Page 1
EstimatingtheSpectralMeasureofa
Multiv ariate StableDistribution viaSpherical
HarmonicAnalysis
MarcusPiv ato
?
andLuisSeco
y
Octob er ??????
Abstract
A newmethod is develop ed for estimatingthesp ectralmeasure of
amultivariatestableprobabilitymeasure?byrepresentingthemeasure
asasumof sphericalharmonics?
In troduction?
Stable probabilitydistributionsarethenaturalgeneralizationsof
thenormal distribution?andsharewithittwokeyprop erties?
?Stability? Thenormaldistribution isstableinthesensethat?
ifXandYareindependentrandomvariables?withidentical
normal distributions?thenX?Yis alsonormal? and
?
?
???
?X?Y?
?
?
distr
X
?
?
distr
Y
Inasimilarfashion? ifX andYareindependent?identically
distributed?i?i?d?stablerandomvariables?thenX?Yisalso
stable?anditsdistributionisthesameasXandYwhenrenor?
malizedby?
????
?The stabilityexponent?rangesfrom?to
??When????we havethefamiliarnormaldistribution?
?
DepartmentofMathematics?University ofToron to?email?
pivato?math?toronto?edu
y
RiskLabandDepartment ofMathematics?University ofToron to?email?
seco?math?toront o? edu
?
Page 2
?RenormalizationLimit?TheCentralLimit Theorem says
thatthenormaldistribution is thenaturallimitingdistribution
ofasuitablyrenormalizedin?nitesumofindep endentrandom
variableswith?nitev ariance?IfX
?
?X
?
???? isasequenceofsuch
variables?thentherandomvariables
?
N
???
N
X
n??
X
n
converge? indensity? toanormaldistribution?Similarly?iffY
k
g
?
k??
areindependentrandomvariableswhosedistributionsdecay ac?
cordingtoapow erlawwithexponent?????thentherandom
variables
?
N
???
N
X
n??
Y
n
conv erge? indistribution?toan??stabledistribution
Thus?stabledistributions modelrandomaggregations ofmany
small?independentperturbations?F orexample? stabledistributions
mo delthemotionsofMarkovian stochasticprocesseswhose incre?
mentsexhibitpowerlaws?Stabledistributionsarisewithsurprising
frequencyincertainsystems? especiallythose inv olvingmanyinde?
pendentinteractingunitswithsensitive dependenciesb etw eenthem?
Theyhaveappeared inmathematical ?nance????????????? ?????????????????
?????????????????????????????Internettra?c statistics???? ???????? ???????
and ariseinmathematical mo delsofrandomscalar ?elds??????????
radar??? ??andsignalprocessing?????????????telecommunications????
and eventhepowerdistributionofoceanwaves????
Forfurtherexamples?see??????????or?????Thede?nitiverefer?
enceonuniv ariatestabledistributionsis?????thede?nitivereference
onmultivariatedistributionsandstableprocessesis?????Otherre?
cent references are?????? ?????? andaforthcomingbookbyNolan????
sligh tlyolderreferences are????and ????
Although one?dimensionalstabledistributions arewell?understood?
there aremanyopenquestionsinthemultivariateregime?Thesim?
plicityofthemultivariateGaussianuniversedoes notextendto non?
Gaussianmultivariatestable distributions?AnN?dimensionalGaus?
sian distribution iscompletelydeterminedbyits N?Ncovariance
?
Page 3
matrix?whichtransformsnicelyunderlinearchangesofcoordinates?
Inparticular?byorthogonallydiagonalizingthematrix?wecan?nd
anorthonormalbasisforR
N
?relativ etowhichthem ultivariatenor?
malv ariableis rev ealedasa sumof independentunivariatenormal
variables?thisis PrincipleComponentA nalysis?
Fora generalmultivariatestabledistribution?however? thesitu?
ationismuchmorecomplex?Sincethemarginals donot have?nite
variance? it does notmakesensetode?nea?covariancematrix?in
theusualway?noneoftheintegralswouldconverge?V ariousmod?
i?ednotionsof?covariance?havebeenproposed?see?forexample?
