HTehavetheequation A(f) S
・!l・ ∠ 二mE
k=。i、F.+i=k
nbi、..in(expG(t・X・ 》 ・・expG(tnXn))S‑k
it+..+in
×{
auai1・...fauln
n(eXpG(u・X・)・ ・eXpG(unXn))u、=S‑・t、,.,u=sn・tn}
xf(expG(s‑1t1×1)..exp
G(s‑ltnXn))
dtldt2...dt
n
・!二 ・{二mE
k=。i、 午 多.+in=kbi、,・..,in(
&1↑ ・・Cn s‑k+n{al
lif(expG(t1×1)..expG(tnXn))at l..atnn
λ(expG(t1×1》 ・ ・expG(t
nXn
expG(st1×1)..expG(st
nXn))
Thereforetheinequality
E
it+..+a.i=mlln・ ・in{at
、i]∂ ≒;≒h(t・'・ ・'tn)}
h(t1'・ ・'t
n)dtr・dtnこ0(2・6》
holdsforeveryhεC((‑a,+a)×...×(‑a,+a)landhencefor
の c
・veryhεCc(Rn)・ByusingPay・ey‑Wiener'sthe・rem(cf・f・r
example[8]p.161),weobtaintherelation
コ コ
iti2in
it+..+i
n=m11",ln12n
(ξ 、'ξ2'…'ξ
n)・ 限n}⊆{x・R・x≧ ・}・
HencethedegreemofDisevenandtheprincipalsymbol
Q̲(D)satisfiesP m Q
P(D)(T)>0
バ
foreveryτ ε 牙,whichcompletestheproofoftheproposition・
§3.Str・ng・ye・ ・iptice・ements・fU(3)C
Definiti・n・Let3beafinite‑dimensi・nalLiealgebra・ver限
・ndllbean・rm・n要.An・n‑zer・e・ementD・fu(3)cis
saidtobestronglyellipticifthereexistsaconstantOくnくsatisfyingthecondition
ム
Re(戸 「mσ
P(D》(・))≧nl・lmf・reveryTE
wheremisthedegreeof ̲D.
工fDisstronglyelliptic,thenthedeqreeofDisevenand D・,D寸 ,D+D・andD+D+area・s・str・ng・yelliptic.
oo
Definition.SupposethatGisaconnectedLiegroupandUG
isarightinvariantHaarmeasureonG.Wedefineanisometric
linearmappingfｺ.一 一一>fofL1(G,UG)ontoitselfbythe
equation
レ ー1
f(q)=△G(q)f(9,forqεG
(cf.Ch.III,pages40‑41).
Therightregularrepresentati・n¶r・ftheBanachalgeb「a
Ll1G,uG)onitselfisgivenbytheequation
ゾ1
π(h)ξ=ξ ★hforh,ξLL(G)・
r
TherightregularrepresentationRofGonL1(G)is
givenbytheequation
Rk(ξ)(x)一 ξ(xk)f・rξ ・L1(G}'k'x・G・
じ コ
Wehavethefollowingequations
ノ ノ
11Rk(h)*f=h*[R ̲1(f)J
k forfhELI(G)kEG;
ノ 〆 〆21(f
禽h》=h★f
forfheLl(G)
Sincetherightrequ・arrepresentati・‑r・fU(4̀)C・n
C:IG)q1(G)satisfiesthec・nditi・n
(dπr》(Xi
]Xi2…Xim}(ξ)(9)
=一
∂t、amate・ ・Dtmt
、=。,.,tm=。 ξ(qexpG(t・Xi、)・ ・eXpG(tmXim))
forξ εC。Q(G}
,9εG(cf.[32】page95)'theequation
C
3)[dπ
r(D》 】1ξ)・ イ=ξ ★{(dπr(D雪))(n》}
h・ldsf・rξ ・nεCξ 気G)・
工nthissectionweprovethefollowingthreeproPosit二ion・
proposition4・3・1・SupPosethatGisaconnectedl」iegroup
WithLiea・gebra・F・rahermitiane・ementD=D★ ・fU(2)
thefollowingtwoquant■t二Lesβ(D)
β(D)=inf{(d・ ・(D)ε'ζ)II
(こ[一 。。,十 〇。)),
Y(D)・=inf{(d¶(D)ξ'ξ)H
C
andY(D)areidentical:
:(TT,H》iSaCOntinUOUS
unitaryrepresent二ationofG,
ξ εH,llξll=・1}
≪,
:(TT,H)iSacontinuous
irreducibleunitaryrepresentation
・fG,ξ εH,llFl乙t}
tL
(ε[一 。。,+・・}).
