Geometric invariant theory : over the real and complex numbers PDF
Preview Geometric invariant theory : over the real and complex numbers
Universitext Nolan R. Wallach Geometric Invariant Theory Over the Real and Complex Numbers Universitext Universitext Series editors Sheldon Axler San Francisco State University Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah École polytechnique, CNRS, Université Paris-Saclay, Palaiseau Endre Süli University of Oxford Wojbor A. Woyczyński CCCaaassseee WWWeeesssttteeerrrnnn RRReeessseeerrrvvveee UUUnnniiivvveeerrrsssiiitttyyy Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approachtotheirsubjectmatter.Someofthemostsuccessfulandestablishedbooks intheserieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at http://www.springer.com/series/223 Nolan R. Wallach Geometric Invariant Theory Over the Real and Complex Numbers Nolan R. Wallach Department of Mathematics University of California, San Diego La Jolla, CA, USA ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-319-65905-3 ISBN 978-3-319-65907-7 (eBook) DOI 10.1007/978-3-319-65907-7 Library of Congress Control Number: 2017951853 Mathematics Subject Classification (2010): 14-XX, 14L24 © Nolan R. Wallach 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Barbara ‘Fair,’‘kind,’and‘true’haveoftenliv’dalone, Whichthreetillnowneverkept seat inone. FromShakespeare’sSonnet105 Nolan Preface This book evolved from lecture notes I wrote for several of my courses on Lie theory,algebraicgrouptheoryandinvarianttheoryattheUniversityofCalifornia, San Diego.Theparticipantsin theseclasses werefacultyandgraduatestudentsin mathematicsandphysicswithdiverselevelsofsophisticationinalgebraicanddif- ferentialgeometry.Thecoursesweremotivated,inpart,bythefactthatthemethods ofinvarianttheoryhavebecomeimportantingaugetheory,fieldtheory,andinmea- suringquantumentanglement.Thelattertheorycanbeunderstoodasanattemptto findnormalformsfortheelementsofatensorproductofmanycopiesofaHilbert space(whichforuswillbefinite-dimensional)undertheactionofthetensorproduct ofthesamenumberofcopiesoftheoperatorsofdeterminantoneoroftheunitary operators.ThisispreciselythetypeofproblemthatledMumfordtoGeometricIn- variantTheory:parametrizetheorbitsofa reductivegroupactingalgebraicallyon avariety.Thatsaid,specialistsingeometricinvarianttheorywillfindthatthisbook emphasizesaspectsofthesubjectthatarenotnecessarilyintheirmainstream.My goalinthisbookistoexplainthepartsofthesubjectthatIandmycoworkersneeded forourresearchbutfoundverydifficulttounderstandintheliterature. Thetermgeometricinvarianttheory(GIT)isduetoMumfordandisthetitleof hisfoundationalbook[Mu].Thisamazingworkbeganwithanexplanationofhow agroupschemeactsonaschemeandlaysthefoundationnecessaryforthedifficult theoryinpositivecharacteristic.IrememberthatwhenIattemptedtoreadthiswork, Irealizedrapidlythatmyalgebraicgeometrywasinadequate.Ishouldmentionthat at that time I was a differential geometer whose background in algebra didn’t go much further than the book by Birkhoff and MacLane. It was only later, when I beganto understandthe problemsin geometrythat involvedmoduliof structures, thatIbegantohaveanideaofthemeaningofGITandhowitdiffersfromclassical invarianttheory(CIT). The purposeof this bookis to developGIT in the contextof algebraic geome- tryoverthecomplexandrealnumbers.InthiscontextIcanexplainthedifference between what I mean by GIT and what I mean by CIT. The emphasis of CIT is twofold:thefirstproblemistofindanicesetofgeneratorsfortheinvariantpolyno- mialsonavectorspaceonwhichagroup(reductivealgebraic)actslinearly,ormore vii viii Preface generally when it acts regularly on an affine variety. A solution is usually called “thefirst fundamentaltheorem.”Thesecondproblemis todeterminethe relations between the invariantswhich is called “the second fundamentaltheorem.”In CIT the second problem makes no sense without a complete solution to the first. GIT studiesthesecondproblemevenbeforethefirsthasbeensolved.Forexample,the Hilbert–Mumfordtheoremisageometriccharacterizationofthesetofzerosofthe homogeneousinvariantpolynomialsof positive degree. One would think