Highlights in Lie Algebraic Methods PDF
Preview Highlights in Lie Algebraic Methods
Progress in Mathematics Volume 295 SeriesEditors HymanBass JosephOesterlé YuriTschinkel AlanWeinstein Anthony Joseph (cid:2) Anna Melnikov (cid:2) Ivan Penkov Editors Highlights in Lie Algebraic Methods Editors AnthonyJoseph IvanPenkov DepartmentofMathematics SchoolofEngineeringandScience WeizmannInstitute JacobsUniversity Rehovot76100 Bremen28759 Israel Germany [email protected] [email protected] AnnaMelnikov DepartmentofMathematics UniversityofHaifa Mt.Carmel,Haifa31905 Israel [email protected] ISBN978-0-8176-8273-6 e-ISBN978-0-8176-8274-3 DOI10.1007/978-0-8176-8274-3 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011940727 Mathematics Subject Classification (2010): 05C50, 05E15, 13F60, 14K05, 14L30, 14M17, 14M27, 16G70,17B10,15B36,17B10,17B22,17B37,17B55,17B67,17B99,22E47 ©SpringerScience+BusinessMedia,LLC2012 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) Preface Atwo-weekSummerSchoolon“StructuresinLieTheory,Crystals,DerivedFunc- tors,Harish-ChandraModules,InvariantsandQuivers”washeldatJacobsUniver- sityBremenduring9–22August2009onboththegeometricandalgebraicaspects of Lie Theory. The participants were mainly from European countries with strong contingentsfromGermany,Russia,andIsrael.Severalhigh-levelgraduatecourses were given, containing recent or original results on topics of particular current in- terest. The detailed notes of five of these lecture courses are reproduced in this volume.Theynotonlyprovideawelcomereminderofthecontentofthesecourses totheparticipants,butenableallthosewhodidnotattendthemeetingtoprofitfrom insights of leading specialists into the latest developments in some exciting fields ofresearch.Eventhoseparticipantswhofollowedthecoursesattentivelymayprofit fromthedetailspresentedinthisbook. Besides the presenters of the graduate courses, some younger researchers were invitedtospeakatthemeetingandtosubmitamanuscriptforpublication.Thusthe presentvolumealsocontainsfiveoriginalarticles. Allthetextsreproducedhereweresubjecttoastrictrefereeingprocedurebefit- tinganymathematicaljournal. TheCourses SphericalVarietiesbyMichelBrion,Grenoble Thiscoursediscussesandinterrelatesseveralclassesofcomplexalgebraicvarieties equippedwithanactionofaconnectedalgebraicgroupG.Forexample,thehomo- geneous varieties (in which G acts transitively), the spherical varieties (for which G is reductive and for which a Borel subgroup of G has an open orbit), the sym- metricspacesG/Gθ (whereagainGisreductive,andθ isaninvolutionofG),and the“wonderfulvarieties”inthesenseofDeConciniandProcesi,areexaminedin thiscourse.Classificationtheoremsarepresented,especiallyinthecaseofcomplete varieties. v vi Preface A significant aspect of this course is the systematic use of the notion of a log homogeneousvarietyX.ThisisasmoothG-varietyequippedwithadivisorDwith normal crossings such that D is G-stable and the associated “logarithmic tangent bundle” is generated by the Lie algebra g (acting on X via vector fields that pre- serveD).ThisconditionimpliesthatX\DisasingleGorbit. A local structure theorem is given for a complete log homogeneous variety X alongaclosedG-orbit,showinginparticularthatXissphericalifGislinear.Con- versely, it is stated that a spherical homogeneous space G/H always admits a log homogeneouscompletion.Moreover,ifH isitsownnormalizerinG,thentheclo- sureoftheG-orbitofhinthevarietyofLiesubalgebrasofgisaloghomogeneous completionwithauniqueclosedG-orbit. An alternative construction of this “wonderful completion” is presented in the caseofasemisimplegroup G withtrivialcenter,viewedasahomogeneousspace under G×G. Let V be a simple G-module with a regular highest weight. Then G×GactsontheprojectivizationP(EndV),andtheorbitthroughtheclassofthe identity is isomorphic to G. By a result of De Concini and Procesi, the closure of thatorbitisthedesiredwonderfulcompletionofGandisindependentofthechoice ofV. TheaboveresultisgeneralizedtoanarbitrarysymmetricspaceG/Gθ asfollows. AssumethatV