Proposition4.3.2.SupposethatGIisaconnectedLiegroup
・ithLiealqebra乎.lfD=D・isastr・nqlyelliptichermitian
・1ement・fU(a)C,thenthereexistanirreducib・ec・ntinu・us
・niヒaryrepresentati・n(π0'H)・fGlandavecto「 ξ0εII..
・atisfyingthec・nditi・nsi》llε0目=1and
ii》(dπ0(D》 ξ0'ξ0)H=inf{(d・1(D》 ξ'f)・11isa
continuousunitaryrepresentationofG2i
ξisaCoo‑vectorforτ1,11ξll=1}.
Proposition4.3.3.SupposethatGisaconnectedLiegroup
withLiealqebra多 ・SupP・sethatthereexistsahermitianelement
D・D・ ・fU(3)CsatisfyingD・P(G)＼P(叫)C)'deg(D)=
mThentnereexistsastronglyelliptichermitianelementD1
・fU(3)CsatisfyingD
、 ・NP(G)＼P(U(3)C》'deg① 、)=m・
[ProofofProposition4.3.3]
Firstweassuretheimp・icati・nD・P(Ul夢 》C》 ⇒D兄 ・P(U(誹)C)
(￡isanarbitrarynaturalnumber).ThisistrivialifQis anevennumber.WesupPosethatQ>3and見isodd.Weset
‑1
(見 一1》 ・SupP・sethat{π,H,カ}isanarbitrary
p=2
・‑representati・n・fU(31C.SinceD・P(U(誤 》C},there・ati。ns
(T・(D2P+1)ξ,ξ)H=(・(D)π(Dp}ξ 一 ・(Dp)E,》H二 ・h・ ・d
foreveryξ ε ノ).ItfollowsfromTheoremlin§30fl26]that
D2P+1isane・ement・fP(U(3)C》.
[cf.Inp.115wegiveanexamp!eofthefollowing
D、'D2εP(U(Cv))'D、D2=D2D、andD、D2￠P(U{タ 》C)・ 】
Secondbyusingtheassertionaboveweprovetheproposition.
ByProposition4.2.1thedegreemofDiseven。Weset几=
2囎1m・WetakeabasisΨ 一{X、,X2,…,X
n'}・f夢 ・Weden・to
ム
by{τ1'τ2'…'・
n}thedualbasis・fOc・rresp・ndindt・
バ
Ψ ・W・definean・rmil・n望bytheequati・n
qξ 、 τ、+ξ2・2+…+ξn・nl=(ξ]2+ξ22+…+ξn2}1/2 .
Weset
D。=[一(X12+X22+…+X
n2》]Q・
Thenthee・ementD。be・ ・ngst・P(U(2)C)⊂;(Gレ ・andthe
P「icipalsymb・1σ
P(D・)ofD・satisfiestheequation σ
P(D。)(τ)=(一 ・)Ql・lmf・revery・E.・
ByTheoremlin§30f[26】thereexistsapositivenumbern
f・rwhichD+nD
。d・esn・tbe・ ・ngt・P(U(31C)・Theprinc・pa・
symb。1σ
Pω+nD・ 》satisfiestheequation σ
plD+nD・)=σP(D)+nσP(D・)・1
gyProposition4.2.1theinequality
(一 ・)㌦P(D)(・)≧ ・h・ ・dsf・revery・1・ ゑ ・
Thereforetheinequality
戸 一mσP(D+nD。)(・ 》 乙徊 ηh・ ・dsf・revery・ ε タ ・
H・nceweobtainastr・nglyelliptichermitianelementD1=D+nDO
satisfyingD、 ・P(G)＼P(U(9、)C),whichc・mp・etesthepr・ ・f
ofProposition4.3.3.
ToprovePropositions4.3.1and4.3.2weuseLanglands'theory
ofstronglyellipticdifferentialoperators.