that one needstoknowthepolynomialsinordertofindtheirzeros.Thesecondfundamental theoremcanbethoughtofasanalgebraicgeometricstructureonthesetofclosed orbits(the simplestGIT quotient).Again,thisquotientcanbe understoodwithout knowingthefullsetofinvariantsprecisely. Havingrestricted my emphasisto the realand complexnumbers,my approach to the subject will be eclectic. Thatis, when an argumentusing special properties of these fields is simpler than an argument that has applications to more general fields, then I will use the simpler argument (for example, the proof of the Borel Fixed Point Theorem).Also, my concentrationon these fields leads to substantial simplificationsinthedetailsofthebasictheoremsofalgebraicgeometryneededto developthetheory. I have, throughout the book attempted to keep the material to the level of my book with Roe Goodman [GW]. I have freely used results from that book (prop- erly referenced). There are occasions when I prove a result that can be found in [GW] (generallywith a differentproof).This is usually whenI feelthat the argu- mentisusefultounderstandingthemethodologyofthisbook.Thereaderwillfind that the material becomes progressively more difficult (i.e., more complicated) as each chapter progresses. The book is not meant to be read from start to finish. I havetakenpainstomakethestatementsofthetheoremsmeaningfulwithoutafull understandingoftheproofs. Theexpositionisdividedintotwoparts.Thefirst,whichismeanttobeusedas aresourceforthesecond,iscalledBackgroundTheory.Itconsistsoftwochapters that should be read as needed for the second part. The first chapter emphasizes the relationshipbetweenthe Zariskitopology(called the Z-topologyin this book) andthecanonicalHausdorfftopology(alsocalledtheclassical,ormetrictopology whichwewillcallthestandardorS-topology)ofanalgebraicvarietyoverC.Igivea completeproofofthesurprisinglyhardtheoremassertingthatasmoothvarietyover C has a canonicalcomplexmanifold structure when endowedwith its S-topology thatiscompatiblewithitssheafoffunctionsasanalgebraicvariety. ThesecondchapterinthispartisadevelopmentoftheinteractionbetweenLie groupsandalgebraicgroups.Therearetwomaintheoremsinthischapter.Thefirst is that a reductive algebraic subgroup is isomorphic with the Zariski closure of a compact subgroup of GL(n,C) for some n; this approach also appears in [Sch]. The method of proof also proves Matsushima’s theorem on the stability group of anaffineorbitofareductivegroup.ThesecondtheoremisavariantofChevalley’s proof of the conjugacy of maximal compact subgroups of a real reductive group. Bothuseaversionofthe“easypart”oftheKempf–NesstheoremoverRwhichis provedinthatchapter. Preface ix Thesecondpartofthebook,calledGeometricInvariantTheory,consistsofthree chapters.ThefirstcentersontheHilbert–Mumfordtheoremandthestructureofthe categorical (or GIT) quotient of a regular representation of a reductive algebraic group over C. I give two proofs of the Hilbert–Mumford characterization of the null cone of a regular representation of a reductive group. The first proof derives the theorem as a consequence of a theorem over R. The second proof gives the generalizationofthetheorem,duetoRichardson,whichisnecessaryfortheproof of the “hard part” of the Kempf–Nesstheorem thatis a characterizationof closed orbits.Myproofsare,perhaps,abitsimplerthantheoriginals.Theanalogueofthe fullKempf–NesstheoremoverRisderivedfromthetheoremoverC. Oneapplicationofthisresultistophysicists’mixedstates.Asecondapplication of the theorem is to a determinationof the S-topology of the categorical quotient of a regular representation (ρ,V) of a reductive algebraic group G over C with maximal compact subgroup K. We use the Kempf–Ness theorem to define a real affineK-varietyX suchthatrelativetotheS-topology,X/K ishomeomorphicwith thecategorical(i.e.,GIT)quotientV//G[RS]. Thischapteremphasizesreductivegroupactionsonaffinevarieties.Itendswith a developmentof Vinberg’s generalization of the Kostant–Rallis theory including a generalizationof theirmultiplicitytheoremontheharmonics;thisis newto this book.Thismaterialmakesupasubstantialpartofthisbook,butitisincludedonly