Mathematical Surveys and Monographs 1986; 328 pp; softcover Volume: 22 Reprint/Revision History: third printing 1999 ISBN10: 0821815237 ISBN13: 9780821815236 List Price: US$45 Member Price: US$36 Order Code: SURV/22
 This book explores some basic roles of Lie groups in linear analysis, with particular emphasis on the generalizations of the Fourier transform and the study of partial differential equations. It began as lecture notes for a onesemester graduate course given by the author in noncommutative harmonic analysis. It is a valuable resource for both graduate students and faculty, and requires only a background with Fourier analysis and basic functional analysis, plus the first few chapters of a standard text on Lie groups. The basic method of noncommutative harmonic analysis, a generalization of Fourier analysis, is to synthesize operators on a space on which a Lie group has a unitary representation from operators on irreducible representation spaces. Though the general study is far from complete, this book covers a great deal of the progress that has been made on important classes of Lie groups. Unlike many other books on harmonic analysis, this book focuses on the relationship between harmonic analysis and partial differential equations. The author considers many classical PDEs, particularly boundary value problems for domains with simple shapes, that exhibit noncommutative groups of symmetries. Also, the book contains detailed work, which has not previously been published, on the harmonic analysis of the Heisenberg group and harmonic analysis on cones. Reviews "Could be used as a text in a course ... many people will find this book valuable as a reference and as an introduction to the literature."  Mathematical Reviews Table of Contents  Some basic concepts of Lie group representation theory
 The Heisenberg group
 The unitary group
 Compact Lie groups
 Harmonic analysis on spheres
 Induced representations, systems of imprimitivity, and semidirect products
 Nilpotent Lie groups
 Harmonic analysis on cones
 \(\mathrm {SL}(2,R)\)
 \(\mathrm {SL}(2, \mathbf C)\), and more general Lorentz groups
 Groups of conformal transformations
 The symplectic group and the metaplectic group
 Spinors
 Semisimple Lie groups
