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Géométrie Et Quantification De Paires De Howe D'actions Symplectiques

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Géométrie Et Quantification De Paires De Howe D'actions Symplectiques Géométrie et quantification de paires de Howe d’actions symplectiques Carsten Balleier To cite this version: Carsten Balleier. Géométrie et quantification de paires de Howe d’actions symplectiques. Mathéma- tiques générales [math.GM]. Université Paul Verlaine - Metz, 2009. Français. NNT : 2009METZ016S. tel-01752633 HAL Id: tel-01752633 https://hal.univ-lorraine.fr/tel-01752633 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AVERTISSEMENT Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie. Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l’utilisation de ce document. D'autre part, toute contrefaçon, plagiat, reproduction illicite encourt une poursuite pénale. Contact : [email protected] LIENS Code de la Propriété Intellectuelle. articles L 122. 4 Code de la Propriété Intellectuelle. articles L 335.2- L 335.10 http://www.cfcopies.com/V2/leg/leg_droi.php http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm UNIVERSITE´ PAUL VERLAINE UNIVERSITAT¨ PADERBORN METZ Ecole´ Doctorale Fakult¨atf¨urElektrotechnik, IAEM Lorraine Informatik und Mathematik THESE` DE DOCTORAT DISSERTATION Discipline : Math´ematiques im Fach Mathematik pr´esent´eepar vorgelegt von Carsten Balleier pour obtenir le grade de zur Erlangung des Doktorgrades Docteur de l’Universit´ePaul Verlaine-Metz der Universit¨at Paderborn Geometry and Quantization of Howe Pairs of Symplectic Actions Soutenance publique Offentliche¨ Verteidigung le 1er Juillet 2009 am 1. Juli 2009 Rapporteurs: Gutachter: Prof. Alan T. Huckleberry, Ruhr-Universit¨at Bochum Juan-Pablo Ortega, Universit´ede Franche-Comt´e`aBesan¸con Jury: Promotionskommission: M. Helge Glockner¨ Professor, Universit¨atPaderborn Mme Simone Gutt Professeur, Universit´ede Metz M. S¨onke Hansen Professor, Universit¨atPaderborn M. Joachim Hilgert Professor, Universit¨atPaderborn M. Alan T. Huckleberry Professor, Ruhr-Universit¨atBochum M. Martin Olbrich Professor, Universit¨atLuxemburg M. Marcus Slupinski Maˆıtrede Conf´erences, Universit´ede Strasbourg M. Tilmann Wurzbacher Professeur, Universit´ede Metz Laboratoire de Math´ematiques et Applications de Metz, Ile du Saulcy, F-57045 Metz Cedex 1 Institut f¨urMathematik, Fakult¨atEIM, Universit¨atPaderborn, Warburger Str. 100, D-33098 Paderborn Abstract Motivated by the representation-theoretic notion of Howe duality, we seek an analo- gous construction in symplectic geometry in the sense that its geometric quantization decomposes in a Howe dual fashion. We find that in the symplectic context, the correct setting is given by two Lie groups acting on a symplectic manifold when these two actions commute and satisfy the symplectic Howe condition, i. e., these actions are Hamiltonian and their collective functions are their mutual centralizers in the Poisson algebra of smooth functions on the symplectic manifold. Once this condition is satisfied, we can describe the orbit structure in detail. In particular, there is a bijection between the coadjoint orbits in one moment image and those in the other moment image – this bijection is what we call the coadjoint orbit correspondence. We study the coadjoint orbit correspondence further and show, if the acting Lie groups are compact and the symplectic manifold is prequantizable, that it preserves integrality of the coadjoint orbits, so to both coadjoint orbits in the correspondence an irreducible representation can be associated. We thus have a bijection between certain parts of the unitary duals of both Lie groups acting on the symplectic manifold. Applying known results about the interchangeability of quantization and reduction, we see that for a K¨ahlermanifold, its quantization (as a representation of the product of both groups acting on the manifold) decomposes into a multiplicity-free direct sum of tensor products of irreducibles of the individual groups, the pairs being given by the bijection obtained before – as one would expect according to Howe duality. This main result is accompanied by a study of the local structure of a manifold carrying two commuting Hamiltonian action which proves a local version of the orbit correspondence and by a discussion about the relation of the coadjoint orbit correspon- dence to the generalized symplectic leaf correspondence in singular dual pairs. Zusammenfassung Motiviert durch den darstellungstheoretischen Begriff