??????butthesedonottransformin anysimplewayunderc hanges of
coordinates? In particular?there is nothinganalogous toa ?principle
componen ts analysis??Instead?the correlationstructure ofastable
distributiononR
D
isdeterminedbyanarbitrary measure??? onthe
sphereS
D??
?
?
?x?R
D
?k?xk??
?
?calledthe sp
?
ectralmeasure?
Theorem??Let????????andlet?bean??stableprobability
measureonR
D
?with center???R
D
? Then?hasFourierTransform?
??
?
???exp???
?
???
where? ?the?logF ouriertransform?? isgiven?
??
?
???
D
???
?
?
E
i?
Z
S
D??
?
???
D
??s
E
d??s????
where?
???
??????j?j
?
?B
?
??
h?i
i? ???
with?
h?i
??
?
sign????j?j
?
if ????
??log j?jif???
andB
?
??
?
tan
?
??
?
?
if????
?
?
?
if???
and where?is some nonnegative Borel measure onS
D??
?
Proof?See?????x????p????or?????
??Theorem??
?
Page 4
?iscalledthespectralmeasureofthedistribution
?
? andises?
sentiallyan ?in?nite?dimensional?data?structure?so itisclearthat?
ingeneral? noN?Nmatrixcanpossiblybeadequateforrepresen ting
it?A?principlecomp onents?typedecomp ositionisonlyvalidwhen
thesp ectralmeasureconsistsof?D antipodalypositionedatoms?
Estimating?ismuchdi?cultthanestimatingacovariancematrix?
Whereastheterms ofacovariancematrixcanbedirectlycomputed
byestimating thecorrelationb etweencoordinates?? isonlyindirectly
visible? theimage of?underasort of?sphericalconvolution? appears
inthelo garithmofthecharacteristicfunctionof thedistribution? there
isnomoredirectwaytoobserveit?
Inthispaper?wedevelopanmethod forestimating?fromthelog?
characteristicfunction??Assumeforsimplicitythatthedistribution
iscentered attheorigin?andletthesphericallog?characteristic
functionbethefunctiong?S
D??
??Cdeterminedbyrestricting?
tothesphere?Then?forall
?
??S
D??
?wehave
g?
?
???
Z
S
D??
?
???
D
?
??s
E
d??s? ???
Thecharacteristicfunctionofadistribution iseasy toestimate
fromempiricaldata?andthus?weassumewehaveagood estimate
ofgon somesuitably ?ne meshoverS
D??
?Hence?the problemis to
recov er? fromg?
Abusingnotation?wemightrewriteequation ???as?g??
???
?
???IfD??orD? ??thenS
D??
isatopologicalgroup?andthis
?convolution?canbeinterpretedliterally?via the form ula?
?
???
? ??
?
???
Z
S
D??
?
???
?
?
??s
??
?d??s??
Inotherdimensions?howev er?S
D??
isnota topologicalgroup?
andtherefore?conv olutionper seisnotwell?de?ned?Wemustinstead
thinkofS
D??
asahomogeneousmanifoldundertheactionofSO
D
?R??
andde?neakindof?conv olution?intermsofthisgroupaction?
Theeigenfunctionsof theLaplacianoperator onS
D??
are called
sphericalharmonics?andform an orthonormalbasis forL
?
?S
D??
??
?
Thisterminologyisstandard? butsomewhatunfortunate?since?isunrelatedto
anyoneofhalfadozenother?spectra?and?spectralmeasures?currently existentin
mathematics?P erhapsitwouldbemoreappropriate tocall?aFeldheim measure?
sinceFeldheim????wasthe?rsttode?neit?
?
Page 5
analogoustotheF ourierbasisforL
?
?S
?
?fromclassical harmonicanal?
ysis? TheexpressionofafunctiononS
D??
intermsofthisbasis is
calleditssphericalF ouriertransform?Afunctionf?L
?
?S
D??
?
iscalledzonal ifitis rotationallyinvariantarounda particularco or?
dinate axis?forexample??
???
iszonal?There isawayof convolv?
ingarbitraryfunctionsbyzonalfunctions?and?just asin classical
harmonicanalysis? convolutionofafunctionfby?translates in to
component wisemultiplication oftheirrespectiveF ouriertransforms?
