Representations of SL2(R)

51 Pages·2014·0.44 MB·English

Lastrevised9:24p.m. November1,2020
Representationsof SL (R)
2
BillCasselman
UniversityofBritishColumbia
[email protected]
InthisessayIhopetoexplainwhatisneededaboutrepresentationsofSL2(R)intheelementarypartsofthe
theoryofautomorphicforms. Onthewhole,Ihavetakenthemoststraightforwardapproach,eventhough
thetechniquesusedaredefinitelynotvalidforothergroups.
Thisessayisaboutrepresentations. Inthecurrentversionitsaysalmostnothingaboutwhatisusuallycalled
invariantanalysis,whichistosayharmonicanalysisofdistributionsinvariantunderconjugationinSL2(R).
Itsaysverylittleaboutorbitalintegrals, andequallylittleaboutcharactersofrepresentations. Nordoesit
prove anything deep about the relationship between representations of G and those of its Lie algebra, or
aboutversionsofFouriertransforms. Thesetopicsinvolvetoomuchanalysistobedealtwithinthislargely
algebraicessay. Ishalldealwiththosematterselsewhere,ifnotinasubsequentversionofthisessay.
Fromnowon,unlessspecifiedotherwise:
G=SL (R)
2
K = themaximalcompactsubgroupSO2ofG
A = thesubgroupofdiagonalmatrices
N = subgroupofunipotentuppertriangularmatrices
P = subgroupofuppertriangularmatrices
=AN
gR=Liealgebrasl2(R)
gC=C RgR.
⊗
Mostofthetime,itwon’tmatterwhetherIamreferringtotherealorcomplexLiealgebra,andI’llskipthe
subscript.
Muchofthisessayisunfinished. Theremightwellbesomeannoyingerrors,forwhichIapologizeinadvance.
Constructivecomplaintswillbewelcome. Therearealsoplaceswheremuchhasyettobefilledin,andthese
willbemarkedbyoneoftwofamiliarsigns:
DETOUR
Thefirstshouldbeself explanatory. Thesecondmeansthatatthemomentthemostconvenientreferenceis
elsewhere.
RepresentationsofSL(2,R) 2
Contents
Introduction
I.Bargmann’sclassification
1. RepresentationsofthegroupandofitsLiealgebra ................................... 5
2. TheLiealgebra .................................................................... 9
3. DifferentialoperatorsandtheLiealgebra ............................................ 12
4. Admissibility ...................................................................... 14
5. Classificationofirreduciblerepresentations .......................................... 17
6. Thealgebraicconstruction .......................................................... 19
7. Duals ............................................................................. 20
8. Unitarity .......................................................................... 21
II.Theprincipalseries
9. Vectorbundles..................................................................... 22
10. Theprincipalseries ................................................................ 24
11. Frobeniusreciprocityanditsconsequences .......................................... 27
12. Intertwiningoperatorsandthesmoothprincipalseries ............................... 32
13. Thecomplementaryseries .......................................................... 33
14. Normalization ..................................................................... 33
15. Appendix.Charactersasdistributions............................................... 34
16. Appendix.TheGammafunction .................................................... 35
17. Appendix.CharactersandtheFouriertransform..................................... 38
18. Appendix.Invariantintegralsonquotients .......................................... 40
III.Thediscreteseries
19. Projectivespacesandrepresentations................................................ 42
20. RestrictiontoK ................................................................... 43
21. Inclassicalterms................................................................... 44
22. Relationwithfunctionsonthegroup ................................................ 46
23. Holomorphicautomorphicforms ................................................... 49
24. Square integrability ................................................................ 50
IV.Matrixcoefficientsanddifferentialequations
25. Matrixcoefficients ................................................................. 52
26. Differentialequations .............................................................. 53
27. Thenon EuclideanLaplacian ....................................................... 56
28. Realizationsofadmissiblemodules.................................................. 59
V.Langlands’classification
29. TheWeilgroup .................................................................... 59
30. Langlands’classificationfortori..................................................... 62
31. Langlands’classificationforSL2 .................................................... 64
VI.VermamodulesandrepresentationsofG
32. Vermamodules .................................................................... 65
33. Vermamodulesandintertwiningoperators .......................................... 66
34. TheBruhatfiltration................................................................ 67
35. TheJacquetmodule ................................................................ 68
36. ThePaley Wienertheoremforsphericalfunctions .................................... 68
37. Thecanonicalpairing .............................................................. 68
38. Whittakermodels.................................................................. 69
39. Themodule ....................................................................... 74
40. WhittakermodelsforSL(2) ......................................................... 75
VII.References
RepresentationsofSL(2,R) 3
Introduction
SupposeGforthemomenttobeanarbitrarysemi simplegroupdefinedoverQ. Itwillhavegoodreduction
atallbutafinitenumberofprimesp,whichmeansthatGdefinesasmoothgroupschemeoverZp. Suppose
thatforeachprimepthegroupKpisacompactopensubgroupofG(Qp),andthatforallbutafinitenumber
ofpitisG(Zp). LetΓbethesubgroupofγinG(Q)suchthatγliesinKpforallp. ThenΓisdiscreteinG(R)
andΓ G(R)hasfinitevolume. ThegroupΓiscalledacongruencesubgroupofG(Q).
