Harmonic Analysis on Exponential Solvable Lie Groups / / by Hidenori Fujiwara, Jean Ludwig

• Autore: Fujiwara Hidenori
• Pubblicazione: Tokyo : , : Springer Japan : , : Imprint : Springer, , 2015
• Edizione: 1st ed. 2015.
• Descrizione fisica: 1 online resource (468 p.)
• Disciplina: 515.2433
• Soggetto topico: Topological groups; Lie groups; Harmonic analysis; Functional analysis; Topological Groups, Lie Groups; Abstract Harmonic Analysis; Functional Analysis
• Persona (resp. second.): LudwigJean
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Preface; Contents; 1 Preliminaries: Lie Groups and Lie Algebras; 1.1 Lie Groups and Lie Algebras; 1.2 Enveloping Algebra; 1.3 Unitary Representations; 2 Haar Measure and Group Algebra; 2.1 The Haar Measure of a Locally Compact Group; 2.2 The Group Algebra; 2.3 Representations; 3 Induced Representations; 3.1 Measures on Quotient Spaces; 3.2 Definition of an Induced Representation; 3.3 Conjugation of Induced Representations; 3.4 The Imprimitivity Theorem; 4 Four Exponential Solvable Lie Groups; 4.1 The Group Rn; 4.2 The Heisenberg Group; 4.2.1 Induced Representations; 4.3 The ``ax +b '' Group ; 4.3.1 Induced Representations4.3.2 The Plancherel Theorem; 4.3.3 dπ for the Enveloping Algebra; 4.4 Grélaud's Group; 4.4.1 Induced Representations; 4.4.2 dπμ for the Enveloping Algebra; 4.4.3 The Imprimitivity Theorem for Grélaud's Group; 4.4.4 The Plancherel Theorem; 5 Orbit Method; 5.1 Auslander-Kostant Theory; 5.2 Exponential Solvable Lie Groups; 5.2.1 Co-exponential Sequence; 5.2.2 The Exponential Mapping for Solvable Lie Groups; 5.3 Kirillov-Bernat Mapping; 5.4 Pukanszky Condition; 6 Kirillov Theory for Nilpotent Lie Groups; 6.1 Jordan-Hölder and Malcev Sequences ; 6.1.1 Unipotent Representations6.1.2 Polynomial Vector Groups; 6.1.3 Unipotent Orbits; 6.2 Schwartz Spaces; 6.3 Intertwining Operator for Irreducible Representations; 6.4 Traces and Plancherel Theorem; 6.5 Parametrization of All Orbits; 7 Holomorphically Induced Representations ρ(f,h,G) for Exponential Solvable Lie Groups; 7.1 First Trial; 7.2 Exponential j-Algebras; 7.3 Exponential Kähler Algebras; 7.4 Structure of Positive Polarizations; 7.5 Non-vanishing of H(f,h,G); 7.6 Irreducibility and Equivalence of ρ(f,h,G); 7.7 Decomposition of ρ(f,h,G); 8 Irreducible Decomposition ; 8.1 Monomial Representations8.2 Restriction of Unitary Representations to Subgroups; 9 e-Central Elements; 9.1 Fundamental Result of Corwin and Greenleaf; 9.2 Monomial Representations and e-Central Elements; 10 Frobenius Reciprocity; 10.1 Frobenius Reciprocity in Distribution Version; 10.2 Realization of the Reciprocity in Nilpotent Case; 10.3 Examples of Semi-invariant Distributions; 11 Plancherel Formula; 11.1 Penney's Plancherel Formula; 11.2 Vergne's Decomposition Theorem; 11.3 Examples of Penney's Plancherel Formula; 11.4 Bonnet's Plancherel Formula ; 12 Commutativity Conjecture: Induction Case12.1 Toward the Commutativity Conjecture; 12.2 Corwin-Greenleaf Functions; Key Lemmas; 12.3 Proof of the Commutativity Conjecture; 13 Commutativity Conjecture: Restriction Case; 13.1 Kernel of Representation; 13.2 Commutativity of the Algebra Dπ(G)K; 13.3 Commutativity of the Centralizer of Lie Subalgebras; References; List of Notations; Index
• Sommario/riassunto: This book is the first one that brings together recent results on the harmonic analysis of exponential solvable Lie groups. There still are many interesting open problems, and the book contributes to the future progress of this research field. As well, various related topics are presented to motivate young researchers. The orbit method invented by Kirillov is applied to study basic problems in the analysis on exponential solvable Lie groups. This method tells us that the unitary dual of these groups is realized as the space of their coadjoint orbits. This fact is established using the Mackey theory for induced representations, and that mechanism is explained first. One of the fundamental problems in the representation theory is the irreducible decomposition of induced or restricted representations. Therefore, these decompositions are studied in detail before proceeding to various related problems: the multiplicity formula, Plancherel formulas, intertwining operators, Frobenius reciprocity, and associated algebras of invariant differential operators. The main reasoning in the proof of the assertions made here is induction, and for this there are not many tools available. Thus a detailed analysis of the objects listed above is difficult even for exponential solvable Lie groups, and it is often assumed that the group is nilpotent. To make the situation clearer and future development possible, many concrete examples are provided. Various topics presented in the nilpotent case still have to be studied for solvable Lie groups that are not nilpotent. They all present interesting and important but difficult problems, however, which should be addressed in the near future. Beyond the exponential case, holomorphically induced representations introduced by Auslander and Kostant are needed, and for that reason they are included in this book. .
• Titolo autorizzato: Harmonic Analysis on Exponential Solvable Lie Groups
• ISBN: 4-431-55288-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910299762903321
• Lo trovi qui: Univ. Federico II
• Serie: Springer Monographs in Mathematics, . 1439-7382