Thus? to deconv olvefand??itsu?cesdividetheF ouriertransform
of??fbythat of??
Theadvantageofthisapproachistw ofold?First?itprovidesa
naturalcontinuousrepresentationofthe sp ectralmeasure?obviating
theneedtoapproximate itwitha sum ofatoms?Second? itiscom?
putationallyfaster? Thecomputationsinvolvedarestillexpensive?
numericallyin tegratingonasphere usingameshofdensity????N
isacomputationof orderO
?
N
?D???
?
? and computingaconv olutionof
twofunctions isth usacomputationoforderO
?
N
??D???
?
?However?
thereis noneedtoexplicitlycompute a matrixin verse?rst? b ecause
aclosed?formexpression existsfor theelements ofthe orthonormal
basis?
Organization ofthisPap er?Inx??wesummarizepreviouswork
on thisproblem? Inx??wedevelopsome backgroundmaterial?treat?
ingS
D??
ashomogeneous manifoldundertheactionofSO
D
?R??and
reviewing zonalfunctions?theeigenfunctions oftheLaplacian? anda
suitablenotionofconvolution?andprovideexplicitformulaeforthe
sphericalharmonics?Inx??wede?nethesphericalFouriertransform
andshow howtocompute?deconv olution?usingthis transform?Inx??
wecharacterizethe rateof conv ergenceofthesphericalFourierseries
asan estimateofthespectralmeasure?andrelate thisto convergence
of theunderlying stabledistribution?
?SummaryofpreviousWork?
Early on?Press????dev elopedan estimationscheme formultivariate
stable distributions?througha straigh tforward generalizationof his
one?dimensionalmethod?Press?s method? however?onlyworksfor
?pseudo?Gaussian?distributions?withlog?characteristicfunctionsof
theform?
?
Page 6
?
X
?
?
???
D
?
????
E
i?
D
?
???
?
?
E
???
where?issomesymmetric?p ositivesemide?nite?covariance ma?
trix??If?has uniteigenv ectors??
?
???????
D
?witheigenvalues ?
?
?????
D
?ie?asa covariancematrix?wehave?principlecomponen ts??
?
??
?
?????
?
?
??
?
??thenthespectralmeasureof thisdistribution issymmetricand
atomic? withatomsateachof???
?
????????
D
?withmasses?
?
?????
D
?inotherw ords?
??
D
X
d??
?
d
?
?
??
d
??
???
d
?
Press prop oses to solve forthe comp onents ofthematrix?byem?
piricallyestimatingthe logc haracteristicfunction atsomecollection
offrequenciesf
?
?
?
?????
?
?
N
g? whereN?D?D? ????? and then solv?
ingasystemofNlinearequations? He claimsthat hismethodwill
generalizetoasumofpseudo?Gaussians?
?
X
?
?
???
D
?
????
E
i?
M
X
m??
D
?
???
m
?
?
E
???
?where?
?
??????
M
arelinearly independent?symmetric?p ositive
semide?nite matrices?? How ever? inthiscase? one no longerendsup
withasystem oflinearequations? so itis notclearthat themethodis
tractable?In anyevent?Press?s methodonlyappliestom ultivariate
distributions withparticularlysimpleatomicspectralmeasures?which
furthermoremustbe symmetricallydistributed?Empiricalevidence
?see? forexample??? ??suggeststhatthestabledistributions foundin
?nancial data aresigni?can tlyskewed? symmetryis notareasonable
assumption?
Cheng?RachevandXin????????developamore sophisticatedmethod?
byexpressingastablerandomv ectorinpolarcoordinates?andthen
examining theorderstatisticsof theradialcomponent? asa function
ofthe angularcomponent?Theyutilizethetheorem ofAraujoand
Gin?e ?Corollary?????b??Chapter??p????of???? statingthatthera?
dialdistributiondecaysmostslowlyinthose angulardirectionswith
the heaviestconcentration ofspectralmass?thesedi?erences indecay
ratearethenusedto estimatethe densitydistribution of thespectral
measure?
?
Page 7
Nolan?Panorska?andMcCullo ch??? ???????developamethodbased
up ona discreteappro ximationofthe spectralmeasure? Ifthespectral
measureistreated asasumofa?niten umberofatoms?
