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Intermediate in level between an advanced textbook and a monograph, this book covers both the classical and representation theoretic views of automorphic forms in a style which is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin-Selberg method and the triple L-function, examining this subject matter from many different and complementary viewpoints. Researchers as well as students will find this a valuable guide to a notoriously difficult subject.

Automorphic forms. --- Representations of groups. --- Lie groups.

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This book is the result of many years of research in Non-Euclidean Geometries and Geometry of Lie groups, as well as teaching at Moscow State University (1947- 1949), Azerbaijan State University (Baku) (1950-1955), Kolomna Pedagogical Col­ lege (1955-1970), Moscow Pedagogical University (1971-1990), and Pennsylvania State University (1990-1995). My first books on Non-Euclidean Geometries and Geometry of Lie groups were written in Russian and published in Moscow: Non-Euclidean Geometries (1955) [Ro1] , Multidimensional Spaces (1966) [Ro2] , and Non-Euclidean Spaces (1969) [Ro3]. In [Ro1] I considered non-Euclidean geometries in the broad sense, as geometry of simple Lie groups, since classical non-Euclidean geometries, hyperbolic and elliptic, are geometries of simple Lie groups of classes Bn and D , and geometries of complex n and quaternionic Hermitian elliptic and hyperbolic spaces are geometries of simple Lie groups of classes An and en. [Ro1] contains an exposition of the geometry of classical real non-Euclidean spaces and their interpretations as hyperspheres with identified antipodal points in Euclidean or pseudo-Euclidean spaces, and in projective and conformal spaces. Numerous interpretations of various spaces different from our usual space allow us, like stereoscopic vision, to see many traits of these spaces absent in the usual space.

Lie groups --- Geometry --- Groupes de Lie --- Géométrie --- 514.747 --- 512.81 --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Mathematics --- Euclid's Elements --- Geometric objects. Representations of Lie groups. Lie differentiation --- 512.81 Lie groups --- 514.747 Geometric objects. Representations of Lie groups. Lie differentiation --- Géométrie --- Geometry. --- Topological groups. --- Lie groups. --- Topological Groups, Lie Groups. --- Groups, Topological --- Continuous groups

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From the reviews: "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..." Zentralblatt für Mathematik und Ihre Grenzgebiete, 1991 "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensable for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer." Nieuw Archief voor Wiskunde, 1992 "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra.... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modern algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers..." The Mathematical Gazette, 1991 "... According to the preface, the book is addressed to "students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics." I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit." Zeitschrift für Kristallographie, 1991.

Algebra. --- Mathematics. --- Math --- Science --- Mathematics --- Mathematical analysis --- Algebra --- Topological Groups. --- K-theory. --- Topological Groups, Lie Groups. --- K-Theory. --- Algebraic topology --- Homology theory --- Groups, Topological --- Continuous groups --- Topological groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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This book is a compilation of several works from well-recognized figures in the field of Representation Theory. The presentation of the topic is unique in offering several different points of view, which should makethe book very useful to students and experts alike.Presents several different points of view on key topics in representation theory, from internationally known experts in the field

Lie groups. --- Representations of groups. --- Group representation (Mathematics) --- Groups, Representation theory of --- Groups, Lie --- Group theory --- Lie algebras --- Symmetric spaces --- Topological groups --- Lie, Groupes de --- Lie, Algèbres de. --- Lie groups --- Representations of Lie groups --- Representations of Lie algebras --- Représentations de groupes de Lie. --- Représentations d'algèbres de Lie. --- Lie, Algèbres de. --- Représentations de groupes de Lie. --- Représentations d'algèbres de Lie.

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Representations of Lie groups --- Representations of Lie algebras --- Représentations de groupes de Lie --- Représentations d'algèbres de Lie --- Nombres, Théorie des --- Formes automorphes --- Number theory --- Automorphic forms --- Représentations de groupes de Lie --- Représentations d'algèbres de Lie --- Nombres, Théorie des