SupposethatGisaconnected,simplyconnectedLiegroup
withtheLiealgebraO・Weden・teby{¶'C。(G)'}theright
「equla「 「ep「esentati・n・fG・ntheBanachspaceCO(G)・We
defineadenselinearsubspace「CO(G)】
。.・fCO(G》andarepresen‑
tati・ndπ
r・fU(7)C・n[C。(G)】..asinpage94・
Theorem[R.P.Langlands[17)page39,Theorem8
Assumethesituationabove.SupPosethatDisastronqlyelliptic
element・fU(7)C・Thenthe・perat・rd1・
r(D》 ・[C。(G》 】..にC。(G}
→ 【CO(G月..にCO(G))iscl・sableintheBanachspaceCO(G) andthereexistrealnumbersa8Msatisfyingthefollowing
conditions
・)Theres・ ・ventset・fthe・perat・r‑d}・
r㈲C・(G)c・ntains
the・penset・ Σ={λ ・C・ λ ≠a,IArq《 λ 一a)1・ θ}
whereθisapositivenumberwithη/2<0≦113 2)Misapositivenumberandsatisfies
)
「1(λ1層 卜d・r(D)C・(G)エ)‑111≦1
λ讐al
f・rλ ε ・ Σ,IArg(λ 一a)1.2‑1(・+2‑111)3
3、Foreveryλ ε Σ(=Cthereexistsauniqueboundedcomplex
Radonmeasurevλ ・nGsatisfyingtheequati・n
(λ ・ 一[‑d町
r(D)C・(G)])一'(f)(x)
=いXg騨1)dv
λ(9}=(f・vλ 》(x)
(fεCO(G)・x・G)・
ByLanglands'theoremthereexistsa1‑parameterconvolution
・emigr・up(ut)t
.0・fb・undedc・mplexRad・nmeasures・nG satisfyingthecondition
exp(‑td・ ・
r(D》C・(G))(f)(x)
=f(xg‑1)du
t(g)̲(f*Ut)(x}G
f・・fεC。(G),xεG(cf・[3・ 〕P・294‑298)・Wesetφ 二2僑1(・+
2.1π)anddefineacurverinCbytheequati。n
r={Y=ρe‑iφ+b=。 。 〉 ρ>0}
){Y一 ρe+iφ'+b、 ・ ≦ ρ …}
wherebisaconstsntsatisfyinga<b<・ 。.Wedefinea
directionofI'sothatIm(Y)increasesalongthepositive
directignofI' .Bythetheoryofholomorphicsemigroups(cf.[3U],
p.295)theequation
exp(顧tdπ
r(D)C・(G》)
=(2πi)‑1ate(λ ・ 一 トa・ ・
r(DプC・IG)】 》‑1dλ(3・ ・)
h・ld・fo「t>0'wherethefamily・fmeasuresvλsatisfies
Ilvλll≦ 十 』 「「(λ の ・
Weassu「ethatthemeasure}1
tisabs・lutelyc・ntinu・uswith re・pecttotheHaarmeasureUG・Byvirtue・ftheequati・n(3・1)'
forthispurposeitsufficestoshowtheabsolutecontinuityof
themeasu「evλ(λ ・ Σ)withrespectt・therightHaarmeasure
uG・Weusethef・ll・winqlemma・
Lemma.[aslightmodificationofTheorem6.3.6in[11]]
LetGbeaconnectedLiegroupF1(G)thespaceofall
functi・ns・fCO(G》whichare
.c・ntinu・uslydifferentiableina neiqhborhoodofeεG.LetTbeaboundedlinearoperatoron
CO(G}definedbytheequati・n
(Tf》(x)=∫
Gf(xg‑q)(f・C。(G}'xεG)
whereuisaboundedcomplexRadonmeasureonG.
・fthe・perat・rTsatisfiesT(C。(G))魯 ε 、(G)'
thenthemeasureUisabsolutelycontinuouswithrespecttothe
leftHaarmeasurepG
,￡onG・
Inotherwords,ifaboundedRadonmeasurevsatisfies
f・v・6、(G)f・reveryf・C。(G)'thenthemeasurevis
ab・ ・lutelyc・ntinu・uswithrespectt・therightHaarmeasurel1G・
Wepr・vethatf・vλ ・ ε 、(G)f・reveryf・C。(G》'λ ・ Σ・
Wefixanelementλ ・fΣ ・Bythedefiniti・1・ ・fvλwehave
theequation
λ ψ ・vλ+dπ
r(D)(ψ)・vλ=耳 ワ
f・reveryψ εC
c(G)・Wetakeane・ementf・fC。(G}andfixit・
W・ch・ ・seasequence{ψ
IIψm鱒f口supナ ・asm+…Sincetheequati・n
(λ ・+d・r(D)C・(G)}‑1dπr(D)(ψ)
∫ 可rC・(G)(入 ・+d・T
r(D)C・(G)ド1(ψ)
h・ldsfo「eve「yψ ・C
c(G)'theequati・n
λ ψ
m畑 λ+d・ ・r(D)(ψm)・vλ
=λ ψ
m・vλ+dTrr(DIC・(G)(ψm・ ∀ λ 》 一 ψ,l
holdsfireverymENThereforewenavetherelations
dπ
r(D)C・(G)(ψm・vλ)一 一 λ ψm・vλ+ψm
‑〉 一 λf・v
λ+fasm‑}w andψ
m★vλ ナf・vλasm‑ro・ ・Ilellcethefuncti・n
f・vλbe・ ・ngst・thed・main・fdefiniti・n・fd・
r(D)C・(G) andtheequation
λf★vλ+dπ
r(D)C・(G}《f・vλ)=f(3.2》
holds・Weseth=f・vλ(・CO《G)}・Bytheequati・n(3・2》
wehavetheequation
Gh(x)ψ(X)dUG(X)+h1G(X)d・r(D・)(ψ 一UG(X)
=いx》 Ψ(x)dμ
G(x)
f・reveryψ εc
c(G》'thatis'thefuncti・nhisas・ ・uti・n
intheditrihutionsenseoftheellipticdifferentialequation
lλ1+D1(ξ)=f.
