An introduction to \(p\)-adic fields, harmonic analysis and the representation theory of \(\text{SL}_2\)
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.
scientific article; zbMATH DE number 1233918
scientific article