Homogeneous Bounded Domains and Siegel Domains
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241
Soji Kaneyuki Scuola Normale Superiore, Pisalltalia Nagoya University, Nagoya!Japan
Homogeneous Bounded Domains and Siegel Domains
Springer-Verlag Berlin · Heidelberg · New York 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich Series: Scuola Normale Superiore, Pisa Adviser: E. Vesenti ni
241
Soji Kaneyuki Scuola Normale Superiore, Pisalltalia Nagoya University, Nagoya!Japan
Homogeneous Bounded Domains and Siegel Domains
Springer-Verlag Berlin · Heidelberg · New York 1971
AMS Subject Classifications (1970): 32-M-1O
ISBN 3-540-05702-1 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-05702-1 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 1971. Library of Congress Catalog Card Number 71-183988. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachlBergstr.
Preface
These notes are based on lectures on "Cayley transforms of homogeneous bounded domaids" given at Scuola Normale Superiore, Pisa during the academic year 197071. In
§
1, 2 and 3 we give definitions and fundamental results on
homogeneous bounded domains, Siegel domains of type II and their affine automorphism groups; Iwasawa subgroups and Iwasawa jalgebras are introduced and considered. In these sections we gave detailed proofs for wellknown facts in (14) and [7] . In §4 we consider the universal and the classifying jalgebras of an Iwasawa jalgebra, due to PiatetskiSapiro (15) . In §5 we
con-
struct the universal domain
of a
D
and the classifying domain
given homogeneous bounded domain
D We discuss the fibring of an arl• bitrary homogeneous bounded domain coming from a decomposition of its Iwasawa jalgebra into a jideal and a jsubalgebra; the fibring of over
D(t)
with the standard fibre
D l
D
is also considered and its uni-
versal property is obtained. The central topics are treated in §6§9. In §6, we generalize the Borel imbedding in the case of symmetric bounded domains to an arbi trary homogeneous bounded domain
D
o
; to do this we need several prop-
erties of the full automorphism group of
D
o
which are proved at the
moment by using the main theorem of Vinberg, Gindikin and PiatetskiSa piro [20] . Contrary to the case of symmetric bounded domains, the com plex homogeneous space in which
D
o
is imbedded holomorphically and
equivariantly is not compact in general. As an application of our Borel imbedding, we obtain a holomorphic imbedding of
D
o
into a complex
projective space. In §7 we obtain var
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