\
ThegroupG(R)actsontherightonΓ G(R),andthisgivesrisetorepresentationsonseveraldifferentspaces
\
of functions and distributions, among them L2 Γ G(R) . Among the most interesting questions in this
\
businessisthis:Whatirreduciblerepresentations(cid:0)ofG(R)o(cid:1)ccurinthedecompositionofL2(Γ G(R))?There
\
isasimpleconjectureconcerningnecessaryconditionsforthis,analogoustoRamanujan’sconjectureforthe
coefficients of certain modular forms. This might also be considered in some sense a problem in number
theory, since it is known not to be true for some discrete subgroups that are not congruence subgroups,
anda resolutionoftheconjecturewill follow fromconjecturesofLanglands regarding certainL functions
associated to automorphic forms. A commutative ring of certain algebraic correspondences called Hecke
operatorsalsoactsonthisquotientandalsointerestingisthis:WhataretheeigenvaluesofHeckeoperators
onthese?AssociatedtothegroupGisitsade`legroupG(A). ItcontainsthediscretesubgroupG(Q)andthe
quotientG(Q) G(A)alsohasfinitevolume. Thequestionsaboveisaresubsumedinthemorecomprehensive
\
one: Whatirreduciblerepresentationsoftheade`legroupG(A)occurintherightregularrepresentationof
G(A)onL2(G(Q) G(A))?Thesequestionsarerelated,sinceifKf isacompactopensubgroupofthefinite
\
ade`legroupG(Af)thenG(Q) G(A)/Kf isaunionofquotientsoftheformΓ G(R).
\ \
The problems raised by these questions are extremely difficult, even perhaps beyond answering in any
complete fashion. They have motivated much investigation into harmonic analysis of reductive groups
definedoverlocalfields. Representationsofsemi simple(andreductive)groupsoverRhavebeenstudied
since about 1945, and muchhas beenlearned aboutthem, butmuchremainstobe done. The literatureis
vast,anditisdifficulttoknowwheretobegininordertogetanyideaofwhatitisallabout.
For p adic groups, the problemsarising that are the most difficult—andthe most intriguing—are those of
numbertheoryandalgebraicgeometry. Butforrealgroupsthereareinadditionproblemsofanalysisthat
donotoccurforp adicgroups. Thesehavecausedmuchannoyanceandsomeseriousdifficulties.
For one thing, there are a number of somewhat technical issues that complicate things. Perhaps the first
is that one does not usually deal with representations of a real reductive group, but rather with certain
representationsofitsLiealgebra. Thishasbeentrueeversincetheoriginsofthesubject,althoughnotreally
understooduntilafewyearslater. Thisisjustifiedbysomerelativelydeeptheoremsinanalysis,andafteran
initialperiodthisshiftinattentioniseasilyabsorbed. TherepresentationsoftheLiealgebra,whicharecalled
Harish-Chandramodules,arerepresentationssimultaneouslyofgandamaximalcompactsubgroupKofG.
TheseareassumedtobecompatibleinthatthetworepresentationsthusassociatedtotheLiealgebrakarethe
same. Inmakingthisshift,onehastomakeachoiceofmaximalcompactsubgroupK,butthedependence
onK isweak,sinceallchoicesareconjugateinG. Theimportantconditionontheserepresentationsisthat
therestrictiontoKbeadirectsumofirreduciblerepresentations,eachwithfinitemultiplicity. Inviewofthe
originalquestionsposedabove,itisgoodtoknowthattheclassificationofirreducibleunitaryrepresentations
ofGisequivalenttothatofirreducibleunitarizable,Harish Chandramodules. Onedoesnotlosemuchby
lookingatrepresentationsoftheLiealgebra.
