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ptmb7tChapter8R12 Compact8RGreoups5_3{
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ptmr7tMost*binnitegroups,Ginpractice,comedressedinanaturalKj
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ptmri7ttopology,withre-
_specttowhichthegroupoperationsarecontinuous.,Allthefamiliargroups|_in2particular,eallmatrixgroups|arelocallycompact;andthismarksthe_naturalboundaryofrepresentationtheoryI.3{
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ptmri7ttopoloIgical7gruoupg cmmi12Gisatopologicalspacewithagroupstructuredened on it,suchthatthegroupoperations NKXQ cmr12(x;yn9)UR!",
cmsy107!xy;
x7!xK cmsy8 |{Y cmr81of multiplicationandin&versionarebothcontinuous.bExamples:1html: html:1._Therealnumbers$
msbm10Rformatopologicalgroupunderaddition,withtheusual_topology denedbythemetric e6d(x;yn9)UR=jx yj:html: html:qq2._Thenon-zerorealsR2
y=\Rnf0gformatopologicalgroupundermultipli-_cation, underthesamemetric.html: html:Q3._TheY strictly-positi3verealsR2+չ=]fx2R:x>0gY formaclosedsubgroup_of R2x,andsoconstituteatopologicalgroupintheiro3wnright.TT_Remarks:html: html:鍍$(a)79qNotethatinthetheoryoftopologicalgroups,¡weareonlyconcerned79qwithclosedsubgroups.9Whenwespeakofasubgroupofatopological79qgroup,ʈitnisunderstoodthatwemeanaclosedsubgroup,unlessthe79qcontrary isexplicitlystated. 1{1 * y html: html:dt *3{
ptmro7t1{2Y i html: html: #[(b)79qNoticeهthatifasubgroupHBURGisopenthenitisalsoclosed.}Forthe 79qcosetsdCgn9HQareallopen;dandsoH V,}Tasthecomplementoftheunionof79qall othercosets,isclosed. 79qSo forexample,thesubgroupR2+
qURR2
xisbothopenandclosed. RecallthataspaceXeissaidtobecompactifitishausdorande3veryopencovering2 ehXFչ=URu
cmex10[22 cmmi8IUithas anitesubcovering: =XFչ=URUiqAa cmr61
:)[UN[Ui ; cmmi6rJ: (ThemspaceXishausdorifgi3venany2pointsx;y2XthereexistopensetsU;VURXsuch that xUR2U;UPyË2VU\V=UR;:AllK@thespaceswemeetwillbehausdor;andwewillusetheterm`space'or`topological space'henceforthtomeanhausdorspace.)In=fIactallthegroupsandotherspaceswemeetwillbesubspacesofeuclideanspaceTE 2n\g.Insuchacaseitisusuallyeasytodeterminecompactness,sinceasubspace XFURE 2n \giscompactifandonlyifzYhtml: html:1._Xis closed;andChtml: html:2._Xis bounded Examples:html: html:1._The orthoIgonalgruoup t6-'N cmbx12O(n)UR=fT2Mat(n;R):TƟ0oT=I g:_HereMat ǹ(n;R)denotesthespaceofalln nrealmatrices;^andTƟ20denotes _the transposeofTƻ: 5T0ڍij
6=URTjviJ:_W
BecanidentifyMatn(n;R)withtheEuclideanspaceE 2n-:2
,}byregardingthe_n22entries tij
JasthecoorVdinatesofTƻ. _W&iththisunderstanding,O(n)isaclosedsubspaceofE 2n-:2
,sinceitisthe_setof`points'satisfyingthesimultaneouspolynomialequationsmakingup_the matrixidentityTƟ20oT=URI .PItisboundedbecauseeachentry RjtijJjUR1: y html: html:dt 1{3Y i <_In fIact,foreachi,Ǎ ~t2ڍ1i
