The Classical Groups and K-Theory
It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invar
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Editors M. Artin S. S. Chern J. Coates J. M. Frohlich H. Hironaka F. Hirzebruch L. Hormander S. MacLane c.c. Moore J.K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt S.Watanabe
Managing Editors M. Berger B. Eckmann S. R. S. Varadhan
Alexander J. Hahn O. Timothy O'Meara
The Classical Groups and K-Theory Foreword by J. Dieudonne
Springer-Verlag Berlin Heidelberg GmbH
Alexander J. Hahn Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA O. TImothy O'Meara Provost University of Notre Dame Notre Dame, IN 46556, USA
Mathematics Subject Classification (1980): 16-XX, 10 Cxx, 20-XX
ISBN 978-3-642-05737-3
Library of Congress Cataloging-in-Publication Data Hahn, Alexander J., 1943The classical groups and K-theory / Alexander J. Hahn, O. limothy O'Meara ; foreword by J. Dieudonne. p. cm. - (Grundlehren der mathematischen Wissenschaften ; 291) Bibliography : p.Includes indexes. ISBN 978-3-642-05737-3 ISBN 978-3-662-13152-7 (eBook) DOI 10.1007/978-3-662-13152-7 1. Linear algebraic groups. 2.K-theory. 1. O'Meara, O. T.(Onorato Tunothy), 1928-. II. litle. III. Series. QA171.H235 1989 512'.55-dc 19 88-11958 CIP This work is subject to copyright. Ali rights are reserved, whether the whole or pan of the material is concemed. specifically those oftranslation. reprinting, re-use of illustrations, recitation. broadcasting. reproduction on microfilms or in other ways, and storage in data banks. Duplication ofthis publication or pans thereofis only permitted under the provisions ofthe German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act ofthe German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Driginally published by Springer-Verlag Berlin Heidelberg New York in 1989
Softcover reprint of the hardcover 1st edition 1989 Typesetting : Thomson Press (India) Ltd., New Delhi 2141/3140-543210 Printed on acid-free paper
To Marianne and Jean
Foreword
It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for th
