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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
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Harmonic Analysis on Exponential Solvable Lie Groups (Paperback, Softcover reprint of the original 1st ed. 2015)
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Harmonic Analysis on Exponential Solvable Lie Groups (Paperback, Softcover reprint of the original 1st ed. 2015)
Series: Springer Monographs in Mathematics
Expected to ship within 10 - 15 working days
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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.
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