EAN: 9784431552871

HARMONIC ANALYSIS ON EXPONENTIAL SOLVABLE LIE G...

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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