??
X
a?A
?
a
?
a
?
then?forany?xed
?
??S
D??
?the function?
???
?
?
?s???
???
D
?
??s
E
ofTheorem? canberestrictedtoafunction?
???
?
?
?A??C?The
setofall discretemeasuressupportedonAisa?nite?dimensional
vectorspace? whichwecan identifywithC
A
?and?
???
?
?
isjustalinear
functional onthisv ectorspace?If??S
D??
issome?niteset?then
wecande?nea linear map?
F?C
A
??C
?
where? for each
?
?? ??
F ???
?
?
?g?
?
???
Z
S
D??
?
???
?
?
d?
The method of Nolanetal? thencomes do wntoinverting this
lineartransformation to recov er?from anempirical estimateofg?
They explicitlyimplemen tedtheirmethod in thetwo?dimensionalcase
?ie? when thesp ectralmeasure liv esona circle?? andtested itagainst
avarietyof distributions?Theyfoundthatitworked fairlywell fora
variety of measuresonthe circle?andconsistentlyoutperformedthe
methodofChenetal?ThemethodsofChenetal? andNolanetal?
arealsodiscussedinx?of?????
Finally
cussion
?
???
Hurd
?
et al????
??
dev
section
elop???
??
?S
D??
asa Homogeneous Riemannian
Manifold?Zonalfunctions?Laplacians?
andConvolution
?The developmentof backgroundmaterialhere lo oselyfollo ws thedis?
inchapterAmorefriendlyapproach is ?????
?
?
Page 8
S
D??
isacompactRiemannianmanifold?andG?SO
D
?R?isa
?nonab elian?compactLie group?acting transitivelyand isometrically
onS
D??
byrotations?Wewilldevelopaversionof harmonicanalysis
onS
D??
asahomogeneousRiemannianmanifold?thistheoryisactu?
allyapplicable toanyhomogeneous Riemannianmanifold?itmaybe
helpfultokeepthisinmind??
LetL
bsg
be the canonicalv olumemeasure inducedonS
D??
byits
Riemannstructure?Forexample?onS
?
?L
bsg
istheusual ?surfacearea?
measure?S
D??
iscompact? soL
bsg
is?nite?assumeL
bsg
isnormalized
to have totalmass ?? LetL
?
?S
D??
??L
?
?S
D??
?L
bsg
?C?? Theaction
ofGonS
D??
inducesa linearG ?act iononL
?
?S
D??
?inthe obvious
wa y? if??L
?
?S
D??
?andg?G?theng???S
D??
??Cis de?ned?
g ???m????g?m??
LetC
?
?S
D??
?be the spaceofsmo oth?complex?valuedfunctions
onS
D??
?L
bsg
is?nite?soC
?
?S
D??
?isalinear subspaceofL
?
?S
D??
?
?thoughnotaclosed subspace??G actssmoothly onS
D??
?soC
?
?S
D??
?
isG?invarian t?Weconsidertherestrictedaction ofGonC
?
?S
D??
??
Let??C
?
?S
D??
???C
?
?S
D??
? is the Laplacianop erator?
Theorem ?? ?TheL aplacian onS
D
?????
Firstconsider thecaseD???Endow thecircleS
?
withtheangular
coordinatesystem?????????sothatanypointonS
?
?
?S
?
?f??? ??g
has coordinates
? cos???? sin?? ??
Iff?S
?
?
??C?then?relativetothiscoordinatesystem?wehave?
?
S
?
f?
?
?
f
??
?
?
Moregenerally?de?neS
D
?
?S
D
n
?
R
D??
???????f?g
?
?and
then de?nethedi?eomorphism
S
D??
?
????????S
D
?
?s??? ???cos???? sin????s?
Thenwehavethe following inductiveformula?
?
Page 9
?
S
D
f?
?
?
f
??
?
??D???cot???
?f
??
?
?
sin???
?
?
S
D??
f????
?
?comm utes withtheisometricGaction?forallg?G?
??g ????g?????
Let???f??C???isaneigenv alueof?g?and foreach??
?? letV
?
??
?
??C
?
?S
D??
????????