The classification of unitarizable Harish Chandra modules has not yet been carried out for all G, and the
classificationthatdoesexistissomewhatmessy. Thebasicideafortheclassificationgoesbacktotheorigins
ofthesubject—firstclassifyallirreducibleHarish Chandramodules,andthendecidewhichareunitarizable.
TheclassificationofallirreducibleHarish Chandramodulesaskedforherehasbeenknownforalongtime.
OneusefulfactisthateveryirreducibleHarish Chandramodulemaybeembeddedinoneinducedfroma
finite dimensionalrepresentationsofaminimalparabolicsubgroup. Thisgivesagreatdealofinformation,
andinparticularoftenallowsonetodetectunitarizability.
RepresentationsofSL(2,R) 4
This technique gives a great deal of information, but some important matters require another approach.
The notable exceptions are those representations of G that occur discretely in L2(G), which make up the
discreteseries. Thesearegenerallydealtwithontheirown,andthenonecanlookalsoatrepresentations
inducedfromdiscreteseriesrepresentationsofparabolicsubgroups. Alongwithsometechnicaladjustments,
thesearethecomponentsofLanglands’classificationofallirreducibleHarish Chandramodules. Oneofthe
adjustmentsis thatone mustsay somethingaboutembeddingsofarbitrary Harish Chandra modulesinto
C∞(G),andsaysomethingabouttheasymptoticbehaviourofcertainfunctionsintheimage. Thisrequires
examiningsolutionsofdifferentialequations.
This essay is mostly just about the group G = SL2(R). Although this is a relatively simple group, it is
instructivetoseethatnearlyallinterestingphenomenamentionedalreadyappear. Ishalltouch,eventually,
onallthemesmentionedabove,butinthepresentversionafewareleftout.
Representationsofcompactgroupshavebeenexaminedforalongtime,butthestudyofunitaryrepresenta
tionsofnon compactsemi simplegroupsbeginwiththeclassic[Bargmann:1947]. Bargmann’sinitialclassi
ficationofirreduciblerepresentationsofSL2(R)alreadymadethestepfromGto(g,K),ifnotrigourously.
EvennowitisworthwhiletoincludeanexpositionofBargmann’spaper,andIshalldoso,becauseitisalmost
theonlycasewhereresultscanbeobtainedwithoutintroducingsophisticatedtools. Butanotherreasonfor
lookingatSL2(R)closelyisthatthesophisticatedtoolsonedoesneedeventuallyareforSL2(R)relatively
simple,anditisvaluabletoseetheminthatsimpleform. Thatismyprimarygoalinthisessay.
InPartIIshallessentiallyfollowBargmann’scalculationstoclassifyallirreducibleHarish Chandramodules
over(sl2,SO(2)). Theunitarizableonesamongthesewillbefound. Thetechniquesusedherewilljustbe
relativelysimplecalculationsintheLiealgebrasl2.
Bargmann’stechniquesbecomeimpossiblydifficultforgroupsotherthanSL2(R). Oneneedsingenerala
waytoclassifyrepresentationsthatdoesnotrequiresuchexplicitcomputation. Thereareseveralwaystodo
this. OneisbymeansofinductionfromparabolicsubgroupsofG,andthisiswhatI’lldiscussinPartII(but
justforSL2(R)). Theserepresentationsarenowcalledprincipalseries. (ThisisatermusedbyBargmann,
butforonlyasubsetofthem.)
It happens, even for arbitrary reductive groups, that every irreducible Harish Chandra module can be
embeddedinoneinducedfromafinite dimensionalrepresentationofaminimalparabolicsubgroup. Thisis
important,butdoesnotanswermanyinterestingquestions. Somegroups,includingSL2(R)possesscertain
irreducibleunitaryrepresentationssaidtolieinthediscreteseries. Theyoccurdiscretelyintherepresentation
ofGonL2(G),andtheyrequirenewmethodstounderstandthem. ThisisdoneforSL2(R)inPartIII.
For SL2(R) the discrete series representations can be realized in a particularly simple way in terms of
holomorphic functions. This is not true for all semi simple groups, and it is necessary sooner or later to
understandhowarbitraryrepresentationscanbeembeddedinthespaceofsmoothfunctionsonthegroup.
