Representation Theory and Geometry of the Flag Variety

136 Pages·2022·3.104 MB·English

by  William M. McGovern

WilliamM.McGovern
RepresentationTheoryandGeometryoftheFlagVariety
De Gruyter Studies in
Mathematics
|
Edited by
Carsten Carstensen, Berlin, Germany
Gavril Farkas, Berlin, Germany
Nicola Fusco, Napoli, Italy
Fritz Gesztesy, Waco, Texas, USA
Niels Jacob,Swansea, United Kingdom
Zenghu Li, Beijing, China
Karl-Hermann Neeb, Erlangen, Germany
René L.Schilling, Dresden, GermanyVolkmar Welker,
Marburg, Germany
Volume 90
William M. McGovern
Representation Theory
and Geometry of the
Flag Variety
|
MathematicsSubjectClassification2020
22E46
Author
Prof.WilliamM.McGovern
UniversityofWashington
DepartmentofMathematics
P.O.Box354350PDLC-450
SeattleWA98195-4350
USA
[email protected]
ISBN978-3-11-076690-5
e-ISBN(PDF)978-3-11-076694-3
e-ISBN(EPUB)978-3-11-076696-7
ISSN0179-0986
LibraryofCongressControlNumber:2022944915
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2023WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Preface
Theaimofthisbook,asindicatedbyitstitle,istodeveloptheconnectionsbetween
therepresentationtheoryofacomplexsemisimpleLiegroupGandthegeometryof
itsflagvarietyX = G/B.Webeginbylookingatfinite-dimensionalrepresentations,
mostlyofclassicalgroups.WethenoutlinetheBeilinson–Bernsteinconstructionof
irreduciblerepresentationsofGfromclosuresoforbitsofsubgroupsofGinX,also
showinghowtogotheotherway,fromrepresentationstoorbitclosures,bymeansof
thecharacteristicvarietyofarepresentation.Thisleadsinanaturalwaytostudying
whichsuchorbitclosureslieinwhichothersandthesingularitiesoforbitclosures
atpointsofsmallerorbits.WediscussKazhdan–LusztigandLusztig–Voganpolyno-
mials, introduced to study both these singularities and representations. We offer a
number of combinatorial characterizations of nonsingular orbit closures, mostly in
theclassicalcases.Finally,inadifferentdirection,westudySpringerfibersinXand
thesingularitiesoftheircomponents.
Inmoredetail,afterlayingoutprerequisitesinapreliminarychapter,wegivea
linearalgebraicconstruction,whichstartsfromthedefiningrepresentationofthegen-
erallineargroupandconstructsfromthisallofitsirreduciblefinite-dimensionalrep-
resentations.Modelsfortheserepresentationsaregivenbythesetsofsemistandard
Young tableaux of a fixed shape with entries from a fixed alphabet of symbols. We
thenextendtheconstructiontotheotherclassicalgroups,concludingwithabriefac-
countoftheBorel–Weilconstructionoffinite-dimensionalirreduciblerepresentations
ofsemisimplegroupsfromtheirhighestweights.
Turningthentoinfinite-dimensionalrepresentations,weusetheBorho–Brylinski
realizationofthequotientU/I oftheenvelopingalgebraU oftheLiealgebragofG
byasuitableminimalprimitiveidealI astheringofalgebraicdifferentialoperators
onXandtheBeilinson–BernsteinlocalizationofafinitelygeneratedmoduleMover
U/I asasheafofmodulesoverthesheafofdifferentialoperatorsonX todefinethe
characteristicandassociatedvarietiesofMinChapter3andthemomentmaptaking
thefirstofthesetothesecond.InthiswaytherepresentationtheoryofGandgeometry
ofXarefirmlytiedtogether.Todocomputations,weneedadecompositionofXinto
finitelymanysuborbits,eitherunderaBorelsubgroupBofGorunderasymmetric
subgroupK.Wealsoneedadescriptionofwhichorbitclosureslieinwhichothers.We
givethisexplicitlyintheclassicalcases.Inthenextchapter,westudythesingularities
ofvarieties,ororbitclosures,inmoredetail;inthefollowingtwochapters,wegive
an account of the Kazhdan–Lusztig and Lusztig–Vogan polynomials, introduced to
measurethesesingularitiesprecisely.Weconsiderthenicestkindofvarieties,namely
therationallysmoothones,showinghowtocharacterizethesecombinatoriallyinthe
classicalcasesandtoacertainextentingeneral.Weconcludewithsomerecentwork
oncomponentsofSpringerfibers.
Theprerequisitesforthisbookconsistofknowledgeofthestructureandfinite-
dimensionalrepresentationtheoryofcomplexsemisimplegroupstogetherwithsome
https://doi.org/10.1515/9783110766943-201
VI | Preface
basicalgebraicgeometryandcertainfactsabouthighestweightandHarish-Chandra
modules.Familiaritywiththeauthor’sjointearlierbookonnilpotentorbitswithDavid
Collingwoodwouldalsobehelpful;someofthematerialfromthisbookisreviewedin
thefirstchapter.ThebookgrewoutofacoursethattheauthorgaveattheUniversity
ofWashingtoninthefallof2021.
