Overview
- Explains topics that have been actively studied in the non-commutative harmonic analysis of solvable Lie groups
- Gives the classical standard results with proof related to the so-called orbit method
- Presents concrete examples that will help provide better understanding and ideas for further progress
Part of the book series: Springer Monographs in Mathematics (SMM)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
Table of contents (13 chapters)
-
Front Matter
-
Back Matter
Keywords
About this book
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, Frobeniusreciprocity, 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 G 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.
Reviews
“The publication of this monograph is obviously an important event for experts in harmonic analysis and especially for researchers working in the theory of representations of solvable Lie groups. … the appearance of this book is an important event which will strongly influence the development of the area.” (Antoni Wawrzyńczyk, Mathematical Reviews, October, 2015)
Authors and Affiliations
Bibliographic Information
Book Title: Harmonic Analysis on Exponential Solvable Lie Groups
Authors: Hidenori Fujiwara, Jean Ludwig
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-4-431-55288-8
Publisher: Springer Tokyo
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Japan 2015
Hardcover ISBN: 978-4-431-55287-1Published: 16 December 2014
Softcover ISBN: 978-4-431-56390-7Published: 02 August 2016
eBook ISBN: 978-4-431-55288-8Published: 05 December 2014
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XI, 465
Topics: Topological Groups, Lie Groups, Abstract Harmonic Analysis, Functional Analysis