Harmonic Analysis on Homogeneous Spaces

This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem.
Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.

Nolan R. Wallach is Professor of Mathematics at the University of California, San Diego. In early 2018, Dover will also publish the second edition of his Symplectic Geometry and Fourier Analysis.

PrefaceSuggestions to the Reader1. Vector Bundles2. Elementary Representation Theory3. Basic Structure Theory of Compact Lie Groups and Semisimple Lie Algebras4. The Topology and Representation Theory of Compact Lie Groups5. Harmonic Analysis on a Homogeneous Vector Bundle6. Holomorphic Vector Bundles over Flag Manifolds7. Analysis on Semisimple Lie Groups8. Representations of Semisimple Lie GroupsAppendix 1. Review of Differential GeometryAppendix 2. Lie GroupsAppendix 3. A Review of Multilinear AlgebraAppendix 4. Integration on ManifoldsAppendix 5. Complex ManifoldsAppendix 6. Elementary Functional AnalysisAppendix 7. Integral OperatorsAppendix 8. The Asymptotics for Certain Sturn-Liouville SystemsBibliographyIndex