becauseitleadstoseveralimportantexamples.Twostrikingexamplesofthemulti- plicityformulaareincludedattheendofthechapter.Alsoincludedinthischapter is a complete proof of the Shephard–Todd theorem and the work of Springer on centralizersofregularelementsofWeylgroups. The second chapter in this part (Chapter 4) studies the orbit structure of a re- ductivealgebraicgroupona projectivevariety.Inthe affinecase the closed orbits tendtobeorbitsofmaximaldimension.Intheprojectivecasetheclosedorbitstend tobetheminimalorbitsorareatleastverysmall.Themainresultsinthischapter involve techniques related to Kostant’s proof of his quadratic generation theorem for the ideal of the minimal orbit of the projectivespace of an irreducibleregular representationof a reductivegroup.We provethe results usingKostant’s amazing formulasinvolvingtheCasimiroperator. Thethirdchapterinthispartstudiestheextensionofclassicalinvarianttheoryto productsofclassicalgroupsandthecorrespondingGIT.Thistheoryisanoutgrowth of recentworkwith Gilad Gour[GoW] forthe case ofproductsof groupsof type SL(n,C)whichshowshowtoconstructallinvariantsofafixeddegree,whichinthe physicsliteraturearecalledmeasuresofentanglement.Thereisasmalldessert.In the last three subsections, we study 4 qubits and 3 qutrits (which is related to the mostinterestingVinbergpairforE ).Inaddition,westudymixedqubitstatesusing 6 resultsderivedfromthetheoryinChapter3. Throughoutthe book examplesare emphasized. There are also exercises that I hopewill addto thereader’sunderstanding.Someof theexercisesarealso neces- sarytocompleteproofs.Theseareenhancedwithhints(asaremanyoftheothers). We also includeasubsectionin Chapter5,5.4.2.1,thatexpressesthequbitresults inbraandketnotationwhichisthenusedliberallyintherestofthechapter. x Preface Acknowledgments Asindicatedabove,thisbookisanoutgrowthofyearsofcoursesonalgebraicand Lie grouptheory.I thank the studentsat RutgersUniversity and the University of California,SanDiego(UCSD),fortheirforbearanceasthematerialevolved.Ihave hadthe goodfortuneto haveanamazinggroupof distinguishedvisitorsatUCSD overtheyears.Ihavelearnedfromallofthem,andtheirlecturesandpersonalcon- versationshaveplayedamajorroleinexpandingmyknowledgebase.Mostnotably IwouldliketothankHanspeterKraftforhishelpovertheyears.IwishthatIcould personallythankBertKostantandArmandBorel.Myone-yearcollaborationwith Dick Grosswas a learningexperienceforbothof us. The beautifulpaper(written byDick)[GrW]thatwastheculminationofourjointworkcontainedtheseedsof mylaterinterestingeometricinvarianttheory,asopposedtoCIT. My one-year collaboration with Gross also contains the solution to a question that David Meyer asked me about quantum entanglement which led to our long collaboration in the study of quantum information theory. Meyer had a visiting postdoctoralfellow,GiladGour,whoseamazingunderstandingofquantumentan- glementhasbeenaninspiration.Inaddition,Iwouldliketothankthepostdoctoral and predoctoral scholars that I have mentored: Laura Barberas, Karin Baur, Sam Evens, Joachim Kuttler and my Ph.D. students over the last 20 years, Allan Kee- ton, Markus Hunziker, Jeb Willenbring, Reno Sanchez, Orest Bucicovschi, Mark Colarusso,OdedJacobi,RaulGomez,AsifShakeel,SeungLeeandJonMiddleton. I wouldalso like to thankElizabethLoewforher earlyencouragementandfor shepherding this book through the publication process. I would especially like to thankAnnKostantforherworkastheeditorofthisbookandalsoforhermanyacts offriendship.Sheencouragedthecompletionofthisbookonmanyoccasionswhen Ihadbalked.Shealsopickedtheworld’sbesttypesetter,BrianTreadway. In October of 2015, I presented the Dean Jacqueline B. Lewis Memorial Lec- turesatRutgersUniversity.Thelectureswereintendedtobeanintroductionofthe methodsandphilosophyofthisbook.IthanktheRutgersmathematicsdepartment for its hospitality and enthusiasm for the materialcoveredduringthe week of my visit. ThemostimportantpersoninvolvedinthisprojectismywifeBarbara.Without hersupportthisbookcouldnothavebeenwritten. NolanWallach DepartmentofMathematics UniversityofCalifornia,SanDiego SanDiego,CA92093
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