This book, "Kac–Moody Groups, their Flag Varieties and Representation Theory" by Shrawan Kumar, is part of the "Progress in Mathematics" series and is published by Springer Science+Business Media, LLC. The author, Shrawan Kumar, is from the Department of Mathematics at the University of North Carolina, Chapel Hill. The book provides a comprehensive treatment of Kac–Moody groups and their flag varieties, focusing on both the algebraic and topological aspects. It covers a wide range of topics, including the basic theory of Kac–Moody algebras, representation theory, Lie algebra homology and cohomology, ind-varieties and pro-groups, Tits systems, and the construction of Kac–Moody groups and their flag varieties. The book also delves into the geometry of Schubert varieties, the BGG and Kempf resolutions, and the topology of Kac–Moody groups and their flag varieties. Key topics include the Weyl–Kac character formula, the Garland–Lepowsky result on n-homology, the Demazure character formula, the Borel–Weil–Bott theorem, and the study of the nil-Hecke ring. The book concludes with a chapter on affine Kac–Moody algebras and groups, which form an important subclass of Kac–Moody algebras beyond the finite case. The author dedicates the book to his parents and acknowledges the contributions of various individuals, including his brothers, wife, and children, as well as his teachers and mentors. The book is suitable for advanced graduate students and researchers in mathematics, particularly those interested in the theory of Kac–Moody groups and their applications.This book, "Kac–Moody Groups, their Flag Varieties and Representation Theory" by Shrawan Kumar, is part of the "Progress in Mathematics" series and is published by Springer Science+Business Media, LLC. The author, Shrawan Kumar, is from the Department of Mathematics at the University of North Carolina, Chapel Hill. The book provides a comprehensive treatment of Kac–Moody groups and their flag varieties, focusing on both the algebraic and topological aspects. It covers a wide range of topics, including the basic theory of Kac–Moody algebras, representation theory, Lie algebra homology and cohomology, ind-varieties and pro-groups, Tits systems, and the construction of Kac–Moody groups and their flag varieties. The book also delves into the geometry of Schubert varieties, the BGG and Kempf resolutions, and the topology of Kac–Moody groups and their flag varieties. Key topics include the Weyl–Kac character formula, the Garland–Lepowsky result on n-homology, the Demazure character formula, the Borel–Weil–Bott theorem, and the study of the nil-Hecke ring. The book concludes with a chapter on affine Kac–Moody algebras and groups, which form an important subclass of Kac–Moody algebras beyond the finite case. The author dedicates the book to his parents and acknowledges the contributions of various individuals, including his brothers, wife, and children, as well as his teachers and mentors. The book is suitable for advanced graduate students and researchers in mathematics, particularly those interested in the theory of Kac–Moody groups and their applications.