Collected Papers: Volume I 1955-1966 PDF
Preview Collected Papers: Volume I 1955-1966
Collected Papers Volume I 1955–1966 Kostant at family event, June 2002 Bertram Kostant Collected Papers Volume I 1955–1966 Anthony Joseph Michèle Vergne Shrawan Kumar Editors 123 Bertram Kostant Massachusetts Institute of Technology Department of Mathematics Cambridge, MA 02139 USA k [email protected] ISBN 978-0-387-09582-0 e-ISBN 978-0-387-09583-7 DOI 10.1007/b94535 Springer Dordrecht Heidelberg London New York Library of Congress Control Number:2009926994 Mathematics Subject Classification (2000): 01A75, 32-06, 14-06 © Bertram Kostant 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer soft-ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Bertram Kostant is to be counted as one of the remarkable mathematicians of the second half of the twentieth century through his fundamental and varied contri- butions to many aspects of Lie theory, a subject which itself pervades almost the wholeofmathematics. Hisworkismarkedbyararesimplicityandcharacteristic elegance, makingiteminentlyreadable, wonderfullyenjoyable, andeasilyunder- standableeventoatruenovice. Itisoftenthroughhisworkthatonemayunderstand manysophisticateddevelopmentsofmodernmathematics. ThemathematicalworkofBertramKostanthasspannedwelloverfiftyyears inwhichhehaspublishedatanalmostconstantrhythmover100papersofwhich morethanafewhavebecomeacornerstoneofrichandfruitfultheories. Here,in briefly listing some of his most important contributions and their eventual ramifi- cationswecangiveonlyasmallglimpseintothedepthandnatureofhislegacy. InoneofhisearliestworksKostantintroducedapartitionfunctiondescribing weight multiplicities. Significantly, it provided a tool which allowed Jantzen to compute certain fundamental determinants concerning weight spaces in Verma modulesinthemuchmoredifficultparaboliccase. Kostant studied extensively the principal three-dimensional subalgebra of a semisimple Lie algebra, extracting from its adjoint action the Betti numbers of the corresponding Lie group. In addition, he thereby obtained a remarkable lin- earizationofthefundamentalinvariants. Geometricallythisrealizesanaffineslice to the regular coadjoint orbits, a result subsequently broadened notably by Luna, Slowody,andPremet,andnowofgreatlyrenewedinterest. Kostant followed this work by a further fundamental paper describing har- monic polynomials in the symmetric algebra. This gave a separation of variables theoremandaratherprecisedescriptionofthenilpotentcone. Theseresultshave been key building blocks in many important results in representation theory, in- cludingthetheoryofprimitiveideals. Kostant discovered a remarkable connection between the regular nilpotent elementandtheCoxeterelementintheWeylgroup. Thiswassubsequentlybrought toaclassificationofnilpotentorbitsbyCarterandElkingtonandthenelevatedtoa mapfromnilpotentorbitstoconjugacyclassesintheWeylgroupbyKazhdanand Lusztig. Kostant himself has returned to the many beautiful themes which may be developedfromtheseearlypapers. TheyincludeanextensivestudyofWhittaker vi Preface modules,themathematicalunderpinningsoftheTodalatticewhichhegeneralized wellbeyondthedreamsofphysicists,andquantumcohomology. Building upon Bott’s geometric results, Kostant computed the Lie algebra cohomology of certain nilpotent Lie algebras arising as nilradicals of parabolic subalgebras, studying at the same time their relationship to the cohomology of the flag variety and its generalizations. This recovered the Bernstein–Gelfand– Gelfand resolution of simple finite-dimensional modules admitting far-reaching generalizations to Kac–Moody and later to Borcherds algebras. Here he inadver- tently rediscovered the Amitsur–Levitski identity, so important in algebras with polynomialidentity,givingitafarsimplerproofandplacingtheresultinthemore generalcontextofsemisimpleLiealgebras. Kostant was one of the first to realize that a construction of Kirillov