Non-Abelian harmonic analysis. Applications of \(SL(2,{\mathbb{R}})\)
type: Item
instance of: scholarly article
title: Non-Abelian harmonic analysis. Applications of \(SL(2,{\mathbb{R}})\)
zbMATH Open document ID: 0768.43001
published in: Universitext
publication date: 1992-09-18T00:00:00Z
review text: Although this book appears in the Springer ``Universitext'' series, this introduction to representation theory and some of its applications is more likely to find favour amongst postgraduates than undergraduates, since it requires quite a breadth of sophistication. The book contains five chapters. The first is a fast introduction to the necessary background: from algebra, some ideas of the theory of Lie groups and Lie algebras, and from analysis, notions of Fourier analysis and spectral theory. The second chapter involves a description of (some of) the indecomposable, rather than irreducible, representations of the Lie algebra \({\mathfrak sl}(2)\). Some of these come as a surprise: they are not submodules of the principal series, and so may well be unfamiliar even to those who have been exposed to \(SL(2,\mathbb{R})\) or more general semisimple groups. Another feature of Chapter 2 is that ``formal eigenfunctions'' are used: if \(H_ 1,X_ 1,Y_ 1\) and \(H_ 2,X_ 2,Y_ 2\) are different standard bases of \({\mathfrak sl}(2)\), then a module which is an algebraic direct sum of eigenvectors for \(H_ 1\) will not usually contain eigenvectors for \(H_ 2\); formal infinite sums of the eigenvectors for \(H_ 1\) must be allowed. The third chapter builds on the results of Chapter 2 to obtain a classification of the irreducible unitary representations of \(SL(2,\mathbb{R})\) and its universal covering group, following Bargmann's original approach, and introduces the oscillator (also well known as Weil) representation. The key fact, for the applications in the following chapter, is that \(\widetilde{SL}(2,\mathbb{R})\) has a unitary representation \(\pi\) on \(L^ 2(\mathbb{R}^ n)\) (or more generally \(L^ 2(\mathbb{R}^{p+q}))\), which commutes with the action of the orthogonal group \(O(n)\) (or the group \(O(p,q)\)), for which the Fourier transform appears as \(\pi(k)\) for a particular element \(k\) of \(\widetilde{SL}(2,\mathbb{R})\). Chapter 4 shows how representation theory …
reviewed by: Michael G. Cowling
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 22-02
zbMATH DE Number: 52848
zbMATH Keywords: Weil representation
zbMATH Keywords: oscillator representation
zbMATH Keywords: Lie groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: Fourier analysis
zbMATH Keywords: spectral theory
zbMATH Keywords: principal series
zbMATH Keywords: semisimple groups
zbMATH Keywords: standard bases
zbMATH Keywords: eigenvectors
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: universal covering group
zbMATH Keywords: orthogonal group
zbMATH Keywords: Fourier transform
zbMATH Keywords: Bochner's periodicity theorem
zbMATH Keywords: local integrability of characters
author: Roger Howe
author: Eng-Chye Tan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 52848