?
be thecorresponding
eigenspace? Th us?V
?
isaG ?inv ariantsubspace?
Theeigenfunctions oftheLaplacianonS
D??
arecalledspherical
harmonics?Further informationon sphericalharmonicscanbe found
inc hapter??section?of?????c hapterIIof ????chapters ?and?of
??? ??c hapters?and? of?????x??andx?? of?????andalsoin?????
??? ??????? ?????? ????? ?? ?????and?????
Lete?????????? ???S
D??
? and de?ne
G
e
?fg?G?g ?e?eg?
thesetof allorthogonaltransformations ofR
D
?xingthee?axis?In
otherwords?G
e
isthesetofall?rotations?oftheremaining?D???
dimensionsaboutthisaxis?hence?thereisanaturalisomorphism
G
e
?
?
SO
D??
?R??G
e
is thusaconnected?compactsubgroupofG?
Theaction ofGuponC
?
?S
D??
?restrictstoanactionofG
e
?andthe
spacesV
?
remain inv ariant under this newaction?
De?nition??Zonalfunction
Afunction??C
?
?S
D??
?iscalledzonal?relativetoGandthe
?xedpointe?S
D??
?ifitisinvariantundertheaction ofG
e
?
F ormally? for anyG
e
?invariantsubspaceV?C
?
?S
D??
??de?ne
Z
e
?V???f??V??g?G
e
?g????g
Thus?thezonalelementsofC
?
?S
D??
?aresmoothfunctionsrota?
tionallyinvariantabouttheeaxis?Clearly?anysuchfunctionmust
be of theform
??x???
?
?x
?
?
where?
?
??????? ??C? andwherex??x
?
?x
?
?????x
D
?isany
elementofS
D??
?
?
Page 10
Proposition??
??IfV?C?S
D??
? isanontrivialG?invariant subspace?thenZ
e
?V?
isnon trivial?
??Ifdim?Z
e
?V?????thenV isanirreducibleG?module?
Proof?
ProofofP art??
Claim??Vcontainsanelement? suchthat??e?????
Pro of?SinceV isnon trivial?thereissome??Vwhichis
nonzerosomewhere?say??x??? ??SinceGactstransitively
onS
D??
? ?ndg?Gso thatg ?e?x? Th us?if??g ???then
??e????g?e????x????? SinceVisG ?inv ariant???V is
theelementwe seek????????????????????????Claim??
Now?G
e
isaclosedsubgroupofthecompactgroupG? thus?G
e
iscompact?soithasa ?nite HaarmeasureH
aar
?De?ne
???
Z
G
e
g??dH
aar
?g?
SinceH
aar
is?nite? thisintegral isw ell?de?ned?SinceVisa
closed?G?invariantsubspace?theelement? is inV?F urthermore?
since??e????e??and??e??? ??weconcludethat?isnontrivial?
Finally?notethat?isG
e
?invariantbyconstruction?inother
words?itiszonal?
ProofofPart??Supp oseV?V
?
?V
?
?whereV
?
?V
?
areG?
invariant?ThenbyPart??wecanconstructlinearlyindependent
zonalfunctions?
?
?Z
e
?V
?
?and?
?
?Z
e
?V
?
??Since?
?
??
?
?
Z
e
?V??thiscontradictsthehypothesisthat dim?Z
e
?V?????
??Proposition??
Theisometricaction ofG
e
on S
D??
inducesalinear?isometric ac?
tion uponthetangent spaceT
e
S
D??
?If?v?T
e
S
D??
isthederivative
ofapath??????????S
D??
with?????e?theng??visthederivative
ofthepath?g????????????S
D??
?TheactionofofG
e
onS
D??
is
rankone?meaningthatG
e
acts transitively onthesetofunittan?
gentvectorsT
e
S
D??
?For anyr???letB?e?r?betheball ofradius
rab oute inS
D??
?relativeto the intrinsicRiemannianmetric?The
following is clear?
??
Page 11
Lemma??Forallr?G
e
actstransitivelyon?B?e?r?inS
D??
?
Proposition??IfS
D??
isofrankone?theneacheigenspace
V
?
of? is anirreducibleG ?module?
Pro of? ByProp osition ??it su?cesto showthat dim?Z
e
?V
?
