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The purpose of the book is to discuss the latest advances in the
theory of unitary representations and harmonic analysis for
solvable Lie groups. The orbit method created by Kirillov is the
most powerful tool to build the ground frame of these theories.
Many problems are studied in the nilpotent case, but several
obstacles arise when encompassing exponentially solvable settings.
The book offers the most recent solutions to a number of open
questions that arose over the last decades, presents the newest
related results, and offers an alluring platform for progressing in
this research area. The book is unique in the literature for which
the readership extends to graduate students, researchers, and
beginners in the fields of harmonic analysis on solvable
homogeneous spaces.
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 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.
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 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.
The aim of the Expositions is to present new and important
developments in pure and applied mathematics. Well established in
the community over more than two decades, the series offers a large
library of mathematical works, including several important
classics. The volumes supply thorough and detailed expositions of
the methods and ideas essential to the topics in question. In
addition, they convey their relationships to other parts of
mathematics. The series is addressed to advanced readers interested
in a thorough study of the subject. Editorial Board Lev Birbrair,
Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann,
Columbia University, New York, USA Markus J. Pflaum, University of
Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen,
Germany Katrin Wendland, University of Freiburg, Germany Honorary
Editor Victor P. Maslov, Russian Academy of Sciences, Moscow,
Russia Titles in planning include Yuri A. Bahturin, Identical
Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G.
Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups,
Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems
for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer,
Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical
Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia
Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces
(2021)
The purpose of the book is to discuss the latest advances in the
theory of unitary representations and harmonic analysis for
solvable Lie groups. The orbit method created by Kirillov is the
most powerful tool to build the ground frame of these theories.
Many problems are studied in the nilpotent case, but several
obstacles arise when encompassing exponentially solvable settings.
The book offers the most recent solutions to a number of open
questions that arose over the last decades, presents the newest
related results, and offers an alluring platform for progressing in
this research area. The book is unique in the literature for which
the readership extends to graduate students, researchers, and
beginners in the fields of harmonic analysis on solvable
homogeneous spaces.
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