Lie Groups

Preface Part I: Compact Groups: Haar Measure Schur Orthogonality Compact Operators The Peter-Weyl Theorem Part II: Lie Group Fundamentals: Lie Subgroups of GL(n, C) Vector Fields Left Invariant Vector Fields The Exponential Map Tensors and Universal Properties The Universal Enveloping Algebra Extension of Scalars Representations of sl(2, C) The Universal Cover The Local Frobenius Theorem Tori Geodesics and Maximal Tori Topological proof of Cartan s Theorem The Weyl Integration Formula The Root System Examples of Root Systems Abstract Weyl Groups The Fundamental Group Semisimple Compact Groups Highest Weight Vectors The Weyl Character Formula Spin Complexification Coxeter Groups The Iwasawa Decomposition The Bruhat Decomposition Symmetric Spaces Relative Root Systems. Embeddings of Lie Groups Part III: Frobenius-Schur Duality: Mackey Theory Characters of GL(n, C) Duality between Sk and GL(n, C) The Jacobi-Trudi Identity Schur Polynomials and GL(n, C) Schur Polynomials and Sk Random Matrix Theory Minors of Toeplitz Matrices Branching Formulae and Tableaux The Cauchy Identity Unitary branching rules The Involution Model for Sk Some Symmetric Algebras Gelfand Pairs Hecke Algebras Cohomology of Grassmannians References

From the reviews:"This book is a nice and rich introduction to the beautiful theory of Lie groups and its connection to many other areas of mathematics." (Karl-Hermann Neeb, Mathematical Reviews, 2005f)"As Lie theory prerequisites can pose a great hurdle to number-theory students attracted to this program, Bump's book will find an enthusiastic clientele even in an already crowded market. It will particularly delight readers who already know some of this material: the many short chapters generally begin with a map of the precise regress necessary to start wherever one ought. Summing Up: Highly recommended." (D.V. Feldman, CHOICE, Vol. 42 (8), April, 2005)"This book is intended for a one-year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups ... and provides a carefully chosen range of material to give the student the bigger picture." (L'Enseignement Mathematique, Vol. 50 (3-4), 2004)"This book aims to be a course in Lie groups that can be covered in one year with a group of seasoned graduate students. ... offers a wealth of complementary, partly quite recent material that is not found in any other textbook on Lie groups. ... this book covers an unusually wide spectrum of topics ... . the entire presentation is utmost thorough, comprehensive, lucid and absolutely user-friendly. ... All together, this graduate text his a highly interesting, valuable and welcome addition ... . (Werner Kleinert, Zentralblatt MATH, Vol. 1053, 2005)"Reductive Lie groups and their representations form a very broad field. The aim of the book is to select essential topics for a year course for graduate students ... . The book is nicely written and efficiently organized. ... The presented book brings a beautiful selection of a number of further important additional topics, which are worth to include into a course. It is a very important addition to existing literature on the subject." (EMS Newsletter, June, 2005)"This