isspherical(i.e.,itadmitsa Gθ-invariantvector)and“regular”for that property. Then, as shown by De Concini and Procesi, the orbit closure of the corresponding point of P(V) is again a wonderful completion of G/Gθ, indepen- dentofV. Finally,thenotionofawonderfulvarietyisdiscussed.Itisstatedthatthereare only finitely wonderful varieties for a given semisimple group G, and there is an ongoingprogramtoclassifythem. ConsequencesoftheLittelmannPathModelfortheStructure oftheKashiwara B(∞) CrystalbyAnthony Joseph, Weizmann Institute This course starts with the observation that the B(∞) crystal of Kashiwara asso- ciatedto aKac–Moodyalgebra g is givenbya purelycombinatorialconstruction. ThepropertiesofB(∞)weredeterminedbytakingaq→0limitofhighestweight modulesoverthequantizedenvelopingU (g). q The Littelmann path construction associates a crystal to a highest weight of g. ThenB(∞)canbeobtainedbyalimitingprocessonthehighestweight.Thisgives apurelycombinatorialwaytoanalyzethestructureofB(∞),whichhastheadvan- tage that there is no need to assume that g is symmetrizable. In particular the key propertiesthatB(∞)isuppernormalandcanonicallydeterminedbyatensorprod- uct construction are established. Furthermore it is shown that B(∞) admits an in- volutionwhichcoincideswiththeKashiwarainvolutioninthesymmetrizablecase. Conversely,itisshownthatLittelmann’scrystalsmayberecoveredfromB(∞),but Preface vii sofaritisnotknownhowtoshowthattheresulting“highestweightcrystals”satisfy tensor product decomposition without appealing to the Littelmann path model (or theLusztig–Kashiwaratheoryofbases). Characterformulasarediscussed,notingthatthereisnotyetapurelycombina- torialproofoftheWeyldenominatorformulaifdimgisinfinite. CombinatorialDemazureflagsaredescribed.Hereitisnotedthatthecorrespond- ingresultforthetensorproductofaDemazuremodule(globalsectionsofsheaves onSchubertvarieties)withaone-dimensionalDemazuremoduleisknownonlyfor semisimple g (Mathieu)oriftherootsystemissimplylaced.Thelatterisnotably byvirtueofapositivityresultinthemultiplicationofcanonicalbasiselementsdue toLusztig. Under a positivity hypothesis, Nakashima and Zelevinsky showed that B(∞) admits an additivestructure (which as yet has no module-theoreticinterpretation). Theirproofisreproduced,notingthatthispositivityhypothesisimpliesuppernor- malityandsoisliabletoberatherdifficulttoestablish. Aside from the Littelmann construction, detailed proofs are given for most of thesecombinatorialresults. Structureand RepresentationTheoryof Kac–Moody Superalgebras, byVeraSerganova, Berkeley The essence of supersymmetry is the introduction of a sign rule in mathematical operations.Thoughdefinitionscarryovertothissituationinaseeminglyinnocent fashion,thisidealeadstoanewtheorywithmanychallengingopenproblems.From themathematician’spointofview,the“signrule”leadstonewnotionssuchassu- permanifold,Liesupergroup,andLiesuperalgebra.Theinterrelatedtheoriesbased on the notions of superanalysis, supergeometry, representation theory of Lie su- pergroups and Lie superalgebras started their rapid development in the works of Berezin,Bernstein,Kac,Kostant,Leites,andothers. In these lectures, Serganova concentrates on the theory of contragredient Lie superalgebras and their representations. She recalls Kac’s classification of finite- dimensional simple Lie superalgebras, gives a brief introduction into Lie super- groups,andthenproceedstothedefinitionofKac–MoodyLiesuperalgebras.The existenceofoddreflections(“odd”elementsoftheWeylgroup)playsanimportant role when describing the structure of Kac–Moody Lie superalgebras. Serganova then presents the classification of finite-growth contragredient Lie superalgebras, due to Van de Leur in the symmetrizable case and to Hoyt and Serganova in the non-symmetrizablecase. The nexttopicdiscussedis integrablehighestweightmodules.In contrastwith the case of contragredient Lie superalgebras, only a few finite-growth contragre- dient Lie superalgebras admit nontrivial highest weight modules; see Theorem 5 duetoKacandWakimoto.SerganovaprovestheWeylcharacterformulafortypical viii Preface integrable highest weight modules. This formula goes back to Kac in the finite- dimensionalcase.