O+t2ڍ2i+UN+t2ڍni|=UR(TƟ0oTƹ)ii =1:<_Thus theorthoIgonalgruoupOS(n)iscompact. fhtml: html:df2._The specialorthoIgonalgruoup|SO (n)UR=fT2O(n):detT=1g_is aclosedsubgroupofthecompactgroupO(n),andsoisitselfcompact.df_Note that }T2URO(n)= )detT=1;1_sinceZ&TƟ0oT=URIFչ= )detTƟ0odetT=1= )(detQTƹ)2V=1;_sincedetMoTƟ20Q=URdetTƻ.ThusO(n)splitsinto2parts:*SOY(n)wheredetT=_1;andqasecondpartwheredet+T=UR 1.IfdetT=UR 1thenitiseasytosee_that thissecondpartisjustthecosetTSOES(n)ofSO(n)inO(n).df_W
BeshallndthatthegroupsSOKF(n)playamoreimportantpartinrepresen-_tation theorythanthefullorthogonalgroupsO(n).fhtml: html:df3._The unitarygruoupǍs
U(n)UR=fT2Mat(n;C):TƟaT=I g:_HereMatM(n;C)ûdenotesthespaceofnznûcomplexmatrices;andTƟ2 _denotes theconjugatetransposeofTƻ: 5Tڍij
6=UR\- z - ӍTjvi7: J_W
BeKcanidentifyMat(n;C)withtheEuclideanspaceE 22n-:2,c^byregarding_the realandimaginarypartsofthen22entriestij
JasthecoorVdinatesofTƻ.df_W&ith[Hthisunderstanding,rU(n)isaclosedsubspaceofE 22n-:2.(Itisbounded_because eachentryhasabsolutev3alue RjtijJjUR1:_In fIact,foreachi,jjt1ij2j+jt2ij2+UN+jtni*j2V=UR(TƟaTƹ)ii =1:_Thus theunitarygruoupU(n)iscompact. y html: html:dt 1{4Y i <_When nUR=1, U(1)UR=fx2C:jxj=1g::/<_Thus U(1)UR=S ן1P-=rT1V=R=Z:<_Notethatthisgroup(whichwecandenoteequallywellbyU(1)orT21)is<_abelian (orcommutati3ve). Phtml: html:Ѝ4._The specialunitarygruoupm|SU (n)UR=fT2U(n):detT=1g_is aclosedsubgroupofthecompactgroupU(n),andsoisitselfcompact.>_Note that 0T2URU(n)= )jdetQTj=1:_sinceX^TƟaT=URIFչ= )detTƟaȹdetT=1= )jdetQTj2V=1;_since detQTƟ2 =UR: z ց detQT+ӻ.>_The mapq6U@(1)SU(n)UR!U(n):(;Tƹ)7!T_is asurjecti3vehomomorphism.PItisnotbijective,sincem IF2URSU5S(n)UR( )n=1:_Thus thehomomorphismhaskIernel Cn=URh!n9i;_where !Ë=URe22I{=nSb.PItfollo3wsthat OU(n)UR=(U(1)SU(n))X=CnP:p}_W
BeshallndthatthegroupsSU(n)playamoreimportantpartinrepre- _sentation theorythanthefullunitarygroupsU(n).html: html:>5._The symplecticgruouppySpy(n)UR=fT2Mat(n;H):TƟaT=I g:_Herev
MatD(n;H)v
denotesthespaceofnnv
matriceswithquaternionen-_tries; andTƟ2 aʻdenotestheconjugatetransposeofTƻ: 5Tڍij
6=xbURTjvi7: y html: html:dt 1{5Y i <_(Recall thattheconjugateofthequaternion* qË=URt+xi+yn9jW{+z k<_is thequaternion dq ý=URt xi yn9jW{ z kg:<_Note thatconjugacyisananti-automorphism,ie ;d z bKq1q2 |l=URd z Kq2
Kd z Kq1@:<_It follo3wsfromthisthat (AB )V=URB[
A <_for*any2matricesA;Bwhoseproductisdened.8Thisinturnjustiesour<_implicit assertionthatS p(n)isagroup:=ݸS;T2URSpUR(n)UR= )(S Tƹ)(ST)UR=TaS ןrS T=TaT=IFչ= )S T2SpUR(n):V<_NoteEbtoothatwhilemultiplicationofquaternionsisnotingeneralcommu-<_tati3ve, ܹ =q o#qË=URqJعn9q gq=t2j+x2+yn92+z 29=URjqn9j2; <_dening thenorm,orabsolutev3alue,jqn9jofaquaternionq.)*<_W
BecanidentifyMat߹(n;H)withtheEuclideanspaceE 24n-:2,Nbyregarding<_the coecientsof1;i;j;kgin then22entriestij
JasthecoorVdinatesofTƻ.<_W&ith2thisunderstanding,
>Sp
>(n)isaclosedsubspaceofE 24n-:2.Itisbounded<_because eachentryhasabsolutev3alue* RjtijJjUR1:<_In fIact,foreachi, jt1ij2j+jt2ij2+UN+jtni*j2V=UR(TƟaTƹ)ii =1:<_Thus thesymplecticgruoupSp (n)iscompact.*<_When nUR=1,<_SpK_(1)UR=fqË2H:jqn9j=1g=ftW+xi+yj8*+z ko:URt2K[+x2+yn92+z 29=UR1g:<_Thus ѶSp (1)PURn:=S ן3r: <_W