Thisisintermsofmatrixcoefficients,explainedlaterinPartIII.Onecorollaryofthisinvestigationwillbe
that any admissible representation of (sl2,SO(2)) can be extended to a smooth representation of G, thus
extendingaresultaboutirreduciblerepresentationsduetoHarish Chandra.
In order to understand the role of representations in the theory of automorphic forms, it is necessary to
understandLanglands’classificationofrepresentations,andthisisexplainedinPartIV.
EventuallyI’llincludeinthisessayanaccountofhowD modulesonthecomplexprojectivelinecanbeused
toexplain many phenomenathat appear otherwisemysterious. Othertopics I’venot yetincludedare the
relationshipbetweentheasymptoticbehaviourofmatrixcoefficientsandembeddingsintoprincipalseries,
andtherelationshipbetweenrepresentationsofSL2(R)andWhittakerfunctions.
The best standard reference for this material seems to be [Knapp:1986] (particularly Chapter II, but also
scattered references throughout), although many have found enlightenment elsewhere. A very different
approachisChapterIof[Vogan:1981]. ManybooksgiveanaccountofBargmann’sresults,butformostof
therestIbelievetheaccountherehasmanynewfeatures.
RepresentationsofSL(2,R) 5
PartI.Bargmann’sclassification
1. RepresentationsofthegroupandofitsLiealgebra
What strikes many on first sight of the theory of representations of G = SL2(R) is that it is very rarely
concernedwithrepresentationsofG! Instead,oneworksalmostexclusivelywithcertainrepresentationsof
itsLiealgebra. Andyet... thetheoryisultimatelyaboutrepresentationsofG. InthisfirstsectionI’lltryto
explainthisparadox,andsummarizetheconsequencesofthesubstitutionofLiealgebraforLiegroup.
I want first to say something first about why one is really interested in representations of G, or at least
somethingclosely relatedtothem. The main applications of representationtheoryofgroupslikeGare to
thetheoryofautomorphicforms. ThesearefunctionsofacertainkindonarithmeticquotientsΓ Goffinite
\
volume,forexamplewhenΓ=SL2(Z). ThegroupGactsonthisquotientontheright,andthecorresponding
representationofGonL2(Γ G)isunitary. WhichunitaryrepresentationsofGoccurasdiscretesummands
\
ofthisrepresentation?AsIhavealreadymentioned,thereisaconjecturalifpartialanswertothisquestion,
atleastforcongruencegroups,whichisananaloguefortherealprimeoftheRamanujanconjectureabout
the Fourier coefficients of holomorphic forms. It was Deligne’s proof of the Weil conjectures that gave at
the same time a proof of the Ramanujan conjecture, and from this relationship the real analogue acquires
immediatelyacertaincachet.
Asforwhyonewindsuplookingatrepresentationsofg,thisshouldnotbeunexpected. Afterall,eventhe
classificationoffinite dimensionalrepresentationsofGcomesdowntotheclassificationofrepresentations
ofg,whicharefareasiertoworkwith. Forexample,onemightcontrasttheactionoftheunipotentupper
triangularmatricesofGonthesymmetricpowerSn(C2)withthatofitsLiealgebra.
The representations one winds up looking at are infinite dimensional. What might be surprising is that,
unlikewhathappensforfinite dimensionalrepresentations,theserepresentationsofgarenotusuallyatthe
sametimerepresentationsofG. ItisthisthatIwanttoexplain.
I’llbeginwithanexamplethatshouldatleastmotivatethesomewhattechnicalaspectsoftheshiftfromG
tog. TheprojectivespaceP=P1(R)isbydefinitionthespaceofalllinesinR2. ThegroupGactsonR2by
lineartransformations, andittakeslinestolines,soitactsonPaswell. Thereisastandardwaytoassign
coordinates on P by thinking of this space as R togetherwith a point at . Formally, every line but one
∞
passes through a unique point (x,1) in R2, and this is assigned coordinate x. The exceptional line is the
x axis,andisassignedcoordinate .
∞
x= =0 = 2
x
−1 x
x=
∞
IntermsofthiscoordinatesystemSL2(R)actsbyfractionallineartransformations:
a b
: x (ax+b)/(cx+d),
(cid:20)c d(cid:21) 7−→
aslongasweinterpretx/0as . Thisisbecause
∞
az+b
a b z az+b cz+d
= =(cz+d)  .
