Group Representations A Survey of Some Current Topics
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388 Ronald L. Lipsman University of Maryland, College Park, MD/USA
Group Representations A Survey of Some Current Topics
Springer-Verlag Berlin· Heidelberg· New York 1974
AMS Subject Classifications (1970): 22-02, 43-02 ISBN 3-540-06790-6 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-06790-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin · Heidelberg 1974. Library of Congress Catalog Card Number 74-4654. Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
To
Irving, Blanche, Lenny and Barry
PREFACE The idea for these notes was conceived as I watched the progress of several thesis students in my department at the University of Maryland.
Although they were all working in representation theory, and
although they talked together regularly, each seemed to have a limited idea of what the other was doing.
In particular, while one
was into semisimple Lie groups, another was involved with nilpotent Lie groups and there was a minimum of mutual understanding.
Now, it
is true that even the most advanced researchers in these areas have undertaken too little intercommunication; so, it is not at all surprising that students in the field should fare no better. With an eye towards remedying this unpleasant situation (at least locally), I conducted a survey course in group representations during the spring of 1973.
The main goal was to open the students
eyes to various different vistas within the general panorama of group representations.
A secondary goal, motivated in part by my current
work, was to present the various interactions and connections between these fields (as it has rarely if ever been done).
A third and
final goal was to help the students in their own work by providing numerous examples and exercises, indications of current problems in the field, and copious bibliographical references. I think the course had a fair degree of success.
In any event,
several students and faculty have encouraged me to put it in writing to see if anyone else might be interested.
The result is these notes.
It is my hope that they might prove useful to students trying to learn the field, workers in one area of representation theory who lack familiarity with another, or as a general source of reference.
VI The topics treated
are the following.
In Chapter I, I present
an introduction to the representation theory of semisimple Lie groups. This is a beautiful and elaborate subject -- due in large part to Barish-Chandra -- and to tell the full story would fill several books (if you don't believe me
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