to clas- sifyunitaryrepresentationsofnilpotentLiegroupsimpliedthatanycoadjointorbit admitted the structure of a symplectic variety, that is, the phase space of classical mechanics. Hequicklyseizeduponthepassagetoquantummechanics,developed by physicists, as a process of “geometric quantization.” The Kostant line bundle detects which symplectic manifolds are quantizable and this result is fundamen- tal in Hamiltonian geometry. When applied to the coadjoint orbits, quantization is designed to produce most unitary representations of the corresponding real Lie group. WithAuslanderheclassifiedtheunitaryrepresentationsforrealclass1sim- ply connected solvable Lie groups, laying the groundwork for the more complete theoryofPukanszky. Thegeneralcase,thoughevenmoreresistant,hascomeunder the intensive study of many mathematicians, including notably Duflo, Rossmann, Vergne and Vogan. In more algebraic terms geometric quantization dramatically laidopenthepathtotheclassificationofprimitiveidealsinitiatedbyDixmierand Gabriel. Furtherdeepmanifestationsofgeometricquantizationareexemplifiedby theDufloisomorphism,theKashiwara–Vergneconjecture, andtheDrinfeldasso- ciatorthemselveshavingbeenunifiedandextendedfollowingnotablythepowerful techniquesofKontsevitchinspiredbyFeynmandiagramsandknottheory. Stepping outside the purely algebraic framework Kostant discovered a far- reachinggeneralizationoftheGolden–Thompsonruleinaconvexitytheoremsub- sequently generalized by Atiyah–Bott and Guillemin–Sternberg. Going beyond zero characteristic, he introduced the “Kostant form” on the enveloping algebra whichhasplayedamajorroleinmodularrepresentationtheory. Moreoveritsgen- eralization for quantum groups and the subsequent development of the quantum Frobenius map due to Lusztig allowed Littelmann to complete the Lakshmibai– Seshadriprogrammeofdescribingstandardmonomialbases. Preface vii In a veritable “tour de force” Kostant calculated certain fundamental deter- minants showing them to have linear factors from which their zeros could be de- termined. This led to a criterion for unitarity which has so far gone unsurpassed. Subsequently, many other such factorizations were achieved, particularly that of ShapovalovandJanztennotedabove;butalsoofGorelik–Lanzmannwhichdeter- minedexactlywhenaVermamoduleannihilatorforareductivesuperLiealgebra wasgeneratedbyitsintersectionwiththecentre. Kostant was one of the first to firmly lay the foundations of supermanifolds andsuperLiegroups. HealsostudiedsuperLiealgebrasprovidingafundamental structure theorem for their enveloping algebras. He initiated the study of the ten- sorproductofaninfinite-dimensionalrepresentationwithfinite-dimensionalones, leading in the hands of Jantzen to the translation principle. The latter, combined with Kostant’s description of the nilpotent cone, led to the Beilinson–Bernstein equivalence of categories which heralded a new enlightenment in Verma module and Harish-Chandra theory. Kostant described (in sl(4)) a non-trivial example of aunitaryhighestweightmodule,thetheoryofwhichwassubsequentlycompleted byEnright,HoweandWallach(andindependentlybyJakobsen). Inthetheoryof characteristicsdevelopedbyGuillemin,QuillenandSternberg,aresultofKostant proves in the finite-dimensional case the involutivity of characteristics ultimately resolvedinfullgeneralitybyGabber. Amongsomeofhisimportantcollaborativeworks,wementionhispaperwith HochschildandRosenbergonthedifferentialformsonregularaffinealgebras,his paper with Rallis on separation of variables for symmetric spaces, his work with KumaronthecohomologyandK-theoryofflagvarietiesassociatedtoKac–Moody groups,hisdeterminationwithR.Brylinskiofinvariantsymplecticstructuresand a uniform construction of minimal representations, and his work with Wallach on Gelfand–Zeitlin theory from the point of view of the construction of maximal Poissoncommutativesubalgebrasparticularlyby“shiftofargument”comingfrom classicalmechanics. Even in