Be leaveittothereadertosho3wthatthereisinfIactanisomorphism dzSp ֳ(1)UR=SU5S(2): &Ѡ y html: html:dt 1{6Y i 0AlthoughcompactnessisbyfIarthemostimportanttopologicalpropertythat agroupcanpossess,Jasecondtopologicalpropertyplaysasubsidiarybutstill important r ^ole|connectivity.0Recall: thatthespaceX+issaidtobedisconnectedifitcanbepartitionedinto 2 non-emptyopensets:~? XFչ=URU[V ;
U\V¹=UR;: W
Be saythatXisconnectedifitisnotdisconnected. 0Therecisacloselyrelatedconceptwhichismoreintuiti3velyappealing,{butis usuallyүmorediculttowIorkwith.5W
BesaythatX2ispathwise-connectedifgi3ven any 2pointsx;yË2URXwecanndapathn9joiningxtoyn9,ieacontinuousmap~? Ë:UR[0;1]!X with n9(0)UR=x;UP(1)=y: It iseasytoseethat }vpathwise-connected X= )URconnected6: For?ifX1=ˮU[ VZisadisconnectionofX ,Oandwechoosepointsu2U;v92Vp, then therecannotbeapathn9joiningutovn9.PIftherewere,then DIFչ=URn9 1ʵU[n9 1V wIouldbeadisconnectionoftheinterv3al[0;1].Butitfollowsfromthebasicprop- ertiesofrealnumbersthattheinterv3alisconnected.
(SupposeIڹ=WUq[0Vp.W
Be may supposethat0UR2U@.PLet fl=URinfFxUR2V2[B: Then wegetacontradictionwhetherweassumethatxUR2Vpor x=2Vp.)0Actually8Y,forHallthegroupswedealwiththe2conceptsofconnectedand pathwise-connectednwillcoincide.Thereasonforthisisthatallourgroupswill turnouttobelocallyeuclidean,'ieeachpointhasaneighbourhoodhomeomor- phicZ2totheopenballinsomeeuclideanspaceE 2n\g.Thiswillbecomeapparent much laterwhenweconsidertheLiealgebraofamatrixgroup.0W
Bencertainlywillnotassumethisresult.Wementionitmerelytopointout thatb+youwillnotgofIarwrongifyouthinkofaconnectedspaceasoneinwhich you cantravelfromanypointtoanyother&,without`takingo '.0TheRfollo3wingresultprovidesausefultoolforshowingthatacompactgroup is connected. #Nhtml: html: 1+ y html: html:dt 1{7Y i +ߌ
ptmb7tPrȹoposition 1.1qSupposethecompactgruoupGactstransitivelyonthecompact space X .PLetx0V2URX;andlet~荒 E1HB=URS (x0)=fgË2G:gn9x0V=gx0g be thecorrVespondingstabilisersubgruoup.PThenoXconnected7K&HVconnected:p= )URG connected3: Pruoofz
msam10I`ByafIamiliararȹgument,theactionofGonXsetsupa1-1correspon- dence betweenthecosetsgn9HVofHinGandtheelementsofX .PInfIact,let &UR:G!X be themapunderwhich CgË7!URgn9x0:$ Then ifxUR=gn9x0, 1\|fxgUR=gn9HF:: html: html:Lemma 1.1?2Each cosetgn9HVisconnected.Pruoof ofLemmaBThemapO hUR7!gn9h:HB!gHis acontinuousbijection.But\HIٻiscompact,sinceitisaclosedsubgroupofG(asH=ڵ2 1\|fx0g).No3w&acontinuousbijectionofacompactspaceK"ontoahausdorspaceYis+\necessarilyahomeomorphism.:dForifUKisopen,v3thenC\M=KengUl@isclosedandthereforecompact.gHence(C ܞ)iscompact,'andthereforeclosed;/andso(U@)n(=Ynn(C ܞ)isopeninYp.EThissho3wsthat2 1#iscontinuous,ieisahomeomorphism.Thus HPB[=Qgn9H V;andsoq|HVconnected:p= )URgn9HVconnected4: CNo3w suppose(contrarytowhatwehavetoprove)thatGisdisconnected,say ?~GUR=U[V ;
U\V¹=UR;:This splitinGwillspliteachcoset:ygn9HB=UR(gH\U@)[(gn9H\Vp):