?????
So?supposethat?
?
??
?
?Z
e
?V
?
?arelinearlyindependent?Since
theyarezonal??
?
?u? and?
?
?u?arefunctionsonlyofthedistance
fromu toe?So?forsomeu?S
D??
with dist?u?e??r?de?ne
z
?
???
?
?u?andz
?
???
?
?u?? andlet? ??z
?
?
?
?z
?
?
?
?Thus?? is
alsozonal?We aimtoshow that? is the zero function?thus??
?
and?
?
arejustscalarmultiplesofoneanother?
Now?byconstruction???u???? and thus???? on?B?e?r??At
thesametime?howev er?? isa linear combination oftwo elements
ofV
?
? hence?it isalso inV
?
?ie?? isa?????eigenfunctionsof
??Fix?? and letrgetsmall?Ifr ismadesmallenough?then
theDirichletboundarycondition?
j?B?e?r?
?? forcesthesmallest
eigenvalue of?tobelargerinabsolutevaluethan??creatinga
contradiction?
??Proposition ??
Oneconsequence of thisirreducibility is
Theorem???Schur?sLemma?????
IfViscomplexBanachspaceandanirreducibleG?module?and
??V??Visacontinuous?complex?linearmapthatcommuteswith
theG?action?then?mustbem ultiplicationbyascalar?
?
No wconsiderthe D ?torusT
D
?equipped with the standardequiv?
ariantmetric?TheeigenfunctionsoftheLaplacianonaretheperiodic
functionsoftheformE
n
?x??exp???i?hn?xi??withn?
c
T
D
?
?
Z
D
?
wherex??????
D
and?????
D
isiden ti?edwithT
D
intheobvious
way?TheseeigenfunctionsformanorthonormalbasisforL
?
?T
D
??
ThesameistrueforarbitraryhomogeneousRiemannian manifolds?
and inparticular?forthesphere?
Theorem??
?If?
?
???
?
?thentheeigenspacesV
?
?
andV
?
?
areorthogonalas
subsetsofL
?
?S
D??
??
??
Page 12
?Theeigenspacesof? spanL
?
?S
D??
??In otherw ords?
L
?
?S
D??
??
M
???
V
?
Proof?Seeforexample????? chapter?? p? ????or???? Theorem?????
p?????Ortreat? asanellipticdi?erentialoperator? anduse
?????x???? Theorem ??p????? Alternately?employ theSpectral
Theoremforunboundedself?adjoint operators?see?????chapter
X?section??p? ?????
??Theorem??
De?nition??EquivariantFunction
If??S
D??
?S
D??
??C?thensaythat? isaG ?equivariantif?
forall m?n?S
D??
andg?G?
??g ?m?g ?n????m?n?
SinceG acts isometricallyandtransitively onS
D??
? thisis equiv?
alentto sayingthat??x?y?isafunction onlyof thedistance
dist?x?y??
Forinstance?ifthefunction?
???
?S
D??
?S
D??
??Cde?nedby
equation???isG?equiv arian t?
G?equivariantfunctions areinterestingbecausewecande?nea
sortofconvolutionwiththem?
De?nition??? Convolution
If? isG?equiv ariant???S
D??
??C?andb othareL
bsg
?integrable?
then de?ne????S
D??
??Cby
??????s??
Z
S
D??
??s???????dL
bsg
???
Forexample?if? isameasure onS
D??
?with Radon?Nikodym
derivative??S
D??
??C?then????S
D??
??Cisde?ned
????s??
Z
S
D??
??s???????dL
bsg
????
Z
S
D??
??s???d????
??
Page 13
Inparticular?if?isaspectral measureand???
???
?thenthis
formulais identicaltoequation???? In otherw ords?
?
???
???g
where
It
gis
out
thespherical
that
log?characteristicfunction?
RecallagainthecaseofT
D
Theeigenfunctionsofthe Laplacian?
?
E
n
?n?Z
D
?
?arew ell?behav edunderconvolution? classicalhar?
monicanalysis tellsusthat
?
?
X
n?Z
D
a
n
E
n
?x?
?
A
?
?
?
X
n?Z
D
b
n
E
n
?x?
?