工tfollowsfromLemma50f[17](page20)(cf.[6】page1704,Th.2
and[18】Ch・3'page194'Th・3・23)thath=f・vλis
c・ntinu・us・ydifferentiab・e'andhencehε ε、(G》 ・
Theref・rethemeasuresvλ(λ ・ Σ 》andpt(t">0)
inpage109areabsolutelycontinuouswithrespecttouG.
Weset
d・'λ(x)=9入(x)duG(x》(λ ・ Σ)
anddUt(x)=ft(x)dUG(x)(t>0),
Nextweprovetheuniversalityoftheholomorphic1‑parameter
f・mi・y{9ゴ}λ,ΣinL1(G,UG》andthe・‑parameterc・nv・ ・uti・n
semigr・up{fご}t.。inL1(G,UG)・Weusethef・rmu・as・ 》,2}and
3)inpage105.
SupPosethat{π,X'}isacontinuousisometricla.nearrepre‑
sentationoftheLiegroupGonacomplexIlanachspaceXWe
・・n・idertherepresentati・n¶ ・ftheBanacha・qebraL'(G,μG)
q'venb㌧
(f)̲
Gf(x川x)dUG(X)(feL1(G)》 ・
ByTheorem80f[17]theoperator‑d1T(D)isclosableinX
anditsclosureqeneratesaholomorphicsemiqrouP。
Foreveryφ'ψ εC
c(G)wehavetheequations
・(φ)(λ ・+der(D))π(9て)π(ψ ノ)
ノ ノ
=π(λ φ+ldπ
r(D1)](φ)》 π(9λ)π(ψ)
・ ・(rλ φ+dπ
r(D†)ゆ)*[ψ.9λ1ソ 》
・ ・(11[λ ψ ・q
λ 】ゾ+φ ・ 【(dπr(D》)(ψ ・9λ 》】ソ)
=π(φ ★ ψ)(λ ε ・ Σ) .
Thereforewehavetheequations
la・ 一[一(der》(D)Xガ1一 π 〔9P(λ ・ Σ 》
ande即(爾t(dπ)(D》X》‑1丁(vft}{t・̀)}・
Inthereasonaboveweusethefollowingsymbolicnotationsfor
astr・nq・ye・ ・iptice・ ・ementD・fU(参)C・
v
ft=exp(‑tD)(εL1(G,りG)》(t・ ・)
andqく=(λ ・ 一[‑D】)‑1(・L1(G'UG》)(λ ε ・ Σ)・
Summingupthearguementsabove,weobtainthefollowingassertion.
Corollary4.3.4.SupposethatGisaconnected,simply
connectedLiegroupwithLiealgebr.a,
エfDisastr・ng・ye・ ・iptice・ement・fU(別C,thenthe
1‑parameterconvolutionsemigroup{exp(‑tD)t>0}
3・rresp・ndingt・Disc・ntainedintheBanacha・qebraL1(G,㌃1G)
endhencein七hegroupC★‑algebraC★(G》.
[ProofofProposition4.3.2]
WemayassumethatGIisthequotientgroupofaconnected,
simplyconnectedLiegroupGbyitsdiscretecentralsubgroupr.
WesupposethatDisastronglyelliptichermitianelementof
U (許)C・Weputh、=exp(‑D)(・L1(G,uG》)・F・revery
continuousunitaryrepresentation(U,H)ofG,(dU)(D}i8
・nessentiallyself‑adゴ ・int・perat・rsatisfyinginfSpec(dU(D)●H)
〉 一..andhenceU(h1)isap・sitiveb・unded・perat・r・n
H.Thereforehl=hl
/2*hl/2isapositivehermitianelement ofthe*‑BanachalgebraL1(G,UG).
〜1 .L(G/r)b
ytheWedefineanelementhofequation1
〜
hl(gl')=Ehl(gY)(gEG).
