**Kac–Moody Groups, Their Flag Varieties and Representation Theory** by Shrawan Kumar is a comprehensive treatment of Kac–Moody algebras, their groups, and representation theory. The book provides a detailed exploration of the structure and properties of Kac–Moody algebras, their associated groups, and the geometry of their flag varieties. It covers a wide range of topics, including the Weyl–Kac character formula, the study of Schubert varieties, the Borel–Weil–Bott theorem, the BGG and Kempf resolutions, and the topology of Kac–Moody groups and their flag varieties. The book also introduces the concepts of ind-varieties, pro-groups, and Tits systems, and discusses the rational smoothness and normality of Schubert varieties. It includes a chapter on the explicit realization of affine Kac–Moody algebras and their groups, which are the most important subclass of Kac–Moody algebras beyond the finite case. The book is written for advanced graduate students and researchers in mathematics, and it is self-contained, with appendices covering algebraic geometry, topology, homological algebra, and spectral sequences. The book is devoted to the treatment of the theory over the complex numbers or an algebraically closed field of characteristic 0, and does not use any characteristic p methods. It is suitable for an advanced graduate course. The book is dedicated to the author's parents and includes acknowledgments to various individuals and institutions who contributed to its completion.**Kac–Moody Groups, Their Flag Varieties and Representation Theory** by Shrawan Kumar is a comprehensive treatment of Kac–Moody algebras, their groups, and representation theory. The book provides a detailed exploration of the structure and properties of Kac–Moody algebras, their associated groups, and the geometry of their flag varieties. It covers a wide range of topics, including the Weyl–Kac character formula, the study of Schubert varieties, the Borel–Weil–Bott theorem, the BGG and Kempf resolutions, and the topology of Kac–Moody groups and their flag varieties. The book also introduces the concepts of ind-varieties, pro-groups, and Tits systems, and discusses the rational smoothness and normality of Schubert varieties. It includes a chapter on the explicit realization of affine Kac–Moody algebras and their groups, which are the most important subclass of Kac–Moody algebras beyond the finite case. The book is written for advanced graduate students and researchers in mathematics, and it is self-contained, with appendices covering algebraic geometry, topology, homological algebra, and spectral sequences. The book is devoted to the treatment of the theory over the complex numbers or an algebraically closed field of characteristic 0, and does not use any characteristic p methods. It is suitable for an advanced graduate course. The book is dedicated to the author's parents and includes acknowledgments to various individuals and institutions who contributed to its completion.