(Arelatedresult,notdiscussedinthelectures,isGorelik’srecent proofofthe“super”denominatoridentity–editorialnote.) In the final Sect. 4, Serganova concentrates on the case of a finite-dimensional contragredient superalgebra. Here she discusses the case of atypical finite-dimen- sional simple modules and, in particular, recalls the Kac–Wakimoto conjecture on thesuperdimensionofsuchamodule. Apointworthyof specialattentionisthediscussionofgeometricmethods:the notion of an “odd associated variety” due to Duflo and Serganova, and the older Bott–Borel–WeilapproachtoflagsupervarietiesgoingbacktoPenkov. Thelecturesareconcludedbyalistofcurrentopenproblems. CategoriesofHarish-Chandra ModulesbyWolfgang Soergel, Freiburg Recall that the Kazhdan–Lusztig polynomials evaluated at q =1 give the Jordan– Hölder multiplicity [M :L] of a simple quotient L in a Verma module M. How- ever the polynomials themselves have two possible interpretations, either as de- scribing the more precise data encoded as the dimensions of the extension groups Extk(M,L)orastheJordan–Höldermultiplicities[M ,L]withM thekthgraded k k componentofM.Heretheresultingmatricesarenotthesamebutrelatedtoonean- otherbymatrixinversionandappropriatesignchanges.(Thiswasfirstsuggestedin O. Gabber and A. Joseph, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261– 302, where the gradation was to be given by the Jantzen filtration. Notably the second interpretation was proven for the socle filtration in R. S. Irving, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 47–65. It had the important consequence that one can thereby describe the Duflo involutions through the Kazhdan–Lusztig polynomials—editorialnote.) The above observation can be interpreted as an inversion formula for the Kazhdan–Lusztigmatrixofpolynomials.InthewellknownpaperofA.Beilinson, V. Ginzburg, and W. Soergel, J. AMS 9 (1996), no. 2, pp. 473–526, this inversion formula has been given a categorical interpretation as Koszul self-duality of the category O. In fact, Soergel sketches a proof of this result in the latter part of his lectures. Themaincontentofthecourseisastatementofaconjectureextendingtheabove ideastothecategoryofHarish-Chandramodules.Thestartingpointistheinversion formulainJ.Adams,D.Barbasch,andD.A.Vogan,ProgressinMathematics,104. BirkhaüserBoston,Inc.,Boston,MA,1992,involvingtwosetsofmatricescoming fromJordan–Hölder(JH)andintersectioncohomology(IC)matrices. Soergel’sconjectureisthatthecategoryofHarish-Chandramodulesisequivalent toacertaincategoryoffinite-dimensionalmodulesdeterminedentirelyintermsof intersectioncohomology.ThisgivesthedesiredrelationshipbetweentheJHandIC matricesandemphasizesthesimilaritybetweenthecategoryO andthecategoryof Preface ix Harish-Chandramodules.Italsoshowshowfar-reachingtheideaofKoszulduality is.Soergel’sconjectureisprovedsofarfortwocases,namelyforSL(2,R)andfor complexgroups. Soergeldiscusseshisconjectureinlightoftiltingmodules.Forexample,inthe O categorytiltingmodulesinterpolatebetweenprojectivesandsimples,andcanbe usedtoestablishtheBernstein–Gelfand–Gelfandreciprocity. GeneralizedHarish-Chandra ModulesbyGreggZuckerman, Yale AmoduleM overaLiealgebragisdefinedtobe“generalizedHarish-Chandra”if thereexistsaLiesubalgebralofgsuchthatM islocallyfiniteasanl-moduleand has finite multiplicities as an l-module in an appropriate sense (see Definition 1.8 inthelectures).Ifgisfinite-dimensionalandsemisimple,listhefixedpointofan involutionθ ofg,andM isfinitelygeneratedoverg,thenthisreducestotheusual definitionofaHarish-Chandramodule. Harish-Chandramodulesfirstaroseinthedescriptionofunitaryrepresentations ofrealLiegroupsandhavebeenintensivelystudiedsincethemonumentalworkof Harish-Chandra.NotablyLanglandsgaveaclassificationofsimpleHarish-Chandra modules. Key questions are to determine the precise multiplicities as an l-module and to determine when (in terms of the Langlands