T YEr WedenotebyAtheabelianC*‑subalgebraofC*(G/1')generated
り
byh1・Wech・ ・sea1‑dimensi・na1★‑representati・n{φ'Cξ0}
。fAs・thatφlh
、)=目h、llC★(G/r)・Wetakeanirreducible
★陶repr
esentati・n{π 。'Fi}・fC★(G/F)f・rwhichξ 。 εHand
π0(a)ξ
0一 φ(a)ξ0{aεA)(cf・[31】ProP・2・3・6andProp・2・3・8・3
[4]Pr・P ・2・ …2)・Wemayassume口 ξ 。ll=・ ・
ThenthevectorξOisaneigenvect・r・ftheself‑adゴ ・int
・perat・r(d・r。)(D》Handsatisfiesthec・nditi・n
(d・ ・。)(D)Hξ 。‑Lξ 。
whereLisarealnumbergivenby
I.=inf{(dπ(D)ξ,ξ):(π,Ii》isacontinuous
unitaryrepresentationofG1=G/1',
ξ ・H.。,日 ξ}1=1}・
ForeverynεH,weconsiderthecontinuousfunctionf
。nGdefinedbyf(q)=い ・0(g)ξ0'n)H(q・G)・
Thenthefunctionfisaweaksolutionofthedifferential
・quati・n(L'1‑dπr(D》}(f)=0・ltf・ll・wsfr・m
Theorem2inp.17040f[6]thatf=fnisanana!ytic
functiononG'thereforethemapPing9{εG)← 一 → π0(q)ξ0
̀εH)isstr・nglyC‑differentiable'andhenceξ0εH ..'
whichcomplerestheproofofProposition4.3.2.
[ProofofProposition4.3.1]
SupP。sethatDisanarbitraryhermitiane・ement・fU(ン}C.
Theinequalityβ(D)≦Y(D》istrivial・Weprovetheinequality
Y(D)〈 β(D).
Wetakeanaturalnumbermsuchthat2m>deb(D)Wetake
・basis{X1,X2,…'X
n}・f多andset D。=【 一(X12+X22+…+Xn2)]m・
WetakearbitraryM>β(D)andε>Oandfixthem。We
chooseacontinuousunitaryrepresentationTryofGanda
vectorξ
(dπ11D》 ξ1'ξ1)≦M・
Wechoosen>Osuchthat(del(nDO)ζ1・ ξ ユ 》 ≦ ・ ・
Thee・ementnD。+D・fU(9)Cisstr・nq・ye・ ・」Pticand
hermitian.ByProposition4.3.2thereexist二acontinuous
'「 「educib'eun'to「y「ep「esentat'on1T・ofGandξ ・ ε(II
・・。}・ ・' 口 ξ 。 日 一1suchthat
(dTrO(nDO+D)ξ0'ξ0)重(d¶1(nDO+D)ζ1'ξ1}
Thenwehavetheinequalities
(dπ0{D)ξ0'ξ0)≦(dπ0(nDO+D)ε0'ξ0)
≦(dTτ1(D)ξ1'ξ1》+ε ≦M+ε ・
SinceMandεarearbitrary,theinequali『 ヒyY《D)≦ β(D}
holds,whichcompletest二heproofofPropositioiz4.3.1.
ｧ4.Step2nilpotentLiegroups
InthissectionweprovethefirsthalfofTheorem4.1.1.
Theorem[S.L.WoronowiCz[33]cf.[15]]
SupposethatGisa3‑dimensionalHeisenberqgroup.Then
v(G)<4
SupP・setha七 多isthe3‑dimensi・nallieisenbergLiealqebra and{X,Y,Z}isabasis・f皇satisfyinq【X'Y]=Z・
[X,Z]=[YiZ]=0.Thent二heelement二
D=(‑X2‑Y2)(‑X2‑Y2‑iZ)
・fU(3)CsatisfiesD・P(G)＼P(U(♀)C)・
Thee・ementsD、=‑X2‑Y2andD2‑X2‑Y2‑iZ
be・ ・ngt・P(U(ク)C》andsatisfytheequati。nD、D2'=D2D、
●Corollary.SupposethatGisaconnectednilpotent,non‑abelian
Liegroup.Thenv(G)く4.
[proof]Everyconnectedsimplyconnectednilpotent,non‑abelian
LiegroupGcontainsaclosed,connectedsubgroupisomorphicto
the3‑dimensionalHeisenberggroup.Thereforetheassertionofthe
corollaryfollowsimmediatelyfromthetheoremaUoveandLemma6
inｧ30f[26].