(cid:20)c d(cid:21)(cid:20)1(cid:21) (cid:20)cz+d(cid:21)
1
 
RepresentationsofSL(2,R) 6
Theisotropysubgroupfixing isP,andPmaybeidentifiedwiththequotientG/P. SinceK P = I
∞ ∩ {± }
andK actstransitivelyonP,asaK spacePmaybeidentifiedwithK/ 1 .
{± }
TheactionofGonPgivesrisetoarepresentationofGonfunctionsonP: Lgf(x)=f(g−1x). (Theinverse
isnecessaryhereinordertohaveLg1g2 = Lg1Lg2,asyoucancheck.) Butthereareinfactseveralspacesof
functionsavailable—forexamplerealanalyticfunctions,smoothfunctions,continuousfunctions,functions
that are locally square integrable, or the continuousduals of any of these infinite dimensional topological
vectorspaces. Whichofthesedowereallywanttolookat? Wecanmakelifealittlesimplerbyrestricting
ourselvestosmooth representations(onesthat are stablewithrespect toderivation byelementsoftheLie
algebrag),inwhichcasethespacesofcontinuousandlocallysquare integrablefunctions,forexample,are
ruledout. Butevenwiththisrestrictionthereareseveralspacesathand.
Hereisthemainpoint:AlloftheserepresentationsofGshouldbeconsideredmoreorlessthesame,atleast
formostpurposes. TherepresentationofK onPisasmultiplicationonK/ 1 . Amongthefunctionson
{± }
thisspacearethosewhichtransformbythecharactersε2nofK where
c s
ε: − c+is.
(cid:20)s c(cid:21)7−→
Thewaytomaketheessentialequivalenceofallthesespacespreciseistoreplaceallofthembythesubspace
offunctionswhichwhenrestrictedtoK areafinitesumofeigenfunctions. This subspace is the same for
allofthesedifferentfunctionspaces,andinsomesenseshouldbeconsideredthe‘essential’representation.
However, it has what seems to be at first one disability—it is nota representation of G, since g takes
eigenvectorsforK toeigenvectorsfortheconjugategKg−1, whichisnotgenerallythesame. Tomakeup
forthis,itisstablewithrespecttotheLiealgebragR =sl2(R),whichactsbydifferentiation.
TherepresentationofSL2(R)onspacesoffunctionsonPisamodelforotherrepresentationsandevenother
reductivegroupsG. Bydefinition,acontinuousrepresentation(π,V)ofaLiegroupGonatopologicalvector
spaceV isahomomorphismπfromGtoAut(V)suchthattheassociatedmapG V V iscontinuous.
× →
TheTVSVisalwaysassumedinthisessaytobelocallyconvex,Hausdorff, andquasi complete. Thislast,
somewhattechnical,conditionguaranteesthatifF isacontinuousfunctionofcompactsupportonGwith
valuesinV thentheintegral
F(g)dg
Z
G
iswelldefined. Onebasicfact aboutthisintegralisthatit containedintheconvexhulloftheimageoff,
scaledbythemeasureofitssupport. Iff isinC∞(G),onecanhencedefinetheoperator valuedintegral
c
π(f): v f(g)π(g)vdg
7−→Z
G
(i.e.takingF(g)=f(g)π(g)v).
If(π,V)isacontinuousrepresentationofamaximalcompactsubgroupK,letV bethesubspaceofvectors
(K)
that are contained in a K stable subspace of finite dimension. These are called the K-finite vectorsof the
representation. IfGisSL2(R),itisknownthatanyfinite dimensionalspaceonwhichK actscontinuously
will be a direct sum of one dimensional subspaces on each of which K acts by a character, so V is
(K)
particularlysimple. SometechnicalproblemsarisebecauseK mightnotbeconnected,butasidefromthat
theirreduciblerepresentationsofK arewellunderstood(andmoreorlesscompletelyclassified).
(1)If(π,V)isanycontinuousrepresentationofK,thesubspaceV isdenseinV.
(K)
Thisisabasicfactaboutrepresentationsofacompactgroup,andaconsequenceoftheStone Weierstrass
Theorem.
RepresentationsofSL(2,R) 7
AvectorvinthespaceV ofacontinuousrepresentationofGiscalleddifferentiableifthelimit
π(exp(tX))v v
π(X)v = lim −
t→0 t
existsinV foreveryXing. TherepresentationiscalledsmoothifallvectorsinV aredifferentiable,inwhich
caseπ(X)isanoperatoronV foreveryX inU(g).
