Representation Theory of Solvable Lie Groups and Related Topics

620 Pages·2021·6.172 MB·English

by  Ali Baklouti, Hidenori Fujiwara, Jean Ludwig

Springer Monographs in Mathematics
Ali Baklouti
Hidenori Fujiwara
Jean Ludwig
Representation
Theory
of Solvable Lie
Groups and
Related Topics
Springer Monographs in Mathematics
Editors-in-Chief
Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study,
Seoul,SouthKorea;MathematicalInstitute,UniversityofWarwick,Coventry,UK
Katrin Wendland, Research group for Mathematical Physics, Albert Ludwigs
UniversityofFreiburg,Freiburg,Germany
SeriesEditors
Sheldon Axler, Department of Mathematics, San Francisco State University, San
Francisco,CA,USA
MarkBraverman,DepartmentofMathematics,PrincetonUniversity,Princeton,NY,
USA
Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton,
NY,USA
TadahisaFunaki,DepartmentofMathematics,UniversityofTokyo,Tokyo,Japan
IsabelleGallagher,DépartementdeMathématiquesetApplications,EcoleNormale
Supérieure,Paris,France
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NewYork,NY,USA
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Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay,
France
AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal
GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy
Ken Ribet, Department of Mathematics, University of California, Berkeley, CA,
USA
RenéSchilling,InstituteforMathematicalStochastics, TechnicalUniversityDres-
den,Dresden,Germany
Panagiotis Souganidis, Department of Mathematics, University of Chicago,
Chicago,IL,USA
EndreSüli,MathematicalInstitute,UniversityofOxford,Oxford,UK
ShmuelWeinberger,DepartmentofMathematics,UniversityofChicago,Chicago,
IL,USA
BorisZilber,MathematicalInstitute,UniversityofOxford,Oxford,UK
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Ali Baklouti (cid:129) Hidenori Fujiwara (cid:129) Jean Ludwig
Representation Theory
of Solvable Lie Groups
and Related Topics
AliBaklouti HidenoriFujiwara
DepartmentofMathematics FacultédeScienceetTechnologiepour
UniversityofSfax l’Humanité
Sfax,Tunisia UniversitédeKinki
Iizuka,Japan
JeanLudwig
InstitutÉlieCartandeLorraine
UniversitédeLorraine
Metz,France
ISSN1439-7382 ISSN2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN978-3-030-82043-5 ISBN978-3-030-82044-2 (eBook)
https://doi.org/10.1007/978-3-030-82044-2
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Preface
RepresentationtheoryofLiegroupsandnoncommutativeharmonicanalysisonLie
groupsandhomogeneousspaceshavewitnessedasignificantgrowthstartingfrom
the 1940swhen research work in these areas began.Taken jointly, they constitute
a pivotal and compelling field within mathematics, due to the tight relationships
withsomanyotherareas—thinkofnumbertheory,algebraicgeometry,differential
geometry,operatoralgebras,PDEs,tonamebutafew—andalsowithfartherfields
likephysics.
WithinthesettingofexponentialsolvableLiegroups,theso-calledorbitmethod
isafundamentaltooltoassociatethroughanelegantwayunitarydualstothespace
ofcoadjointorbits.Still,theharmonicanalysisonthecorrespondinghomogeneous
spacesremainsadifficultsubject.
Thepurposeofthisbookistodiscussthelatestadvancesinthisarea,provenovel
resultsonnoncommutativeharmonicanalysisonsolvablehomogeneousspaces,and
providemanyapplications.Thetextoffersthemostrecentsolutionstoanumberof
openquestionsposedoverthelastdecades,presentsthenewestresearchresultson
thematter,andprovidesanalluringplatformforprogressinginthisresearcharea.
Throughoutthe text, unless otherwiseexplicitly stated, G will always denote a
connected and simply connected exponentialsolvable Lie group with Lie algebra
g. This means the exponential mapping exp : g −→ G is a diffeomorphism.
