia n technique s ar e use d t o obtai n ne w Capelli-typ e identitie s (for the orthogonal and symplectic Lie algebras). Differen t approache s to the highe r Capelli identitie s fo r Qi(ri) wer e develope d b y A . Okounkov .

6. Th e ai m o f the pape r "Hinge s an d th e Study-Semple-Satake-Furstenberg -De Concini-Procesi-Oshim a boundary " b y Yu . Nereti n i s t o propos e a unifie d el -ementary geometri c descriptio n fo r variou s boundarie s an d completion s o f group s and symmetri c spaces—th e Satake-Furstenber g boundary , th e Marti n boundary , the Karpelevich boundary , complet e symmetri c varietie s in the sense of De Concin i and Procesi , compactincation s o f Bruhat-Tit s buildings , etc . Th e ke y elemen t o f the author' s construction s i s th e ne w concep t o f a "hinge " ( a finit e collectio n o f points o f a Grassman n manifol d subjec t t o certai n conditions) .

7. Th e pape r "Multiplicitie s an d Newto n pol y topes" b y A . Okounko v deal s with Newto n polytopes associate d i n the author' s recen t pape r (Invent . Math . 12 5 (1996), 405-411 ) t o G-space s X , wher e G i s a connecte d recluctiv e group ,

(*) X C P(V r), X i s closed , irreducibl e an d G-stable ,

and V i s a finite-dimensional representatio n o f G . Th e first resul t o f th e pape r is the explici t computatio n o f th e polytop e fo r th e cas e whe n G i s the symplecti c group, G = 5p(2n) , and X i s the flag variety. Th e polytope thus obtained coincide s with th e Gelfand-Zetlin-typ e polytop e tha t appear s i n th e well-know n descriptio n (due t o Zhelobenko ) o f weigh t multiplicitie s fo r th e reductio n schem e Sp(2n) j . Sp(2n — 2 ) I • • • . Thi s give s ye t anothe r proo f an d a geometri c interpretatio n o f Zhelobenko's theorem . Th e secon d resul t i s tha t th e polytope s correspondin g t o all the differen t G-equivarian t embedding s (* ) o f X ca n b e arrange d int o a conve x cone. Thi s gives a strengthening of the theorem from the author's paper cited above: the semi-classica l limi t o f weigh t multiplicitie s fo r th e actio n (* ) i s a log-concav e function o f bot h th e weigh t an d th e G-linearize d invertibl e shea f tha t define s th e embedding (*) .

8. Th e pape r "Shifte d Schu r function s II . Th e binomia l formul a fo r character s of classical groups and its applications" b y A. Okounkov and G . Olshanski continue s the authors ' previou s wor k (referre d t o a s Par t I ) bu t ca n b e rea d independently .

PREFACE x n i

Note tha t th e shifte d Schu r function s ar e a modificatio n (o f a specia l case ) o f fac -torial Schu r polynomials . Th e result s o f Par t I hav e a direc t relationshi p t o th e groups GL(n); th e ai m o f Par t I I wa s to find thei r counterpart s fo r th e orthogona l and symplecti c groups . Th e pape r start s wit h th e binomia l formula , whic h i s a kind o f Taylo r expansio n fo r finite-dimensional characters . Thi s i s a simpl e result , which ha s a numbe r o f importan t consequences . Fo r instance , i t suggest s th e defi -nition o f a distinguished linea r basi s i n Z(Q), th e cente r o f the universa l envelopin g algebra U(Q), wher e Q stands fo r a n orthogona l o r symplecti c Li e algebra . Th e basis element s ca n the n b e characterize d i n severa l differen t ways . Not e tha t thei r images unde r th e Harish-Chandr a isomorphis m ca n b e expresse d throug h certai n factorial Schu r polynomials . A natura l basi s i n / ( B ) , th e subalgebr a o f invariant s in the symmetri c algebr a 5(g) , i s also examined. Bot h base s turn ou t t o b e relate d via th e "specia l symmetrizatio n map " S(Q) — * C/(g) , a n equivarian t linea r isomor -phism, whic h differs fro m th e usual symmetrization map . Mor e involved versions of the binomia l formul a an d th e combinatoric s o f "generalize d symmetrizatio n maps " are studie d i n subsequen t work s b y th e sam e author s (cite d i n Par t II) .

G. Olshansk i

Moscow, 199 7

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