Elliptic Operators and Compact Groups
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401 Michael Francis Atiyah Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford/England
Elliptic Operators and Compact Groups
Springer-Verlag Berlin· Heidelberg· New York 1974
AMS Subject Classifications (1970): 55B15, 58G10 ISBN 3-540-06855-4 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-06855-4 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 74-17529. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Preface
These lecture notes are based on a series of lectures I gave at the Institute for Advanced Study in 1971.
The
lectures were written up by John Hinrichsen, and I am very grateful to him for undertaking the task.
I am also
indebted to George Wilson who helped me revise and improve them.
CONTENTS
Page 1
Introduction Lecture 1:
Transversally Elliptic Operators .................•...
Page 3
Lecture 2:
The Index of Transversally Elliptic Operators ........... .
Page 9
Lecture 3:
The Excision and ~ltiplicative Properties ................... .
Page 18
Lecture 4:
The Naturality of the Index and the Localization Theorem ..... .
Page 27
Lecture 5:
The Index Homomorphism for G
= s1
••••••••••••••••••••••••
a-
-...
.s1
Page 36
Lecture 6:
The Operators
Lecture 7:
Toral Actions with Finite Isotropy Groups .............. .
Page 55
Lecture 8:
The Index Homomorphism for G = ~ ...................... ..
Page 65
The Cohomology Formula
Page 69
Lecture 9:
for (j "
Page 44
Lecture 10: Applications
Page 81
References
Page 92
Introduction These lectures, based on joint work with I, M. Singer, will describe an extension of the index theory of elliptic operators beyond that developed in [7 ], [8J, [9].
In those papers we studied an elliptic operator P
compact Lie group G.
invariant under a
Its index is a character of G defined by
*
index P
character (Ker P) - character (Ker P )
and a general formula for it was obtained in terms of the geometrical data.
In
fact, by density arguments, this is really a result about finite groups and G enters essentially in an algebraic way.
In the present lectures we shall consider
a more general situation in which G enters analytically.
Roughly speaking we
shall produce a synthesis of the index theory of elliptic operators with the Fourier analysis of compact groups. Given an action of G on a compact manifold X we shall consider a differential (or pseudo-differential) operator P
on X which is G-invariant
and is elliptic in the directions tra
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