The first chapter aims to build the branching laws of mixed representations
(or representations of mixed type), showing that they obey the orbital spectrum
formula.Let A and H be two closed connectedsubgroupsof G, and πandσ two
unitaryirreduciblerepresentationsofGandH,respectively.Mixedrepresentations
are unitary representations of the form (indGσ)| (up-down representation) or
H A
indGH(π|H) (down-up representation). We shall provide explicit formulas for the
multiplicity function occurring in the disintegration of such representations, and
detailed informationregardingits behavior.We shall studymany situationswhere
the multiplicity function is uniformly infinite or finite and bounded, and discuss
alongthewaynecessaryandsufficientconditionsforfiniteness.Onemajormotiva-
tionforthisworkgoesbacktothetheoryofergodicLiegroupactions.Anotheris
v
vi Preface
theneedtobuildsmoothdisintegrationsofthespaceL2(G)onaLiegroupGfrom
arepresentation-theoreticalpointofview.
Now, one of the most important and difficult problems in the disintegration
theory of group representations is to write down explicit unitary operators that
intertwine unitary representations and their decomposition into irreducibles. This
is, as a matter of fact, the basic aim of Chap.3. The answer to this question
can help to solve several important problems, for example regarding the study
of the solvability of invariant differential operators on homogeneous spaces, or
the Frobenius reciprocity problem related to the corresponding multiplicities, the
disintegrationofgeneralizedinvariantvectors,etc.We shallconstructintertwining
operators of induced representations (sometimes concerning discrete subgroups)
andrestricted representationsofarbitrarynilpotentLie groups.Thiswill elicitthe
analysis of the tensor product of two unitary representations. Even in the case of
exponential solvable Lie groups these problems are still not fully solved, and we
shallonlydiscussthecaseofinducedrepresentationsfromnormalsubgroups.
Withtheaboveinmind,weestablishtheBonnet-Plancherelformulaassociated
with a monomial representation of a nilpotent Lie group and, more generally,
of an exponential solvable Lie group when the inducing subgroup is normal. In
Chap.4,wewillalsodiscussavariantofthePenney-Plancherelformulaforinduced
representationsandrestrictions.
The orbit method of Chap.2, which goes back to the early 1970s, asserts that
two monomial representations π = indGχ (H = exp(h ),1 ≤ i ≤ 2) of
i Hi f i i
G are irreducible and mutually equivalent. Here h and h are two polarizations
1 2
of g at f ∈ g∗ that satisfy the Pukanszky condition. We construct an explicit
intertwining operator between these representations in the absence of a complete
proof of the fact that the product H H is a closed subset of G. A third Vergne
1 2
polarization will come into play and give rise to a unitary intertwining operator.
We shall prove a composition theorem involving the Maslov index of the three
polarizations. The fundamental results of Auslander-Kostant [3] and Pukanszky
[144]aboutholomorphicallyinducedrepresentationsofsolvableLiegroups,which
hold without any knowledge of real polarizations for a given linear form in the
dual Lie algebra, live outside the world of exponential Lie groups. There clearly
remainsa massive numberof openquestionsto be solved.Chapter 6 will provide
the occasion to study certain intertwining operators and real polarizations in the
aforementionedrathergeneralsetting.
TheaimofChap.5istopresenttwoopenconjecturesconcerningcertainalgebras
of invariantdifferentialoperatorson the space of inducedand restricted represen-
tations of nilpotent Lie groups. The claim is that whenever these representations
havefinitemultiplicities,thecorrespondingalgebrasareisomorphictothealgebra
ofinvariantpolynomialfunctionsonsomerelatedgeometricobject(manifold).We
willfurnishsolutionsin manyspecialcasesandlay outthelatest advancesin this
direction.
Chapter7isasortofgeneralizationofChaps.2and4tothesettingofexponential
groups G. We shall study in every detail the finite and infinite multiplicity of
Preface vii
discrete-typemonomialrepresentations.Ourinterestlies inthePenney-Plancherel
formulaformonomialrepresentationsofdiscretetype,andinaquestionofMichel
Duflo (cf. [54]) regarding the commutativity of the algebra of G-invariant differ-
entialoperatorsonlinebundlesassociatedwiththerepresentationinquestion.We
shallemphasizetheresultwherebysuchalgebraturnsouttobecommutativeinthe
caseofmonomialrepresentationsofdiscretetypeandhencegivesacounterexample
toDuflo’squestion.Theorbitmethod(cf.[70])willbeafundamentaltechniquein
thestudy.
