Armand Borel

Nolan R. Wallach

Deals with two types of cohomology spaces pertaining to reductive Lie group G and a discrete cocompact subgroup. Material presented here naturally divides into two parts, one devoted mainly to real Lie groups, the other to locally compact totally disconnected groups, in particular reductive p-adic groups, or products of real Lie groups and totally disconnected groups. Each part in turn contains roughly three main items: general results on the cohomology used, specific results for cohomology and representations of reductive groups, and applications to discrete cocompact subgroups. This second edition reports on developments in the field since 1980. Annotation copyrighted by Book News, Inc., Portland, OR.

Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups

Roe Goodman

Presents an updated version of Weyl's invariant theory of the classical groups, together with many of the important recent developments.

Representations and Invariants of the Classical Groups

Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.

Symmetry, Representations, and Invariants

Edward Frenkel

Masaki Kashiwara

Alain Valette

Michael Cowling

David A. Vogan

Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.

Lectures given at the C.I.M.E. Summer School held in Venice, Italy, June 10-17, 2004

Representation Theory and Complex Analysis

Lectures Given at the C.I.M.E. Summer School Held in Venice, Italy, June 10-17, 2004

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

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