An introduction to \(p\)-adic fields, harmonic analysis and the representation theory of \(\text{SL}_2\)
type: Item
instance of: scholarly article
title: An introduction to \(p\)-adic fields, harmonic analysis and the representation theory of \(\text{SL}_2\)
zbMATH Open document ID: 0924.22009
DOI: 10.1023/A:1007583108067
author: Paul J. jun. Sally
published in: Letters in Mathematical Physics
publication date: 1998-12-21T00:00:00Z
review text: The study of harmonic analysis, originated in Fourier series and integrals, has been generalized on locally compact groups, and transformed into rich theories, including the theory of unitary representations of Lie groups. Especially, the techniques of harmonic analysis provide powerful and fruitful tools in mathematics and physics. In this expository article the author develops in considerable detail harmonic analysis on \(p\)-adic fields, from the nature of the underlying topology of \(Q_p\) to the representation theory of the \(p\)-adic \(SL_2\), and illustrates many important computational methods for \(p\)-adic harmonic analysis. This article is organized as follows: 0. Introduction, 1. The Definition of \(p\)-adic Fields, 2. Additive Structure of \(Q_p\), 3. Multiplicative Structure of \(Q_p\), 4. Characters of Locally Compact Abelian Groups, 5. Additive Characters, 6. Fourier Transform on \(Q_p^+\), 7. Multiplicative Characters, 8. Fourier Transforms on \(Q_p^\times\), 9. Finite Abelian Extension, 10. Special Functions, 11. Representations of Locally Compact Groups, 12. The Unitary Principal Series of \(p\)-adic \(SL_2\), 13. Discrete Series, 14. Complementary Series and the Steinberg Representation, 15. Closing Remarks. The reader, who is not familiar with \(p\)-adic fields and representation theory, can study basic results from 1 to 3, 9, and from 4, 11, respectively, and can easily understand the rest of the contents. Moreover, the reader, who has carefully studied many important methods described in this article, will be prepared to work on \(p\)-adic harmonic analysis. In this sense, as the author's previous expository article [Stud. Harmon. Anal., Proc. Conf. Chicago 1974, MAA Stud. Math. 13, 224-256 (1976; Zbl 0338.43011)], this article is very readable and wonderful.
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1233918
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: \(p\)-adic \(SL_2\)
zbMATH Keywords: Fourier transform
zbMATH Keywords: unitary principal series
zbMATH Keywords: Fourier series
zbMATH Keywords: locally compact groups
zbMATH Keywords: unitary representations
zbMATH Keywords: harmonic analysis
MaRDI profile type: Publication
full work available at URL: a:1007583108067
OpenAlex ID: W2912879977
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description: scientific article; zbMATH DE number 1233918