Ineachsection,theorems,lemmas,corollaries,anddefinitionsarenumberedsep-
arately,sothatitispossible,forexample,foratheoremandacorollaryinthesame
sectiontohavethesamenumber.
WilliamM.McGovern
Contents
Preface|V
1 Preliminaries|1
1.1 Structureofg|1
1.2 Theflagvariety|1
1.3 Finite-dimensionalandhighestweightrepresentations|2
1.4 SymmetricsubgroupsofG|4
1.5 Harish–ChandramodulesanddifferentialoperatorsonG/B|5
1.6 Nilpotentorbitsandorbitalvarieties|7
2 Polynomialrepresentationsandtheflagvariety|10
2.1 Schurmodules|10
2.2 IdentificationandirreducibilityofthemodulesVλ|14
2.3 Definingequationsoftheflagvariety|16
2.4 Cohomologyoftheflagvariety|19
2.5 Symplecticgroups|24
2.6 Orthogonalgroups|24
2.7 Thegeneralcaseandlinebundles|26
3 Associated,characteristic,andorbitalvarieties|27
3.1 Definitionsofthecharacteristicvarietyandmomentmap|27
3.2 K-orbitsinX |29
3.3 TheclosureorderonK-orbits|33
3.4 OrbitalvarietiesandSpringerfibers|37
4 Singularitiesoforbitclosures|46
4.1 Smoothandrationallysmoothpoints|46
4.2 Criterionforrationalsmoothness|47
4.3 Symmetricvarieties|50
5 Kazhdan–LusztigpolynomialsandtheHeckealgebra|53
5.1 TheHeckealgebraandtheR-polynomials|53
5.2 Kazhdan–Lusztigpolynomials|55
5.3 Anotherconstruction|58
5.4 InverseKLpolynomials|60
5.5 Cones,cells,andW-graphs|61
5.6 Applicationstorepresentationtheory|65
5.7 Applicationstogeometry|72
VIII | Contents
6 Lusztig–VoganpolynomialsandHarish-Chandramodules|77
6.1 ℤ/2ℤdata|77
6.2 Circleandcrossactions|80
6.3 TheHeckemodule|82
6.4 Applications|85
7 Patternavoidanceandsingularitiesoforbitclosures|88
7.1 PatternavoidanceforclassicalSchubertvarieties|88
7.2 Rootsubsystempatternavoidance|90
7.3 Patternavoidanceforsymmetricvarieties|94
7.4 RichardsonvarietiesandsymmetricorbitsintypeAIII|99
7.5 Intervalpatternavoidance|101
8 Springerfibers|105
8.1 Thetwo-columncase|105
8.2 Particularcomponents|107
8.3 CohomologyofSpringerfibers|110
8.4 Orbitalvarietiesingeneral|111
Bibliography|115
Index|123
1 Preliminaries
Inthischapter,weassemblesomebasicfactsaboutcomplexsemisimpleLiealgebras,
flagvarietiesoftheiradjointgroups,andnilpotentorbitsinthealgebras,underthe
actionsofboththeadjointgroupanditssymmetricsubgroups.Wealsoreviewsome
morespecializedmaterialonhighestweightandHarish–Chandramodulesoversuch
algebras.Ourbasicreferencesare[23,34,66,62,73,137].Throughoutthebook,gde-
notesacomplexsemisimpleLiealgebra(orsometimesareductiveone),Uitsenvelop-
ingalgebra,Gitsadjointgroup,baBorelsubalgebraofg,n=[b,b]itsnilradical,Bthe
BorelsubgroupofGcorrespondingtob,andXthequotientG/B,whichidentifieswith
theflagvarietyofallBorelsubalgebrasofg.Further,weletθdenoteaninvolutiveauto-
morphismofGandK =Gθitsfixedpointsubgroup,calledasymmetricsubgroupofG.
Thedifferentialofθisthenaninvolutiveautomorphismofg,alsodenotedbyθ;wede-
notebyk+pthe(Cartan)decompositionofgasthesumofthe+1-and−1-eigenspaces
ofθ.ForfixedG,thereareonlyfinitelymanychoicesforθuptoconjugacyinG;we
willbrieflyrecallthesebelowintheclassicalcases.
1.1 Structureofg
Werecallthatgisdetermineduptoisomorphismbyits(crystallographic)rootsystem
Φ,afinitecollectionofnonzerovectorsinaEuclideanspaceℝnsuchthat
1. Φspansℝn,
2. Φ∩2Φ=0,
(β,α)
3. 2(α,α) ∈ℤforα,β∈Φ,and
(β,α)
4. sα(β)=β−2(α,α) ∈Φforα,β∈Φ,
where (⋅,⋅) denotes the usual dot product in ℝn. Such systems are classified in [66,
11.4];everyonearisesastherootsystemofasemisimpleLiealgebra.
