Generalized Functions, Volume 6

Representation Theory and Automorphic Functions

I. M. Gel'fand, M. I. Graev, I. I. Pyatetskii-Shapiro

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English

American Mathematical Society
30 September 2025

Calculus & mathematical analysis

Series: AMS Chelsea Publishing

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel'fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. The unifying theme of Volume 6 is the study of representations of the general linear group of order two over various fields and rings of number-theoretic nature, most importantly over local fields ($p$-adic fields and fields of power series over finite fields) and over the ring of adeles. Representation theory of the latter group naturally leads to the study of automorphic functions and related number-theoretic problems. The book contains a wealth of information about discrete subgroups and automorphic representations, and can be used both as a very good introduction to the subject and as a valuable reference.

By: I. M. Gel'fand, M. I. Graev, I. I. Pyatetskii-Shapiro
Imprint: American Mathematical Society
Country of Publication: United States [Currently unable to ship to USA: see Shipping Info]
ISBN: 9781470481155
ISBN 10: 1470481154
Series: AMS Chelsea Publishing
Pages: 426
Publication Date: 30 September 2025
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Forthcoming

Chapters Chapter 1. Homogeneous spaces with a discrete stability group Chapter 2. Representations of the group of unimodular matrices of order 2 with elements from a locally compact topological field Chapter 3. Representations of adele groups Guide to the literature