D-Modules, Perverse Sheaves, and Representation Theory

407 Pages·2008·2.162 MB·English

by  Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki (auth.), Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki (eds.)

Progress in Mathematics
Volume236
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Ryoshi Hotta
Kiyoshi Takeuchi
Toshiyuki Tanisaki
D-Modules,
Perverse Sheaves,
and Representation Theory
Translated by
Kiyoshi Takeuchi
Birkha¨user
Boston • Basel • Berlin
RyoshiHotta KiyoshiTakeuchi
ProfessorEmeritusofTohokuUniversity SchoolofMathematics
Shirako2-25-1-1106 TsukubaUniversity
Wako351-0101 Tenoudai1-1-1
Japan Tsukuba305-8571
[email protected] Japan
[email protected]
ToshiyukiTanisaki
DepartmentofMathematics
GraduateSchoolofScience
OsakaCityUniversity
3-3-138Sugimoto
Sumiyoshi-ku
Osaka558-8585
Japan
[email protected]
MathematicsSubjectClassification(2000):Primary:32C38,20G05;Secondary:32S35,32S60,17B10
LibraryofCongressControlNumber:2004059581
ISBN-13:978-0-8176-4363-8 e-ISBN-13:978-0-8176-4523-6
Printedonacid-freepaper.
c2008,EnglishEditionBirkha¨userBoston
c1995,JapaneseEdition,Springer-VerlagTokyo,DKaguntoDaisugun(D-ModulesandAlgebraic
Groups)byR.HottaandT.Tanisaki.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit-
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Preface
D-Modules, Perverse Sheaves, and Representation Theory is a greatly expanded
translation of the Japanese edition entitled D kagun to daisugun (D-Modules and
Algebraic Groups) which was published by Springer-Verlag Tokyo, 1995. For the
newEnglishedition,thetwoauthorsoftheoriginalbook,R.HottaandT.Tanisaki,
haveaddedK.Takeuchiasacoauthor. Significantnewmaterialalongwithcorrections
andmodificationshavebeenmadetothisEnglishedition.
In the summer of 1982, a symposium was held in Kinosaki in which the sub-
ject of D-modules and their applications to representation theory was introduced.
At that time the theory of regular holonomic D-modules had just been completed
and the Kazhdan–Lusztig conjecture had been settled by Brylinski–Kashiwara and
Beilinson–Bernstein. Thearticlesthatappearedinthepublishedproceedingsofthe
symposiumwerenotwellpresentedandofcoursethesubjectwasstillinitsinfancy.
Several monographs, however, did appear later on D-modules, for example, Björk
[Bj2],Boreletal.[Bor3],Kashiwara–Schapira[KS2],Mebkhout[Me5]andothers,
all of which were taken into account and helped us make our Japanese book more
comprehensiveandreadable. Inparticular,J.Bernstein’snotes[Ber1]wereextremely
usefultounderstandthesubjectinthealgebraiccase;ourtreatmentinmanyaspects
follows the method used in the notes. Our plan was to present the combination of
D-moduletheoryanditstypicalapplicationstorepresentationtheoryaswebelieve
thatthisisanicewaytounderstandthewholesubject.
Let us briefly explain the contents of this book. Part I is devoted to D-module
theory, placing special emphasis on holonomic modules and constructible sheaves.
TheaimhereistopresentaproofoftheRiemann–Hilbertcorrespondence. PartII
isdevotedtorepresentationtheory. Inparticular,wewillexplainhowtheKazhdan–
LusztigconjecturewassolvedusingthetheoryofD-modules. Toacertainextentwe
assumethereader’sfamiliaritywithalgebraicgeometry,homologicalalgebras,and
sheaftheory. Althoughweincludeintheappendicesbriefintroductionstoalgebraic
varietiesandderivedcategories,whicharesufficientoverallfordealingwiththetext,
thereadershouldoccasionallyrefertoappropriatereferencesmentionedinthetext.
ThemaindifferencefromtheoriginalJapaneseeditionisthatwemadesomenew
chapters and sections for analytic D-modules, meromorphic connections, perverse
vi Preface
sheaves,andsoon. WethusemphasizedthestrongconnectionsofD-moduleswith
variousotherfieldsofmathematics.