(2)Smoothvectorsaredenseinanycontinuousrepresentation.
IffliesinC∞(G)andvinV,thenπ(f)vissmooth. ThespaceofsuchvectorsiscalledtheGa˚rdingsubspace.
c
In some circumstances, it is the same as the subspace of smooth vectors. Applying a Dirac sequence, any
vectormaybeapproximatedbysuchvectors.
AcontinuousrepresentationofGonacomplexvectorspaceV iscalledadmissibleifV isadirectsumof
(K)
charactersofK,eachoccurringwithfinitemultiplicity.
(3)If(π,V)isasmoothrepresentationofg,thesubspaceV isstableunderg.
(K)
Ifπissmooth,thenthereisacanonicalmapfromg V toV: X v π(X)v. IfU isK stablethensois
⊗ ⊗ 7→
theimageofg U,whichcontainsallπ(X)uforuinU.
⊗
AcontinuousrepresentationofGonacomplexvectorspaceV iscalledadmissibleifV isadirectsumof
(K)
charactersofK,eachoccurringwithfinitemultiplicity.
(4)If(π,V)isanadmissiblerepresentationofG,thevectorsinV aresmooth.
(K)
Suppose (σ,U) to be an irreducible representation of K, ξ the corresponding projection operator in
C∞(K). For example, if K = SO2 and σ is a character, then ξ amounts to integration against σ−1
over K. The σ component Vσ of V is the subspace of vectors fixed by π(ξ). If (fn) is chosen to be
a Dirac sequence of smooth functionson G (i.e. non negative, having limit the Dirac distribution δ1),
then π(fn)v v for all v in V. The operators π(ξ)π(f)π(ξ) = π(ξ f ξ) are therefore dense in the
→ ∗ ∗
finite dimensional space End(Vσ), hence make up the whole of it. But ξ f ξ is also smooth and of
∗ ∗
compactsupport. AnyvectorvinVσ maythereforebeexpressedasπ(f)vforsomef inCc∞(G).
Asforthesecondassertion,ifvliesinVσ thenπ(g)vliesinthedirectsumofspacesVτ asτ rangesover
theirreduciblecomponentsofthefinite dimensionalK stablespaceg Vσ.
⊗
A unitary representation of G is one on a Hilbert space whose norm is G invariant. One of the principal
goals of representation theory is toclassify representations that occuras discrete summands of arithmetic
quotients. Thisisanextremelydifficulttask. Buttheserepresentationsareunitary,andclassifyingunitary
representationsofGisafirststeptowardscarryingitout.
(5)AnyirreducibleunitaryrepresentationofGisadmissible.
This is the most difficult of these claims, requiring serious analysis. It is usually skipped over in
expositions of representation theory, probably because it is difficult and also because it is not often
needed in practice. One accessible reference is 4.5 (on ‘large’ compact subgroups) of [Warner:1970].
§
Otherreferencesare[Atiyah:1988]andmyownnotes[Casselman:2014]onunitaryrepresentations.
ForG = SL2(R),everyirreducibleunitaryrepresentationofGrestrictstoadirectsumofcharactersof
SO2,eachoccurringatmostonce. Weshallseeaproofofthislateron.
Infact,admissiblerepresentationsareubiquitous. Atanyrate,weobtainfromanadmissiblerepresentation
ofGarepresentationofgandK satisfyingthefollowingconditions:
(a) asarepresentationofK itisadirectsumofsmoothirreduciblerepresentations(offinitedimension),
eachwithfinitemultiplicity;
(b) therepresentationofkassociatedtothatassubalgebraofgandthatasLiealgebraofK arethesame;
RepresentationsofSL(2,R) 8
(c) forkinK,X ing
π(k)π(X)π−1(k)=π(Ad(k)X).
Anyrepresentationofthepair(g,K)satisfyingtheseiscalledadmissible. MostofthetimeIshallbemost
interestedinthosewhicharefinitelygeneratedasmodulesovertheenvelopingalgebraU(g). Theseareof
finitelength,inthesensethattheypossessafinitefiltrationbyirreduciblerepresentations.
