PROOF OF THE TADIC CONJECTURE U0 ON THE

Vincent Sécherre

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PROOF OF THE TADIC CONJECTURE U0 ON THE UNITARY DUAL OF GLm(D) by Vincent Secherre Abstract. — Let F be a non-Archimedean local field of characteristic 0, and let D be a finite-dimensional central division algebra over F. We prove that any unitary irreducible representation of a Levi subgroup of GLm(D), with m > 1, induces irreducibly to GLm(D). This ends the classification of the unitary dual of GLm(D) initiated by Tadic. Introduction Let F be a non-Archimedean locally compact non-discrete field of characteristic zero (that is, a finite extension of the field of p-adic numbers for some prime number p) and let D be a finite-dimensional central division algebra over F. In [22], Tadic gave a conjectural classification of the unitary dual of GLm(D), with m > 1, based on five statements U0,. . . ,U4. In the same article, he proved U3 and U4. In [4], Badulescu and Renard proved U1, and it is known that U0 and U1 together imply U2. In this paper, we prove the remaining conjecture U0, which asserts that any unitary irreducible representation of a Levi subgroup of GLm(D) induces irreducibly to GLm(D).

• division algebra over


• all unitary


• trivial group


• then ?


• unitary irreducible


• let nm


• glmd


• representation

Sujets

• Humboldt University of Berlin
• Trivial group
• Représentation

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

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PROOFOFTHETADIC ´ CONJECTUREU0ONTHE UNITARYDUALOFGL m (D) yb VincentSe´cherre Abstract.—LetFbeanon-Archimedeanlocalfieldofcharacteristic0,andletDbea finite-dimensionalcentraldivisionalgebraoverF.Weprovethatanyunitaryirreducible representationofaLevisubgroupofGL m (D),withm>1,inducesirreduciblytoGL m (D). ThisendstheclassificationoftheunitarydualofGL m (D)initiatedbyTadic´. Introduction LetFbeanon-Archimedeanlocallycompactnon-discretefieldofcharacteristiczero (thatis,afiniteextensionofthefieldofp-adicnumbersforsomeprimenumberp)andletD beafinite-dimensionalcentraldivisionalgebraoverF.In[22],Tadic´gaveaconjectural classificationoftheunitarydualofGL m (D),withm>1,basedonfivestatements U0,...,U4.Inthesamearticle,heprovedU3andU4.In[4],BadulescuandRenard provedU1,anditisknownthatU0andU1togetherimplyU2.Inthispaper,weprove theremainingconjectureU0,whichassertsthatanyunitaryirreduciblerepresentation ofaLevisubgroupofGL m (D)inducesirreduciblytoGL m (D).Theproofisbasedon Bushnell-Kutzko’stheoryoftypes(see[12]),andmorepreciselyontheirtheoryofcovers, whichallowsonetocompareparabolicinductioninGL m (D)withparabolicinductionin affineHeckealgebras. TheresearchforthispaperwaspartiallysupportedbyEPSRCgrantGR/T21714/01.

2VINCENTSE ´ CHERRE TheproofconsistsofreducingtothecasewhereDiscommutative,forwhichtheresult isalreadyknown(Bernstein[8],seeTheorem1.1below).Thiscanbedonebyusing particulartypesofGL m (D),theso-calledBushnell-Kutzkosimpletypes(see[11,19]). TheirHeckealgebrasarewellknownandisomorphictoaffineHecke-Iwahorialgebras oftypeA,whichallowsonetotransportourinductionproblem,viatheHeckealgebra isomorphismsof[19],toaveryspecialcase,inwhichtheconjectureisknowntobetrue. Thismethodhasbeenalreadyusedin[10,11]. NotethatinthecasewhereDiscommutative,henceequaltoitscentreF,Bernstein’s proofofU0isbasedontheproofofKirillov’sconjecture,whichassertsthatif P denotes thesubgroupofGL m (D)madeofelementswithlastrow(0,...,0,1),thentherestriction to P ofanyunitaryirreduciblerepresentationofGL m (D)isirreducible.Thisfactis nomoretruewhenDisnon-commutative.SeealsoTadic´[21]foraclassificationofthe unitarydualofGL m (F). Ourproofcanbedecomposedintothreeparts.Inthefirstpart( § 3.1),wereduceto thecasewheretheunitaryirreduciblerepresentationoftheLevisubgroupissimplein thesenseof[12]:theinertialclassofitscuspidalsupportcontainsacuspidalpairofthe form(GL k (D) r ,ρ ⊗ r ),withm=krandwhereρisacuspidalirreduciblerepresentationof thegroupGL k (D)(seealsoDefinition1.3).Thisspecialcaseoftheconjectureisdenoted .0Syb Inthesecondpart,wetranslatetheproblemintermsofinductionofmodulesover Heckealgebras.Moreprecisely,wereducetheproofofS0toprovingthat,givenaninteger r>1andaLevisubgroupMofGL r (F),anyunitaryirreduciblemoduleovertheHecke- IwahorialgebraofM(thatis,theHeckealgebraofMrelativetosomeIwahorisubgroup) inducesirreduciblytotheHecke-IwahorialgebraofGL r (F)(seeProposition3.3).This stepdemandstheexistenceofsimpletypesforanyirreduciblesimplerepresentationof GL m (D).Suchsimpletypeshavebeenconstructedin[17,18,19,20]. ThelastpartoftheproofconsistsofprovingProposition3.3(seeabove).Thisstepis basedonaresultofBarbasch-Moy(see[5,6])whichassertsthatthefunctorofIwahori- invariantvectorsinducesaone-to-onecorrespondencebetweenunitaryirreduciblerepre- sentationsofGL r (F)havinganon-zerovectorinvariantunderanIwahorisubgroup,and unitaryirreduciblemodulesovertheHecke-IwahorialgebraofGL r (F).