Representation Theory of Solvable Lie Groups and Related Topics PDF
Preview Representation Theory of Solvable Lie Groups and Related Topics
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 SinanGüntürk,CourantInstituteofMathematicalSciences,NewYorkUniversity, NewYork,NY,USA ClaudeLeBris,CERMICS,EcoledesPontsParisTechMarnelaVallée,France 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 Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe “state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive toremainvaluablereferencesformanyyears.Besidesthecurrentstateofknowledge initsfield,anSMMvolumeshouldideallydescribeitsrelevancetoandinteraction with neighbouringfields of mathematics, and give pointers to future directionsof research. Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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