Letmemakesomemoreremarksonthisdefinition. Itisplainlynaturalthatwerequirethatgact. Butwhy
K? Thereareseveralreasons. WhatwearereallyinterestedinarerepresentationsofG,ormoreprecisely
•
representationsofgthatcomefromrepresentationsofG. Butarepresentationofgcanonlydeterminethe
representationoftheconnectedcomponentofG. ThegroupK meetsallcomponentsofG,sothatrequiring
K toactfixesthisproblem. ForgroupslikeSL2(R),whichareconnected,thisproblemdoesnotoccur,but
forthegroupPGL2(R),withtwocomponents,itdoes. OnepointofhavingKactistodistinguishSL2(R)
•
fromothergroupswiththesameLiealgebra. Forexample,PGL2(R)hasthesameLiealgebraasSL2(R),
butthestandardrepresentationofitsLiealgebraonR2doesnotcomefromoneofPGL2(R). Requiringthat
K act picks out a unique group in the isogeny class, since K contains the centre of G. Any continuous
•
representationofK,suchastherestrictiontoK ofacontinuousrepresentationofG,decomposesinsome
senseintoadirectsumofirreduciblefinite dimensionalrepresentations. ThisisnottrueoftheLiealgebra
k, which is after all the same as the Lie algebra of R or R×. So this requirement picks out from several
possibilitiestheclassofrepresentationswewant.
Admissible representations of (g,K) are not generally semi simple. For example, the space V of smooth
functionsonPcontainstheconstantfunctions,buttheyarenotaG stablesummandofV.
If(π,V)isasmoothrepresentationofGassignedaG invariantHermitianinnerproduct,thecorresponding
equationsofinvariancefortheLiealgebraarethat
Xu•v = u•Xv
−
forX ingR,or
Xu•v = u•Xv
−
for X in the complex Lie algebra gC. An admissible (gC,K) module is said to be unitary if there exists a
positive definite Hermitian metricsatisfying thiscondition. In some sense, unitary representationsare by
far themostimportant. At present, themajor justification ofrepresentationtheoryisin itsapplications to
automorphicforms,andthemostinterestingonesencounteredareunitary.
Hereissomemoreevidencethatthedefinitionofanadmissiblerepresentationof(gC,K)isreasonable:
(a) admissiblerepresentationsof(g,K)thatarefinite dimensionalareinfactrepresentationsofG;
(b) ifthecontinuousrepresentation(π,V)ofGisadmissible,themaptakingeach(g,K) stablesubspace
U V(K)toitsclosureinV isabijectionbetween(gC,K) stablesubspacesofV(K)andclosedG stable
⊆
subspacesofV;
(c) everyadmissiblerepresentationof(g,K)isV forsomecontinuousrepresentation(π,V)ofG;
(K)
(d) everyunitaryadmissiblerepresentationof(gC,K)isV(K)forsomeunitaryrepresentationofG;
(e) thereisanexactfunctorfromadmissiblerepresentationsof(g,K)tosmoothrepresentationsofG.
Item(a)isclassical.
For(b),refertoThe´ore`me3.17of[Borel:1972].
Iamnotsurewhatagoodreferencefor(c)is. Forirreduciblerepresentations,itisaconsequenceofatheorem
of Harish Chandra that every irreducible admissible (g,L) module is a subquotient of a principal series
representation. Thisalsofollowsfromthemainresultof[Beilinson Bernstein:1982].
Thefirstproofof(d)wasprobablybyHarish Chandra,butIamnotsureexactlywhere.
For(e),referto[Casselman:1989]or[Bernstein Kro¨tz:2010].
RepresentationsofSL(2,R) 9
Mostoftherestofthisessaywillbeconcernedonlywithadmissiblerepresentations(π,V)of(g,K),although
itwillbegoodtokeepinmindthatthemotivationforstudyingthemisthattheyarisefromrepresentations
ofG.
Curiously,althoughnowadaysrepresentationsofSL2(R)areseenmostlyinthetheoryofautomorphicforms,
thesubjectwasintroducedbythephysicistValentineBargmann,inprettymuchtheformweseehere. Ido
notknowwhatphysicsproblemledtohisinvestigation.
2. TheLiealgebra
The Lie algebra of G is the vector space sl2(R) of all matrices in M2(R) with trace 0. Its Lie bracket is
[X,Y] = XY YX. There are twouseful bases of the complexifiedalgebra gC, mirroring a duality that
−
existsthroughoutrepresentationtheory.