parameters) a simple Harish- Chandramoduleisunitarizable(stillanopenquestion!). Recently there has been considerable interest in extending this well-developed theory to generalized Harish-Chandra modules, for which the course provides an introduction.Afirstcaseof(genuinely)generalizedHarish-Chandramodulesoccurs whengissemisimpleandlisaCartansubalgebra.Theclassificationofthesimple modules is basically due to Fernando and Mathieu. An important fact is that the setg[M]ofelementsofafinite-dimensionalLiealgebragactinglocallyfinitelyon a g-module M forms a Lie subalgebra. (This assertion generally fails for infinite- dimensionalLiealgebras—editorialnote.) Considerable emphasis in these lectures is put on the use of the Zuckerman functor,whichroughlyis the right derivedfunctor of the functor of passing to the submoduleofl-finitevectors.Animportantprocedureiscohomologicalinduction, wheretheZuckermanfunctorisappliedtoaparabolicallyinducedorproducedmod- ule.SomegeneralpropertiesoftheZuckermanfunctoraredescribed,forexample, commutationwiththetensorproductbyafinite-dimensionalg-moduleandwiththe actionofthecenterofU(g).Thesepropertiesarecrucialtothe“translationprinci- ple.”Afinitemultiplicitytheoremisproved,whichallowsZuckermantoapplyhis functorstotheconstructionofgeneralizedHarish-Chandramodules.Someinforma- tiononmultiplicitiesisprovidedbystandardhomologicalarguments,forexample, Frobenius reciprocity and the Euler principle. In the last section Zuckerman de- scribes his jointwork withPenkovclassifying simplegeneralizedHarish-Chandra moduleswith“generic”minimall-typewhengissemisimpleandlisreductiveing. (InthecaseofsimpleHarish-Chandramodules,thenotionofminimall-typeandits x Preface roleintheclassificationtheorywasextensivelystudiedinD.A.Vogan,Proc.Nat. Acad.Sci.USA74(1977),no.7,2649–2650—editorialnote.) Finally, it should be noted that generalized Harish-Chandra modules have been studiedalsobygeometricmethods(D-modules).Inparticular,geometricmethods havebeenhelpfulinthecomputationbyPenkov,Serganova,andZuckermanofthe Liesubalgebrag[M]forcertainM,aswellasintherecentconstructionofbounded multiplicity(g,l)-modulesbyPenkovandSerganova. ThePapers B-Orbitsof2-NilpotentMatricesandGeneralizations, byMagdalena Boosand MarkusReineke,Wuppertal Let B be the standard Borel subgroup of GL(n,C). Melnikov showed that the set ofB-orbitsinthesetofuppertriangularmatricesofsquarezeroisfinite,classified themintermsofinvolutionsinthesymmetricgroupS ,andshowedthattheinclu- n sion relations of their closures can be defined by link patterns. The authors obtain similarresultsfor(thelargerfamilyof)B-orbitsinthesetofallmatricesofsquare zero.TheirmethodsarequitedifferenttothoseofMelnikov.Indeed,theauthorsuse arelationofindecomposablerepresentationsofquiverstotheseorbits.Theformer areclassifiedusingtheAuslander–Reitenquiver.TheyuseresultsofZwaratotrans- latetheconditionofinclusionunderorbitclosureintomoduletheoreticterms.This gives the minimal orbit closure inclusion via oriented link patterns. Finally they discuss generic B-orbits in the set of all nilpotent matrices as well as describing semi-invariantsfortheaction.BoosandReinekesuggestthatthesemaygenerateall thesemi-invariants. WeylDenominatorIdentityforFinite-DimensionalLie Superalgebras, byMariaGorelik,Weizmann Institute TheWeyldenominatoridentityforthealgebrasinthetitlewasformulatedbyKac andWakimoto,whoalsoproveditinsomecases.ThebasicideaofGorelik’sproof (whichispurelycombinatorial)istoshowthatbothsidesoftheidentityareskewin- variantfortheWeylgroupW andthateachsideformsjustoneorbitsum.However, unliketheLiealgebracase,thereareseveraltechnicaldifficultieseventoformulate theidentity. In general, one side of the identity is just a product over the positive roots as in the Lie algebra case but with sign changes corresponding to the odd roots. The second side is given by two sets of data W# and S. The former is a subgroup of the Weyl group, and the latter is a certain maximal isotropic set of simple roots.
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