This volume, "Harmonic Analysis on Semi-Simple Lie Groups I," by Garth Warner, is part of the "Grundlehren der mathematischen Wissenschaften" series. It provides a comprehensive introduction to the representation theory of semi-simple Lie groups, a field largely developed by Harish-Chandra. The book is structured into several chapters, each covering specific aspects of the theory, including the structure of real semi-simple Lie groups, the universal enveloping algebra of a semi-simple Lie algebra, finite-dimensional representations, infinite-dimensional group representation theory, induced representations, and spherical functions. The author emphasizes the importance of the Plancherel measure and the discrete series in harmonic analysis on semi-simple Lie groups. The book also includes appendices that provide additional results and notational conventions, making it a valuable resource for researchers and students in the field of harmonic analysis and representation theory.This volume, "Harmonic Analysis on Semi-Simple Lie Groups I," by Garth Warner, is part of the "Grundlehren der mathematischen Wissenschaften" series. It provides a comprehensive introduction to the representation theory of semi-simple Lie groups, a field largely developed by Harish-Chandra. The book is structured into several chapters, each covering specific aspects of the theory, including the structure of real semi-simple Lie groups, the universal enveloping algebra of a semi-simple Lie algebra, finite-dimensional representations, infinite-dimensional group representation theory, induced representations, and spherical functions. The author emphasizes the importance of the Plancherel measure and the discrete series in harmonic analysis on semi-simple Lie groups. The book also includes appendices that provide additional results and notational conventions, making it a valuable resource for researchers and students in the field of harmonic analysis and representation theory.