Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.
Armand Borel’s mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel’s work influenced some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan, and Steenrod. In 1992, Borel was awarded the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas." He wrote more than 145 articles before 1982, which were collected in three volumes published in 1983. A fourth volume of subsequent articles was published in 2001. Volume I collects the papers written from 1948 to 1958.
Armond Borel’s influential seminar on transformation groups from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.
Algebraic groups and Lie groups are important in most major areas of mathematics, occuring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory. In this book, Professor Borel looks at the development of the theory of Lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. As the starting point of this passagefrom local to global, the author takes Lie's theory of local analytic transformation groups and Lie algebras. He then follows the globalization of the process in its two most important frameworks: (transcendental) differential geometry and algebraic geometry. Chapters II to IV are devoted to the former,Chapters V to VIII, to the latter.The essays in the first part of the book survey various proofs of the full reducibility of linear representations of $SL 2M$, the contributions H. Weyl to representation and invariant theory for Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large.The second part of the book starts with Chapter V describing the development of the theory of linear algebraic groups in the 19th century. Many of the main contributions here are due to E. Study, E. Cartan, and above all, to L. Maurer. After being abandoned for nearly 50 years, the theory was revived by Chevalley and Kolchin and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on various aspects of the works of Chevalley on Lie groupsand algebraic groups and Kolchin on algebraic groups and the Galois theory of differential fields.The author brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep appreciation of the underlying mathematics. His lifelong involvement and his historical research in the subject give him a special appreciation of the story of its development.
This book is a publication in Swiss Seminars, a subseries of Progress in Mathematics. It is an expanded version of the notes from a seminar on intersection cohomology theory, which met at the University of Bern, Switzerland, in the spring of 1983. This volume supplies an introduction to the piecewise linear and sheaf-theoretic versions of that theory as developed by M. Goresky and R. MacPherson in Topology 19 (1980), and in Inventiones Mathematicae 72 (1983). Some familiarity with algebraic topology and sheaf theory is assumed.
Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology
It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades.
This text describes the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in the extended Dynkin diagram of the simply connected cover, together with the co-root integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the moduli space.
This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researchers in analytic number theory will find much to interest them in this book.
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