Includes bibliographical references (pages535-544) and index.
Contents:
Compact groups
Haar measure
Schur orthogonality
Compact operators
The Peter-Weyl theorem
Compact lie groups
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 s1 (2, C)
The universal cover
The local Frobenius theorem
Tori
Geodesics and maximal tori
The Weyl integration formula
The root system
Examples of root systems
Abstract Weyl groups
Highest weight vectors
The Weyl character formula
The fundamental group
Noncompact lie groups
Complexification
Coxeter groups
The borel subgroup
The Bruhat decomposition
Symmetric spaces
Relative root systems
Embeddings of lie groups
Spin
Duality and other topics
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
The Cauchy identity
Random matrix theory
Symmetrie group branching rules and tableaux
Unitary branching rules and tableaux
Minors of Toeplitz matrices
The involution model for Sk
Some symmetric algebras
Gelfand pairs
Hecke algebras
The philosophy of cusp forms
Cohomology of grassmannians
Appendix : Sage
References
Index.
Summary:
"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 (which is the basis of many texts) and provides a carefully chosen range of material to give the student the bigger picture. For compact Lie groups, the Peter-Weyl theorem, conjugacy of maximal tori (two proofs), Weyl character formula and more are covered. The book continues with the study of complex analytic groups, then general noncompact Lie groups, including the Coxeter presentation of the Weyl group, the Iwasawa and Bruhat decompositions, Cartan decomposition, symmetric spaces, Cayley transforms, relative root systems, Satake diagrams, extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a "topics" section giving depth to the student's understanding of representation theory, taking the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme, with many applications in diverse areas such as random matrix theory, minors of Toeplitz matrices, symmetric algebra decompositions, Gelfand pairs, Hecke algebras, representations of finite general linear groups and the cohomology of Grassmannians and flag varieties.