Proposition4.4.1.SupposethatGisaconnected,simply
・・nnectedLiegr・upw・thLiealgebraaf・rwhich[3'ク 】 ≠
{0'}andr3,[〃,弁 】]一{0}・Supp・sethatX、'X2'…'X
m
arevect・rs・fglinearlyindependentm・duユ ・[郭3]and
satisfythec・nditi・nRX、+RX2+…+凪X
m+「 、タ'タ 】=3・
SupP。sethatfisa・inearfuncti・na・ ・f}satisfyinq
f(Xゴ)=011≦ ゴ ≦m)・
Denotebyu(f)=Oi
,ゴ)・ ≦i,ゴ ≦mtheskew‑sylnmeし 「ic「eal
mat「ix(f([Xi'X])》 ・
≦i,ゴ ≦m・Denoteby71fthei「 「educible
unitaryrepresentationofGassociatedtofintheKirillov‑
Bernatcorrespondence.
Thenthe・perat・r(dπf》(一[X21+X22+…+X
m2】 》isan essentiallyself‑adjointoperatorandsatisfies
r minSpec(一(dTrf)(X1+X2+...+XmZ))
=2‑11h・(f)II、=2鰯1(1・ ・ 、 国 ・ ・21+…+IV
ml)
wherev1'v2'…'v
marealltheeigenvalues・fl1(f}'thatis'
m
det(λ1‑u(f》)=
ゴ塁、(λ 一vゴ)'
(Proof]Firstweshowthattheproofofthepropositionis
reducedt・thatinthecasethecenter・f7is1‑dilnensi・nal.Wemayassumethatf≠0.Weput
3f={x・3・f([x'Y1)一 ・f・reveryY・ タ}・
Thenwehavetheinclusions
[3'許]Llthecenter・f参}重2f・
Since[3f'弁]皇[あ 皇]'メfisanidea・ ・fρ ・Weset
π={Xε み ・f(X)=0}・
Th・nπisahyperplane・f弁f・Sincef([X・Y]》e・f・revery
X・ 診f'Y・3'thespacenisanidea・ ・fO・ ・fX+πis
・centrale・ement・f3/π,thenwehavetheequati・nf(【X,YD
・ ・f・reveryY・ 〃's・Xisane・ement・fgf・Theref・re
th・center・fof/ 'Lis・‑dimensi・na・andtheLiea・zebraa/π
isisomorphictothe(2k+1)‑dimensionalHeisenbergLiealgebrawhere
Q=2‑1(dim乎 一dim誹f)・Weden・tebyNtheana・yticsub‑
groupofGcorrespondingtonThenNisa(connected)closed
subgroupofG(cf.[321Th.3.18.12)Viehavetheequation
πf(n》=If・reve「ynεN・
Wec・nsiderthe・‑h・m・m・rphismρ ・fUゆC・nt・U(a/η)C
definedbyρ(X》=X+π(X・ 勲 ・Thenthevect・rsρ(Xゴ 》
(1ニ ゴ ≦m}satisfythec・nditi・n
Rρ(X、)+IRp(X2)+…+・(Xm)+【 〃'し'J/it]‑O/n・
Wech・ ・seanelementZε[撃'a】withf(Z)=1・Thenwehave
theequations
【 ρ(Xi)'ρ(Xゴ)】=μi」 ρ(Z》(1≦i'」 ≦m)
and[3/π,O/n】 一(thecenter・fタ/n)=Rρ(Z)・
Wedefinea・inearfuncti・na・f・f多/πbytheequati・n
N
f(ρ(X)》=f(X)(XE)・
Weden・tobyNftheirreducibleunitaryrepresentati・n・fG/N
〜
associatedtofintheKirillov‑Ilernatcorrespondence.Wehave
theequations
陛(gN)=11f(9)f・reveryg・G
anddπ 老(ρ(X、2+X22+…+X
m2》)
‑dπ
f(X、2+X22+…+Xm2)・
Wecharacterizethequantityll(Ui
,j)・ ≦i,j<mli・ ・We
t・keaskew‑symmet「ic「ealmat「ix(b.1
.」)1≦i,ゴ ≦msuchthat li(bi
,ゴ)・ ≦i,ゴ ≦mll。P=max{1vl・visaneigen‑value・f
thematrix(b̲̲)}=1
1,]m and
1,ゴ ー、bi'ゴUi'ゴ=ll(Ui'J)・ ≦i'ゴ ≦mll・ ・
Thenwehavetheequations
dπf卜[X・2+X22+…+X
m2】+
1<1ゴ ≦mbi'」(XiXゴXIXi))
=dπ
f(一[X・2+X22+…+Xm2】)一 、≦1.ゴ ≦mbi,ゴf((x,'Xゴ]}・ ・
=dπ
f(一 【X、2+X22+…+Xm2】 》‑2剛11111(f》 日 ゴ ・ ・
Sincethee・ement‑(X・2+X22+…+X
m2)+/享iく1 ‑b.lll1,ゴ(XiXゴ ーXゴXi)
b…ngst・P(U(汐 》C),the・perat・r
dπf(一[x、2+x22+…+xm2])‑2蘭11h1(f}11、 ・ ・
isapositivesymmetricoperator.ByLanglands'theoremtheoperator
ほ
dπf(一[X、2+…+X
m2]‑Z2)=d咤(一 【PIXI)2+…+ρ(Xm}2】)+・
isessentiallyself‑adjoint .Thereforeweobtaintheinequality
minSpec(髄dπf(X、2+X22+…+Xm2》)≧2‑11ilf)、
andtheequati・n2輪11h1(f川 、
=max{f(Y)・Y・[掃 】f・rwhich‑(X12+X22+…+X
m2) +戸Y・P̀噂)C)}.