Thelastchapter(Chap.8)isanapplicationofearlierresultsofthebook.Wewill
characterize bounded, topologically irreducible representations of an exponential
solvable Lie group G on Banach spaces by using a triple involving topologically
irreducible weighted representations. We shall determine the simple modules of
the group algebra L1(G). These are essentially obtained in the same way as for
nilpotentgroups,exceptthatonehastogeneralizetheinducedrepresentations.This
isnolongertruefortopologicallyirreduciblerepresentations,asshownin[117].We
willshowthatifthegroupGisnotsymmetric,anewtypeofbounded,irreducible
representationofGonBanachspacescropsup.Thecorrespondingrepresentationof
L1(G)isfundamentallydifferentfromtheinducedrepresentation.Theconstruction
ofsuchrepresentationsisrelatedtotheinvariant-subspaceproblem.
Sfax,Tunisia AliBaklouti
Iizuka,Japan HidenoriFujiwara
Metz,France JeanLudwig
Nomenclature
ρ(G,H,A,σ) up-downrepresentation
E(G,K) thespaceofcontinuousfunctionsξ onGwithcompactsupport
moduloK
(H−∞)K,λ spaceofK-semi-invariantgeneralizedvectors
ρ
β Penneydistribution
φ
exp:g−→G Theexponentialmapping
λ leftregularrepresentation
G
C((cid:10))K fieldofinvariantrationalfunctions
C((cid:11) )H fieldofinvariantrationalfunctions
τ
C[(cid:10)(π)]K algebraofinvariantpolynomialfunctions
C[(cid:11) ]H algebraofinvariantpolynomialfunctions
τ
b realkernelpart
d realpart
e complexpart
∗
g /G spaceofcoadjointorbits
S(N) theSchwartz-algebraofN
C compositionsequence
H(f,h,N) spaceofholomorphicallyinducedrepresentation
H(f,η ,h ,G) spaceofholomorphicallyinducedrepresentation
f i
S goodsequenceofsubalgebras
U(g,τ) infinitesimalinvariantoperator
ω fundamentalweight
U
π ×π(cid:7) outertensorproductoftworepresentations
π|H arestrictionofarepresentationtoasubgroup
ρ(f,h,N) holomorphicallyinducedrepresentation
ρ(f,η ,h,G) holomorphicallyinducedrepresentation
f
ρ(G,H,π) down-uprepresentation
indGν inducedrepresentation
H
(cid:16) modularfunction
G
(cid:17) algebrahomomorphism
(cid:17) theKirillov-Bernatmapping
G
ix
x Nomenclature
A(cid:2)top equivalenceclassesofirreducibleBanachspacemodulesofA
a(hi,hj,hk) =ei4πτijk
a pointwisePenneydistribution
(cid:18)
ak Penney’sdistribution
π
c(i,j,k) Maslovindex
D (G)K algebraofinvariantdifferentialoperator
π
D (G/H) algebraofinvariantdifferentialoperators
τ
G·l coadjointorbit
H(cid:2)G(cid:3)B adoublecoset
H(x,y) Hermitianform
H (x,y) Hermitianform
f
I formalintertwiningintegral
h2h1
J complexstructure
K(s,m) integral
P+ =P+(f,n) setofpositivepolarizations
P imagefunction
X
R retract
a,G/P,(cid:18)
S(e) setofjumpindices
S(x,y) symmetricform
Simple(A) equivalenceclassesofsimplemodules
T(e) setofnon-jumpindices
T(f,η ,h,G) inducedrepresentation
f
T intertwiningoperator
h2h1
U layer
e
Xfin finite-ranksubspace
δ(cid:18)(s)=δ(cid:18),S(s) weightfactor
a kernel
τ
ν (s) actionofanautomorphism
h
τ Maslovindex
ijk
θ (s) density
h
(cid:21) intertwiningisomorphism
h