We express our cordial thanks to A. D’Agnolo, C. Marastoni, Y. Matsui,
P. Schapira, and J. Schürmann for reading very carefully the draft of the English
version and giving us many valuable comments. Discussions with M. Kaneda,
K.Kimura,S.Naito,J.-P.Schneiders,K.Vilonen,andotherswerealsoveryhelpful
incompletingtheexposition. M.NaguraandY.Sugikigreatlyhelpedusintyping
andcorrectingourmanuscript. Thanksalsogotomanypeopleforusefulcomments
onourJapaneseversion,inparticulartoT.Ohsawa. Lastbutnotleast,wecannotex-
aggerateourgratitudetoM.Kashiwarathroughouttheperiodsince1980onvarious
occasions.
2006March R.Hotta
K.Takeuchi
CollegeofMathematics,UniversityofTsukuba
T.Tanisaki
FacultyofScience,OsakaCityUniversity
Contents
Preface ......................................................... v
Introduction..................................................... 1
PartI D-ModulesandPerverseSheaves
1 PreliminaryNotions .......................................... 15
1.1 Differentialoperators ....................................... 15
1.2 D-modules—warmingup.................................... 17
1.3 InverseanddirectimagesI................................... 21
1.4 SomecategoriesofD-modules ............................... 25
1.5 InverseimagesanddirectimagesII ........................... 31
1.6 Kashiwara’sequivalence .................................... 48
1.7 Abasechangetheoremfordirectimages ....................... 53
2 CoherentD-Modules ......................................... 57
2.1 Goodfiltrations ............................................ 57
2.2 Characteristicvarieties(singularsupports)...................... 59
2.3 Dimensionsofcharacteristicvarieties.......................... 62
2.4 Inverseimagesinthenon-characteristiccase.................... 64
2.5 Properdirectimages ........................................ 69
2.6 Dualityfunctors............................................ 70
2.7 Relationsamongfunctors.................................... 76
2.7.1 Dualityfunctorsandinverseimages .................... 76
2.7.2 Dualityfunctorsanddirectimages...................... 77
3 HolonomicD-Modules........................................ 81
3.1 Basicresults............................................... 81
3.2 FunctorsforholonomicD-modules ........................... 83
3.2.1 Stabilityofholonomicity.............................. 83
viii Contents
3.2.2 HolonomicityofmodulesoverWeylalgebras ............ 86
3.2.3 Adjunctionformulas.................................. 91
3.3 Finitenessproperty ......................................... 93
3.4 Minimalextensions......................................... 95
4 AnalyticD-ModulesandthedeRhamFunctor ................... 99
4.1 AnalyticD-modules ........................................ 99
4.2 SolutioncomplexesanddeRhamfunctors...................... 103
4.3 Cauchy–Kowalevski–Kashiwaratheorem ...................... 105
4.4 Cauchyproblemsandmicro-supports.......................... 107
4.5 Constructiblesheaves ....................................... 110
4.6 Kashiwara’sconstructibilitytheorem .......................... 113
4.7 AnalyticD-modulesassociatedtoalgebraicD-modules .......... 119
5 TheoryofMeromorphicConnections............................ 127
5.1 Meromorphicconnectionsintheone-dimensionalcase ........... 127
5.1.1 SystemsofODEsandmeromorphicconnections.......... 127
5.1.2 Meromorphicconnectionswithregularsingularities ....... 130
5.1.3 RegularityofD-modulesonalgebraiccurves............. 135
5.2 Regularmeromorphicconnectionsoncomplexmanifolds......... 139
5.2.1 Meromorphicconnectionsinhigherdimensions .......... 139
5.2.2 Meromorphicconnectionswithlogarithmicpoles ......... 143
5.2.3 Deligne’sRiemann–Hilbertcorrespondence.............. 147
5.3 Regularintegrableconnectionsonalgebraicvarieties............. 153
6 RegularHolonomicD-Modules ................................ 161
6.1 Definitionandmaintheorems ................................ 161
6.2 Proofofmaintheorems ..................................... 163
7 Riemann–HilbertCorrespondence.............................. 171
7.1 CommutativitywithdeRhamfunctors......................... 171
7.2 Riemann–Hilbertcorrespondence ............................. 174
7.3 Comparisontheorem........................................ 178
8 PerverseSheaves............................................. 181
8.1 Theoryofperversesheaves .................................. 181
8.1.1 t-structures ......................................... 181
8.1.2 Perversesheaves..................................... 191
8.2 Intersectioncohomologytheory .............................. 202
