Symmetry, Representations, and Invariants - Roe Goodman, Nolan R. Wallach.pdf

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Roe Goodman · Nolan R. Wallach
Symmetry,
Representations,
and Invariants
13
Graduate Texts in Mathematics
255
Editorial Board
S. Axler
K.A. Ribet
For other titles published in this series, go to
http://www.springer.com/series/136
Roe Goodman
Nolan R. Wallach
Symmetry, Representations,
and Invariants
123
Roe Goodman
Department of Mathematics
Rutgers University
Piscataway, NJ 08854-8019
USA
goodman@math.rutgers.edu
Nolan R. Wallach
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093
USA
nwallach@ucsd.edu
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
axler@sfsu.edu
K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA
ribet@math.berkeley.edu
ISBN 978-0-387-79851-6
e-ISBN 978-0-387-79852-3
DOI 10.1007/978-0-387-79852-3
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2009927015
Mathematics Subject Classification (2000): 20G05, 14L35, 14M17, 17B10, 20C30, 20G20, 22E10,
22E46, 53B20, 53C35, 57M27
Roe Goodman and Nolan R. Wallach 2009
Based on
Representations and Invariants of the Classical Groups,
Roe Goodman and Nolan R. Wallach,
Cambridge University Press, 1998, third corrected printing 2003.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Preface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Organization and Notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1
Lie Groups and Algebraic Groups
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 General and Special Linear Groups . . . . . . . . . . . . . . . . . . . . .
1.1.2 Isometry Groups of Bilinear Forms . . . . . . . . . . . . . . . . . . . . .
1.1.3 Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Quaternionic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 General and Special Linear Lie Algebras . . . . . . . . . . . . . . . .
1.2.2 Lie Algebras Associated with Bilinear Forms . . . . . . . . . . . . .
1.2.3 Unitary Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Quaternionic Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Lie Algebras Associated with Classical Groups . . . . . . . . . . .
1.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Closed Subgroups of
GL(n,
R)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Lie Algebra of a Closed Subgroup of
GL(n,
R)
. . . . . . . . . . .
1.3.4 Lie Algebras of the Classical Groups . . . . . . . . . . . . . . . . . . . .
1.3.5 Exponential Coordinates on Closed Subgroups . . . . . . . . . . .
1.3.6 Differentials of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . .
1.3.7 Lie Algebras and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Lie Algebra of an Algebraic Group . . . . . . . . . . . . . . . . . . . . .
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