the fifth and sixth decades of his career, Kostant has continued to produce results of astonishing beauty and significance. We cite here his work on theTodalatticeandthequantumcohomologyoftheflagvariety,hisre-examination of the Clifford algebra deformation of “wedge n” with its beautiful realization of the module whose highest weight is the sum of the fundamental weights, his introductionofthecubicDiracoperator,hisgeneralizationoftheBorel–Bott–Weil theoremwithitsconnectiontoEulernumbermultipletsofrepresentations,hiswork on the set of abelian ideals in the nilradical of a Borel (wonderfully classified by viii Preface certainelementsintheaffineWeylgroupbyDalePeterson),hislongstandinglove affair with the icosahedron, his work with Wallach mentioned above and most recently an attempt to pin down the generators of the centralizer of a maximal compactsubalgebra. After receiving his Ph.D. from the University of Chicago in 1954, Bertram KostantbeganhisacademiccareerasanAssistantProfessorin1956attheUniver- sityofCalifornia,Berkeleyrosetofullprofessorin1962andshortlyaftermoved toMIT.HetaughtatMITuntilretiringin1993. Kostant was elected to the National Academy of Sciences U.S.A. in 1978 and has received many other honours and prizes, including election as a Sackler InstituteFellowatTel-AvivUniversityin1982,amedalfromtheColle`gedeFrance in1983,theSteeleprizeoftheAmericanMathematicalSocietyin1990,andseveral honorarydoctorates. Kostant maintains his physical fitness after the strains of University life by a weight lifting regime well beyond the capabilities of many younger colleagues. Thismaybeonesecrettohisscientificlongevity. WeizmannInstituteofScience,Rehovot,Israel AnthonyJoseph UniversityofNorthCarolinaatChapelHill,US ShrawanKumar E´colePolytechnique,Palaiseau,France Miche`leVergne April,2008 Editors Contents: Volume I Preface .................................................................v Contents: VolumeI ..................................................... ix CollectedPapers ........................................................xi Acknowledgments .....................................................xix [1] HolonomyandtheLieAlgebraofInfinitesimalMotions ofaRiemannianManifold .........................................1 [2] OntheConjugacyofRealCartanSubalgebrasI ......................16 [3] OntheConjugacyofRealCartanSubalgebrasII .....................20 [4] OnInvariantSkew-Tensors ........................................ 28 [5] OnDifferentialGeometryandHomogeneousSpacesI ................32 [6] OnDifferentialGeometryandHomogeneousSpacesII ...............36 [7] OnHolonomyandHomogeneousSpaces ............................40 [8] ATheoremofFrobenius,aTheoremofAmitsur–Levitski andCohomologyTheory ........................................ 64 [9] ACharacterizationoftheClassicalGroups ..........................92 [10] AFormulafortheMultiplicityofaWeight .........................109 [11] ThePrincipalThreeDimensionalSubgroupandtheBettiNumbers ofaComplexSimpleLieGroup .................................130 [12] ACharacterizationofInvariantAffineConnections ..................190 [13] LieAlgebraCohomologyandtheGeneralizedBorel–WeilTheorem ..206 [14] (withG.HochschildandA.Rosenberg),DifferentialForms onRegularAffineAlgebras .....................................265 [15] (withG.Hochschild),DifferentialFormsandLieAlgebra CohomologyforAlgebraicLinearGroups ........................291 [16] LieGroupRepresentationsonPolynomialRings ....................309 [17] LieGroupRepresentationsonPolynomialRings ....................318 x Contents: VolumeI [18] LieAlgebraCohomologyandGeneralizedSchubertCells ............396 [19] EigenvaluesofaLaplacianandCommutativeLieSubalgebras ........469 [20] Orbits,SymplecticStructuresandRepresentationTheory ............482 [21] GroupsOverZ ..................................................483 Kostant’sCommentsonPapersinVolumeI ..............................493
The list of books you might like