THE SPLIT BASIS. Thesimplestbasisisthis:
1 0
h =
(cid:20)0 1(cid:21)
−
0 1
ν =
+ (cid:20)0 0(cid:21)
0 0
ν =
− (cid:20)1 0(cid:21)
withdefiningrelations
[h,ν ] = 2ν
± ±
±
[ν ,ν ]= h.
+ −
ThegroupGpossessesaninvolutoryautomorphismθ,takinggtotg−1. TheinvolutioninducedontheLie
algebratakesXto tX. Thisfixeselementsofkandactsasmultiplicationby 1onthesymmetricmatrices.
− −
Thus ν+θ = −ν−. It is sometimesmore convenient to have the basis stable with respect to θ and, for this
reason,sometimesintheliterature ν−isusedaspartofastandardbasisinsteadofν−.
−
THE COMPACT BASIS. Thisisabasisthat’susefulwhenwewanttodocalculationsinvolvingK. Thefirst
elementofthenewbasisis
0 1
κ= − ,
(cid:20)1 0(cid:21)
whichspanstherealLiealgebrakofK. TherestofthenewbasisistobemadeupofeigenvectorsofK. The
groupK iscompact. Itscharactersareallcomplex,notreal,soinordertodecomposetheadjointactionofk
ongwesowemustextendtherealLiealgebratothecomplexone. ThekR stablespacecomplementarytokR
ingRisthatofsymmetricmatrices
a b
.
(cid:20)b a(cid:21)
−
anditscomplexificationdecomposesintothesumoftwoconjugateeigenspacesforκspannedby
1 1 i
x+ = 2·(cid:20) i −1(cid:21)
− −
1 1 i
x =
− 2·(cid:20) i 1(cid:21)
−
withrelations
[κ,x ] = 2ix
± ±
±
[x ,x ]= iκ.
+ −
−
RepresentationsofSL(2,R) 10
By definition, the space of an admissible representation of (g,K) is spanned by eigenvectors for K. The
followingisanelementaryconsequenceoftheformulasabove:
2.1. Lemma.SupposeV tobeanyrepresentationof(g,K),andsupposevtobeaneigenvectorforKwith
eigencharacterεn.Thenπ(κ)v =nivandπ(κ)π(x±)v =(n 2)iπ(x±)v.
±
Proof. Since
d
π(κ)v = eintv =niv
dt(cid:12)
(cid:12)t=0
(cid:12)
and (cid:12)
π(κ)π(x )v =π(x )π(κ)v+π([κ,x ])v.
± ± ±
THE CAYLEY TRANSFORM. ThegroupGpossessestwomaximaltori,thesubgroupAofdiagonalmatrices
andthecompactsubgroupK. TheyarecertainlynotconjugateinG,buttheybecomeconjugateinG(C)=
SL2(C). Thiscanbeseengeometrically. ThegroupG(C)actsbyfractionallineartransformationsonP1(C).
Thediagonalmatricesfixtheorigin,andthecomplexmatrices
c s
−
(cid:20)s c(cid:21)
fixi. TheCayleytransform
C: z (z i)/(z+i)
7−→ −
takesito0,andtherefore
1 i
CK(C)C−1 =A(C) C = − .
(cid:18) (cid:20)1 i(cid:21)(cid:19)
IntheLiealgebra
Ad κ= hi, Ad x =iν .
C C ± ∓
−
Inclassifying algebraictoriinarbitrarysemi simplegroupsdefinedoverR, copiesoftheCayleymatrixC
arecrucial.
THE CASIMIR. letZ(g)bethecentreofU(g). Suppose(π,V)tobeanirreducibleadmissiblerepresentation
of(g,K). ThenanyelementX ofthecentreofU(g)willactasascalarmultiplication,saybyζ(X),onallof
V. ThehomomorphismfromZ(g)takingX toζ(X)isanimportantcharacteristicofπ,called(forreasons
thatescapeme)itsinfinitesimalcharacter.
WhatisthecentreofU(g)?
2.2. Proposition.ThecentreofU(g)isthepolynomialalgebrageneratedbytheCasimiroperator
h2 ν ν ν ν
+ − − +
Ω= + + .
4 2 2
Ithasalternateexpressions
h2 h
Ω= +ν ν
+ −
4 − 2
h2 h
= + +ν ν
− +
4 2
h2 h
= +ν2 ν κ
4 − 2 +− +
κ2 x x x x
− + + −
= + +
− 4 2 2
κ2 κi
= +x x
− +
− 4 − 2
κ2 κi
= + +x x .
+ −
− 4 2