Harmonic Analysis on Semi-Simple Lie Groups I is a comprehensive work by Garth Warner, published in 1972. It provides a self-contained introduction to Harish-Chandra's theory of representations of semi-simple Lie groups. The book is structured into chapters, sections, and numbers, with detailed theorems, proofs, and examples. It includes appendices with additional results and a guide to the literature. The book begins with an overview of the subject, discussing the representation theory of locally compact groups, with a focus on semi-simple Lie groups. It introduces the concept of Plancherel measure, which is central to harmonic analysis on these groups. The measure is defined in terms of unitary equivalence classes of irreducible unitary representations and is crucial for determining the structure of the representation theory. The text covers the structure of real semi-simple Lie groups, the universal enveloping algebra of a semi-simple Lie algebra, and representations on Banach and Hilbert spaces. It discusses differentiable and analytic vectors, large compact subgroups, and induced representations. The book also addresses the theory of spherical functions, the topology on the dual, and the discrete series for semi-simple Lie groups. The work includes detailed proofs and systematic exposition, with references to various mathematical concepts and theorems. It is aimed at mathematicians with a background in algebra and analysis, and it provides a foundation for understanding the representation theory of semi-simple Lie groups. The book is complemented by a second volume, which continues the discussion of spherical functions and other related topics. The text is well-organized, with a clear structure and detailed explanations, making it an essential resource for researchers and students in the field of harmonic analysis and representation theory.Harmonic Analysis on Semi-Simple Lie Groups I is a comprehensive work by Garth Warner, published in 1972. It provides a self-contained introduction to Harish-Chandra's theory of representations of semi-simple Lie groups. The book is structured into chapters, sections, and numbers, with detailed theorems, proofs, and examples. It includes appendices with additional results and a guide to the literature. The book begins with an overview of the subject, discussing the representation theory of locally compact groups, with a focus on semi-simple Lie groups. It introduces the concept of Plancherel measure, which is central to harmonic analysis on these groups. The measure is defined in terms of unitary equivalence classes of irreducible unitary representations and is crucial for determining the structure of the representation theory. The text covers the structure of real semi-simple Lie groups, the universal enveloping algebra of a semi-simple Lie algebra, and representations on Banach and Hilbert spaces. It discusses differentiable and analytic vectors, large compact subgroups, and induced representations. The book also addresses the theory of spherical functions, the topology on the dual, and the discrete series for semi-simple Lie groups. The work includes detailed proofs and systematic exposition, with references to various mathematical concepts and theorems. It is aimed at mathematicians with a background in algebra and analysis, and it provides a foundation for understanding the representation theory of semi-simple Lie groups. The book is complemented by a second volume, which continues the discussion of spherical functions and other related topics. The text is well-organized, with a clear structure and detailed explanations, making it an essential resource for researchers and students in the field of harmonic analysis and representation theory.