1.2 Theflagvariety
ABorelsubalgebraofgisamaximalsolvablesubalgebra;anytwosuchareconjugate
undertheadjointgroupG,andthestabilizerofaBorelsubalgebrabisitsadjointgroup
B[66,16.4].Wehavethewell-knownBruhatdecompositionG = ⋃w∈W̃ BwBofGinto
doublecosetsofBor,equivalently,oftheflagvarietyX =G/BintoB-suborbits.HereT
isamaximaltorusofGcontainedinB,Nisitsnormalizer,thequotientW =N/Tisthe
WeylgroupofG,andW̃ consistsofoneelementfromeachcosetofT inN [137,10.2].
TheLiealgebrahofT isaCartansubalgebraofg.AtypicaldoubleclosetBwBcorre-
spondstoaB-orbitX inX,calledaSchubertcell;byabuseofnotationweusually
w
regardSchubertcellsasindexedbyelementsofW ratherthancosetrepresentatives
https://doi.org/10.1515/9783110766943-001
2 | 1 Preliminaries
ofTinN.TheorbitXwisisomorphictotheaffinespaceℂℓ(w),whereℓ(w)isthelength
ofw,i.e.,theminimumnumberofsimplereflectionswithproductwor,alternatively,
thenumberofpositiveroots(relativetob)thatwsendstonegativeroots[137].Inpar-
ticular, the Schubert cell X attached to the longest element w of W is open and
w0 0
denseinX,whereasthecellX attachedtotheidentityelementisasinglepoint.The
1
closureV ofX inX (ineithertheZariskiortheEuclideantopology)iscalledthe
w w
Schubertvarietyindexedbyw.AnytwoBorelsubgroupsB ,B ofGhaveacommon
1 2
maximaltorusT,whichisalsoamaximaltorusofG,sothatsomeelementofW =N/T
takesB1toB2[66].ThustheorbitsofBorelsubalgebrasb1×b2ofG×Grelativetothe
diagonalactionofGarealsoindexedbyelementswofW.Inparticular,givenBand
−
amaximaltorusT containedinit,thereisauniqueoppositeBorelsubgroupB toB
−
whoseintersectionwithBisexactlyT;wedenotebyB itsadjointgroup.Theorbits
−
Yw ofB inG/BarecalledoppositeSchubertcells;theyarealsoindexedbyelements
w∈W.TheclosureZwofYwiscalledanoppositeSchubertvariety.
IntheclassicalcasesG = SL(n,ℂ)andG = GL(n,ℂ),wemayrealizeanyBorel
subalgebraofgasthestabilizerofa(complete)flag inℂn,thatis,ofachainV0 ⊂
V1 ⊂ ⋅⋅⋅ ⊂ Vn = ℂnofsubspacesViofV = ℂnsuchthatdimVi = i.Weoftenidentify
thesubalgebrawiththeflag,thusidentifyingXwiththesetofallsuchflags.Herethe
WeylgroupW isthesymmetricgroupSnactingonℂnbypermutingthecoordinates.
Intheotherclassicalcases,whereGisthegroupofisometriesofℂnwithrespecttoa
nondegeneratebilinearform(⋅,⋅),anyBorelsubalgebraisthestabilizerofanisotropic
flag,thatis,achainV0 ⊂V1 ⊂⋅⋅⋅⊂VmofsubspacesofVsuchthateachViisisotropic
with respect to (⋅,⋅),dimVi = i, and m = ⌊n/2⌋, the greatest integer to n/2 [137]; in
thiscase,weidentifyXwiththesetofallsuchisotropicflags.Itisoftenconvenientto
⊥
extendanyisotropicflag{Vi}canonicallytoacompleteflagbysettingVn−i =Vi ,the
orthogonalsubspacetoViwithrespecttotheambientform(⋅,⋅),foralli≤ n2.Intypes
BandCtheWeylgroupWisthehyperoctahedralgroupHnactingonℂnbypermuting
′
andchangingthesignsofthecoordinates;intypeD ,W isthesubgroupH ofindex
n n
twoinH consistingofallelementspermutingthecoordinatesandchangingevenly
n
manyoftheirsigns.
1.3 Finite-dimensionalandhighestweightrepresentations
FixingaBorelsubalgebrabofgandaCartansubalgebrahcontainedinb,werecall
thatanyfinite-dimensionalrepresentationofgisadirectsumoffinite-dimensional
irreduciblerepresentations[66,Thm.6.3]andthatsuchrepresentationsareparame-
terizedbytheirhighestweightsλ[66,21.1,21.2].Inmoredetail,anyfinite-dimensional
irreduciblemoduleVisthedirectsumofitsweightspacesV ,whereV consistsofall
μ μ
∗
v∈Vwithhv=μ(h)vforallh∈h,andμrunsthroughthespaceh oflinearfunctions
∗
onh.Relativetothestandardinnerproductonh (transferredfromtheKillingform
ong),theweightsμwithVμ ≠0areallintegral,thatis,theyalltakeintegervalueson