A
?
X
n?Z
D
?a
n
?b
n
?E
n
?x?
turnsthisphenomenongeneralizes toarbitraryhomo?
geneous Riemannianmanifolds?
Prop osition??? ?Convolutionand Eigenfunctions?
Let??S
D??
?S
D??
??CbeG?equiv arian t?Fix???and
??Z
e
?V
?
?? andde?neA
?
?C
D
b y?
A
?
??
???? ??e?
??e?
?
Then?forany??V
?
?????A
?
???
Proof?LetT
?
?C
?
?S
D??
???C
?
?S
D??
?bede?ned?T
?
????????
Claim??Theop eratorT
?
commuteswiththeG?action?for
allg?G?T
?
?g ????g?T
?
????
Proof?F oranym?S
??
?
T
?
?g ????m??????g?????m?
?
Z
S
D??
??m?n???g ?n?dL
bsg
?n?
?
???
Z
S
D??
??m?g
??
?n
?
???n
?
?dL
bsg
?n
?
?
?
???
Z
S
D??
??g?m?n
?
???n
?
?dL
bsg
?n
?
?
???????g?m?
?g???????m??
??
Page 14
???wheren
?
??g ?n
???Because?isG?equivariant????????????? ?Claim??
Claim ??T
?
comm uteswith??
Proof?Foreachy?S
D??
?de?ne?
y
?S
D??
??Cby?
y
?x??
??y?x????x?y ?? Thus?
??????x??
Z
S
D??
??x?y????y?dL
bsg
?y?
?
Z
S
D??
??y???
y
?x?dL
bsg
?y?
Hence????????x???
Z
S
D??
??y???
y
?x?dL
bsg
?y?
?
Z
S
D??
??y????
y
?x?dL
bsg
?y????
b ecause? isa linearoperator?
Claim??????
y
?x????
x
?y??
Proof?Find someg?G sothatg?x?yandg?y?x?
Thusforanym?S
D??
?
?
x
?m????x?m????g ?x?g?m????y?g ?m?
??
y
?g?m???g??
y
??m?
In otherwords??
x
??g??
y
??
Thus???
x
???g??
y
??g????
y
??
Inparticular???
x
?y??g????
y
??y????
y
?g?y ?
???
y
?x??
??????????????????????????????????????Claim????
Hence?wecanrewriteexpression???as?
Z
S
D??
??y????
x
?y?dL
bsg
?y?
??
Page 15
ButS
D??
isa manifold withoutboundary?so?isself?adjoint
?see?forexample?????chapter???Hence?
Z
S
D??
??y????
x
?y?dL
bsg
?y??
Z
S
D??
???y???
x
?y?dL
bsg
?y?
?
Z
S
D??
??x?y?????y?dL
bsg
?y?
????????x?
????????????????????????????????????????????Claim ??
Itfollowsfrom Claim? thatT
?
must leave inv ariant alleigenspaces
of?? inotherwords?forall????V
?
isinvariant underT
?
?
Butby Claim??therestrictedmap
?T
?
?
j
?V
?
??V
?
isthen anisomorphism of linearG? mo dules? SinceGacts irre?
duciblyonV
?
?byProp osition???it followsfromSchur?s Lemma
thatT
?
mustactonV
?
by scalarmultiplication?thus? thereis
someA
?
?Cso that?forall??V
?
?
T
?
????A
?
??
????inotherw ords?????A
?
???Inparticular?if??Z
e
?V
?
??
then????A
?
??? hencewemusthaveA
?
?
????e?
??e?
?
??Proposition???
Corollary???Let??Z
e
?V
?
?beazonaleigenfunction?nor?
malizedso thatk?k
?
???De?neZ?S
D??
?S
D??
??Cby
Z?x?y????g
??
x
?y?
whereg
x
?Gis anyelementsothatg
x
?e?x?ThenZisw ell?
de?ned?indep endentofthe choiceofg
x
?andisG?e quiv ariant?Ifwe
thende?neP
?
?L
?
?S
D??
???L
?
?S
D??
?by
P
?
??????e???Z???
thenP
?
istheorthogonalprojectionfromL
?
?S
D??
?on to the
eigenspaceV
?
?
??
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