Weset竜=RX、+IRX2+…+IRX
m・Wech・ ・seabasis
{Y、'Y2'…'Ym}・f必s・that{Yゴ ・1茎 」 二P}isabasis・f
託n3f(P≧ ・)・Werepresentthee・ementX12+X22+…+Xm2 2̲̲2̲̲2m
asX・+X2+'●'+X
m=1
,ゴ=、ai'ゴY.Yゴwhe「e(ai'ゴ)1≦i'ゴ ニm
isastrictlypositivedefiniterealsymmetricmatrix.Then
wehavetheequations
2‑11h・(f川 、‑max{f(Y)・Y・[瀞1f・rwhich m
E
i,jニP+、ai,ゴYiYゴ 頑YεP(U(3)C)'}
anddπf(X・2+X22+…+Xm2)=d・f(∴
=P+、ai,ゴYiYう)・
Therefore,toprovethepropositionthefollowingassumptions
donotlosethegenerality
1)Gisthe(2R,+1)‑dimensionalHeisenbergLiegroupwith
theLiealgebra弁 ラ
2)ZisaP・ri‑zer・element・fthecenter・fク3
3)(thecenter・f3》=[3,3]=RZ;
4){X、'X2'…X2Q
,,Z}isabasis・f許7 5)【Xi'Xゴ]UiゴZ(1≦i,j≦2Q)'det((}1i
,ゴ)1<i,ゴ ニ2P)≠03 6}fisthelinearfuncti・nal・f3satisf̲yinqf(Z)̲1,
f(X.♪=0(1≦コ ー‑」≦2Q}.
Nextweprovethepropositionunder『 ヒheassumptionsabove・
The「eexistsa「ealorth・9・nalmatrix(ui
,ゴ)・ ≦i,ゴ ≦2見such that2Q
PIq=、ui'pPP'qu」'q=si,ゴ(ユ ≦i'」 ≦29)
whe「ethemat「ix(si
,ゴ 》 ・ ≦i,ゴ ≦2Q,satisfiesthec・nditi・ns
Si
'Q+i>0'㍉+i,i=‑si,Q,+iく0(1<i<Q)
andSi
,ゴ;見Ofo「1'嗣 ゴi≠Q・
WesetY.7=
k=、uゴ,kXk(1≦'≦2見)・Thenwehavethe equations
2̲̲22‑‑2‑‑2‑‑2Y 1+Y2+…+Y2Q=X1+X2+…+X2￡'
[Yi'Y」]=si」zq≦i'ゴ<2Q)'f(Yう 》=0(1≦ 」≦2￡)
andllu(f}11]=ll(si
,ゴ)、 ≦i,ゴ ≦2Qi1、
Q
=2Σs,..
コ,Q+〕
ゴ=1
Thentheirreducibleunitaryrepresentationcifisrealizedas
thef・ …wing・ ¶f(9)εB(L2(R㌧dx、dx2…dxQ》 》f・r
everyqεGand
dπf(Yゴ 》 ≒
xゴ(1≦ ゴ≦見 》'
dπf(YQ+ゴ)=石sゴ
,死+」Xゴ(・ ≦ゴ≦Q)・
Theref・retheessentia・ ・yse・f‑ads・int・perat・rderf(一 【Y、2+Y22+..+Y21】)
lsq■venbytheequatユ.on
d・f(一[Y・2+Y22+…+2Y2k])=
ゴ1、 ←a2aX2+Sゴ'2Q,一+一 ゴXゴ2)・
TheC‑elementξOforπfqivenbytheequation
見 ̲12
ξ・(x・'x2'…'XQ)=
ゴ塁、exp(‑2sゴ 調xゴ)
isane・gen‑vect・r・fthese・f‑adj・int・perat・rd・ ・fl‑[Y21+・ ・+Y2Q]
fo「theeigen騨values・
,Q+・+s2,￡+2+…+s死,2￡
2‑11h1(f川 、,whichc・mp・etesthepr・ ・f・fthepr・P・siti。n .