der Howe-Dualit¨at, suchen wir eine analoge Konstruktion in der symplektischen Geometrie. Analog bedeutet hierbei, dass die geometrische Quantisierung eine Zerlegung mit Howe-Dualit¨at besitzen soll. Wir stellen fest, dass die im symplektischen Kontext korrekte Situation gegeben ist durch zwei Lie-Gruppen, die auf derselben symplektischen Mannigfaltigkeit wirken, wenn diese Wirkungen kommutieren und die symplektische Howe-Bedingung erfullen,¨ d. h. beide Wirkungen sind Hamiltonsch und die kollektiven Funktionen beider Wirkun- gen sind gegenseitig ihre Zentralisatoren in der Poisson-Algebra der glatten Funktionen auf der symplektischen Mannigfaltigkeit. Ist diese Bedingung erfullt,¨ dann sind wir in der Lage, die Bahnenstruktur detailliert zu beschreiben und zu zeigen, dass eine Bi- jektion zwischen den koadjungierten Bahnen im Bild der ersten Impulsabbildung und denen im Bild der zweiten Impulsabbildung existiert – es ist diese Bijektion, die wir im folgenden als Korrespondenz koadjungierter Bahnen bezeichnen. Wir setzen die Untersuchung der Korrespondenz koadjungierter Bahnen fort und zeigen, dass fur¨ Wirkungen kompakter Lie-Gruppen auf pr¨aquantisierbaren symplek- tischen Mannigfaltigkeiten die Integralit¨at der koadjungierten Bahnen erhalten bleibt, und daher beiden koadjungierten Bahnen gleichzeitig irreduzible Darstellungen zuge- ordnet werden k¨onnen. Somit besteht eine Bijektion zwischen bestimmten Teilmen- gen der unit¨aren Duale beider auf der symplektischen Mannigfaltigkeit wirkenden Lie- Gruppen. Wendet man nun bekannte Resultate uber¨ die Vertauschbarkeit von Quanti- sierung und symplektischer Reduktion an, dann erkennen wir, dass die Quantisierung einer K¨ahler-Mannigfaltigkeit (betrachtet als Darstellung des Produktes beider auf der Mannigfaltigkeit wirkender Gruppen) in eine multiplizit¨atenfreie direkte Summe von Tensorprodukten der irreduziblen Darstellungen beider Gruppen zerf¨allt, wobei die Paare durch die zuvor beschriebene Bijektion gegeben sind – wie man es im Sinne der Howe-Dualit¨at erwartet. Dieses Hauptresultat wird begleitet von der Untersuchung der lokalen Struktur einer Mannigfaltigkeit, auf der zwei Hamiltonsche Wirkungen gegeben sind, die eine lokale Version der Bahnenkorrespondenz liefert, sowie von einer Betrachtung der Bezie- hung der Korrespondenz koadjungierter Bahnen zur Korrespondenz verallgemeinerter symplektischer Bl¨atter in singul¨aren dualen Paaren. R´esum´e Motiv´epar la dualit´ede Howe dans la th´eoriedes repr´esentations de groupes de Lie, on cherche une construction analogue en g´eom´etriesymplectique, c’est-`a-direon souhaite que sa quantification g´eom´etrique d´ecompose de mani`ereHowe-duale. On trouve que dans le contexte symplectique, le cadre correct est donn´epar deux groupes de Lie agissant sur la mˆemevari´et´esymplectique si ces actions commutent et satisfont la condition de Howe symplectique, i. e., ces actions sont hamiltoniennes et leurs fonctions collectives sont leurs centralisateurs mutuelles dans l’alg`ebre de Poisson des fonctions lisses sur la vari´et´esymplectique. Une fois cette condition est remplie, nous pouvons d´ecrirela structure d’orbites en d´etail. En particulier, il y a une bijec- tion entre les orbites coadjointes dans une image d’application moment et celles dans l’image de l’autre application moment – or, il est cette bijection que nous appelerons la correspondance d’orbites coadjointes. On poursuit l’´etudede la correspondance d’orbites coadjointes et on montre que, si les groupes de Lie qui agissent sont compacts et la vari´et´esymplectique est pr´equanti- fiable, l’integralit´eest pr´eserv´eepar la correspondance. Ainsi, il est possible d’associer en mˆemetemps des repr´esentations irr´eductibles aux deux orbites de la correspondance. Donc, nous avons une bijection entre certaines parties des duaux unitaires des deux groupes de Lie qui agissent sur la vari´et´esymplectique. En appliquant des r´esultats connus qui assurent que la quantification et la r´eduction commutent, nous consta- tons que la quantification d’une vari´et´ek¨ahlerienne(vue comme une repr´esentation du produit des deux groupes qui agissent sur la vari´et´e)admet une d´ecomposition en somme direct sans multiplicit´esde produits tensoriels des repr´esentations irr´eductibles des deux groupes, les paires ´etant donn´eespar la bijection obtenue pr´ec´edemment – parfaitement en accord avec la dualit´ede Howe. Ce r´esultatprincipal est accompagn´epar l’´etude de la structure locale d’une vari´et´e avec deux
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