8.2.1 Introduction......................................... 202
8.2.2 Minimalextensionsofperversesheaves ................. 203
8.3 Hodgemodules ............................................ 217
8.3.1 Motivation.......................................... 217
8.3.2 Hodgestructuresandtheirvariations.................... 217
8.3.3 Hodgemodules...................................... 220
Contents ix
PartII RepresentationTheory
9 AlgebraicGroupsandLieAlgebras ............................. 229
9.1 Liealgebrasandtheirenvelopingalgebras...................... 229
9.2 SemisimpleLiealgebras(1).................................. 231
9.3 Rootsystems .............................................. 233
9.4 SemisimpleLiealgebras(2).................................. 236
9.5 Finite-dimensionalrepresentationsofsemisimpleLiealgebras..... 239
9.6 AlgebraicgroupsandtheirLiealgebras ........................ 241
9.7 Semisimplealgebraicgroups ................................. 243
9.8 Representationsofsemisimplealgebraicgroups ................. 248
9.9 Flagmanifolds............................................. 250
9.10 Equivariantvectorbundles................................... 253
9.11 TheBorel–Weil–Botttheorem................................ 255
10 ConjugacyClassesofSemisimpleLieAlgebras.................... 259
10.1 Thetheoryofinvariantpolynomials ........................... 259
10.2 Classificationofconjugacyclasses ............................ 263
10.3 Geometryofconjugacyclasses ............................... 266
11 RepresentationsofLieAlgebrasandD-Modules .................. 271
11.1 Universalenvelopingalgebrasanddifferentialoperators.......... 271
11.2 Ringsoftwisteddifferentialoperatorsonflagvarieties ........... 272
11.3 ProofofTheorem11.2.2..................................... 276
11.4 ProofofTheorems11.2.3and11.2.4 .......................... 280
11.5 EquivariantrepresentationsandequivariantD-modules .......... 283
11.6 ClassificationofequivariantD-modules ....................... 286
12 CharacterFormulaofHighestWeightModules ................... 289
12.1 Highestweightmodules ..................................... 289
12.2 Kazhdan–Lusztigconjecture ................................. 295
12.3 D-modulesassociatedtohighestweightmodules................ 299
13 HeckeAlgebrasandHodgeModules ............................ 305
13.1 WeylgroupsandD-modules ................................. 305
13.2 HeckealgebrasandHodgemodules ........................... 310
A AlgebraicVarieties ........................................... 321
A.1 Basicdefinitions ........................................... 321
A.2 Affinevarieties............................................. 323
A.3 Algebraicvarieties.......................................... 325
A.4 Quasi-coherentsheaves ..................................... 326
A.5 Smoothness,dimensionsandlocalcoordinatesystems............ 328
x Contents
B DerivedCategoriesandDerivedFunctors........................ 331
B.1 Motivation ................................................ 331
B.2 Categoriesofcomplexes..................................... 333
B.3 Homotopycategories ....................................... 335
B.4 Derivedcategories.......................................... 339
B.5 Derivedfunctors ........................................... 343
B.6 Bifunctorsinderivedcategories .............................. 347
C SheavesandFunctorsinDerivedCategories...................... 351
C.1 Sheavesandfunctors ....................................... 351
C.2 Functorsinderivedcategoriesofsheaves ...................... 355
C.3 Non-characteristicdeformationlemma......................... 361
D FilteredRings ............................................... 365
D.1 Goodfiltration ............................................. 365
D.2 Globaldimensions.......................................... 367
D.3 Singularsupports........................................... 370
D.4 Duality ................................................... 372
D.5 Codimensionfiltration ...................................... 375
E SymplecticGeometry ......................................... 379
E.1 Symplecticvectorspaces .................................... 379
E.2 Symplecticstructuresoncotangentbundles..................... 380
E.3 Lagrangiansubsetsofcotangentbundles ....................... 381
References ...................................................... 387
ListofNotation .................................................. 397
Index........................................................... 403