Proposユ ーt■on4・4・2・SupPosethatGisaconnected,simply‑
c・nnectedLiegr・upwith・'iealgebra}f・rwhichthe
equati・n[X'[Y'Z]]‑Oh・ldsf・reveryX・Y・ZC・SupP・sethatDisahermitiane・ement・fU(夢)C・fdegree2with
theprincipalsymbolX72(D)satisfyingthecondition一 σ
2(D)(τ)≧Of・reverynTE・Thenthef・n・w'nqtw・
コ コ
quantitiesm(D)andm(D)areidentical:
m(D}=sup{β 、R、D二 β 、=㌦.・D,f。rs。me
ゴ=1コ 〕
D、,D2'…'DkεU(9)C},
m(D)=inf{(dπ ①)ξ'ξIH・(π'IDisac・ntinu・us
unitaryrepresentationofG,ξ εH}.
[Proof】Theinequality(一 。。 ≦)m(D)≦m(D}(〈+。
。 》
istrivial.Weprovetheequationm(D)=m(D).
L・fρisanaut・m・rphism・ftheLiea・qebra2,thenρi8
・xtendedunique・yt・a・‑aut・m・rphism。fU(
O)C.M・re・verthere
N
existsauniqueautomorphismpoftheLiegroupGforwhichthe
equati・nρ(exp
G(X))=expG(ρ(X》)h・ldsf・reveryXt̲・
Thereforewehavetheequations
NN
m(D)=m(p(D))andm(D)=m(p(D)).
・fτisa・‑dimensi・na・ ★‑representati・n・fU(3)C,七hatis,
τ(rX ,Y】 》=Of・reveryX,Y・ 許and・(宴)CFfR,then
τinducesa・‑aut・m・rphism・
,・fUく2)Cbytheequati・ns O
τ(・)=・'0,(X》=X+τ(X)1(Xε 〃).
TherepresentationTinducesa1‑dimensionalcontinuousunitary
〜
representation・rofGbytheequation.c(expG(X))=exp(・r(X))
(εC)f・reveryXε 乎 ・Weden・teby6theset・fa・1
equivalenceclassesofcontinuousirreducib!eunitaryrepresentations
ム
ofG.PledenotebyΨthebisectionofGontoitselfdefined T
ヘノ ハ
bYΨ,(巨])一[τ の 司f・revery[・ ・]・G・
Thenwehavetheequation
バ
[d(τ ⑧ π 》1(D)=(dπ)(O
τ(D》)f・revery[π1εG・
Thereforeweobtaintheequations
N
m(D)=m(0 ̲(D))andm(D)=m(()̲(D)).
T'I
ByProposition4.3.3toprovetheproposit=ionwemayassume
thatDisastronglyelliptichermitianelementoforder2.
W・mayassumethat[9'3]≠{0}・Wech・ ・seabasis{X1'・ ・'X
m' Xm+、'…'Xn}・f7s・that{Xm+、'…'X}nisabasis・f[夕'3]・
ThentheelementDisrepresentedasfollows:
nn
D=‑
1ゴ ー、ai」XiXゴ+戸 ゴ三、bゴX.+cI
where(ai
,7)1<i,j<nisastrictlypositivedefiniterealsymmetric matrixandcandb.'sarerealconstants.Weconsiderthe
コ automorphismρof}givenbytheequations
n
ρ(Xi)=Xi∵
m+、bi'ゴXゴfo「1<j<m
andρ(X.)=X,form+1≦i≦n.1‑1=̲
Wedefinean×nmatrixC=(ci
,ゴ 》 ・ ≦i,ゴ ≦nbytheequations
C.1
,]=δi,ゴfo「1≦i'ゴ<mandfo「m+1≦i'ゴ ≦n
c,.eb.,for1≦i≦m,m十1≦■ ゴ ≦n
,〕 一 一}㎝
■,〕
andc,.=Oform+1≦i≦n,1≦■ ゴ≦m.
,コ ー 一 一 一
Thentheelementp(D)isrepresentedasfollows:
ρ(D》=一 ∴
=、a.i,ゴxixゴ+戸 ∴ 忌ゴxゴ+C・
〜
whe「e(ai
,ゴ)・ ≦i,ゴ ≦nisthestrictlyp・sit.ivedefinitematrix
ti‑‑
givenby(ai
,ゴ)=tC・(ai,ゴ)・Candb.7'sarerea・
コ
constants,Since,forasuitablechoiceofthecoefficientsbi
rd's, themat「ix(a.
1.7)・ ≦i,ゴ ≦nis・fthef・rm
〜
(ai ,ゴ)・ ≦i,ゴ ≦mの(ai,ゴ)m+・ ≦i,ゴ ≦n'
wemay ̲assumethattheelementDisrepresentedasfollows