Harmonic Analysis on Exponential Solvable Lie Groups

  • Hidenori Fujiwara
  • Jean Ludwig

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Hidenori Fujiwara, Jean Ludwig
    Pages 1-28
  3. Hidenori Fujiwara, Jean Ludwig
    Pages 29-51
  4. Hidenori Fujiwara, Jean Ludwig
    Pages 53-81
  5. Hidenori Fujiwara, Jean Ludwig
    Pages 83-115
  6. Hidenori Fujiwara, Jean Ludwig
    Pages 117-166
  7. Hidenori Fujiwara, Jean Ludwig
    Pages 167-208
  8. Hidenori Fujiwara, Jean Ludwig
    Pages 289-315
  9. Hidenori Fujiwara, Jean Ludwig
    Pages 317-331
  10. Hidenori Fujiwara, Jean Ludwig
    Pages 333-341
  11. Hidenori Fujiwara, Jean Ludwig
    Pages 343-382
  12. Hidenori Fujiwara, Jean Ludwig
    Pages 383-430
  13. Hidenori Fujiwara, Jean Ludwig
    Pages 431-453
  14. Back Matter
    Pages 455-465

About this book

Introduction

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.

 

Keywords

Exponential solvable Lie group Induced representation Nilpotent Lie group Orbit method Restriction of representation

Authors and affiliations

  • Hidenori Fujiwara
    • 1
  • Jean Ludwig
    • 2
  1. 1.Dpt. of Info. and C.S.Kinki UniversityIizukaJapan
  2. 2.Université de MetzMetzFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-4-431-55288-8
  • Copyright Information Springer Japan 2015
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-4-431-55287-1
  • Online ISBN 978-4-431-55288-8
  • Series Print ISSN 1439-7382
  • Series Online ISSN 2196-9922
  • About this book