Buildings of Spherical Type and Finite BN-Pairs
These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is to present the solution of the following two problems: (A
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386
Jacques Tits
Buildings of Spherical Type and Finite BN-Pairs
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Jacques Tits College de France, 11, pI. Marcelin Berthelot 75231 Paris Cedex 05, France
1st Edition 1974 2nd Corrected Printing 1986
Mathematics Subject Classification (1970): Primary: 20G 15, 20G40, Secondary: 05B25, 05B30, 14M 15, 15A63, 17C40, 17005, 20005, 20H 15,50-00, 50A 15, 50A20, 50025,50030,50045, 50050,50099
ISBN 3-540-06757-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06757-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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 "VerwertungsgesellschaftWort", Munich. © by Springer-Verlag Berlin Heidelberg 1974 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Table of contents
Introduction 1.
Complexes
2.
Coxeter complexes
3.
Buildings
4.
Reduction
5.
The building of a semi-simple algebraic group
1
13
60 75
6.
Buildings of type
An
7.
Buildings of type
C • I. Polar spaces n
102
8.
Buildings of type
C • II. Projective embeddings of polar spaces n
117
9.
Buildings of type
C • III. Non-embeddable polar spaces n
177
10.
Buildings of type
F
200
11.
Finite BN-pairs of irreducible type and rank
12.
Appendix 1.
Shadows
232
Appendix 2.
Generators and relations
249
D
n
,E
92
n
4
>3
218
Addenda
274
Bibliography
277
Index of notations
287
Index
288
Introduction
These notes are a slightly revised and extended version of mimeographed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968.
Their main purpose is to present the solution of the
following two problems:
(A)
Determination of the buildings of rank
spherical type, other than ~
and H
group", about the excluded types
("of spherical type" means "with finite Weyl
4
H,
>; and irreducible,
cf. the addenda on p. 274).
Roughly speaking,
those buildings all turn out to be associated to simple algebraic or classical groups (cf. 6.;, 6.1;, 8.4.;, 8.22, 9.1, 10.2).
An easy application provides
of all finite groups with BN-pairs of irreducible type and rank
the enumeration
>;, up to normal
subgroups contained in B (cf. 11.7).
(B)
Determination of all isomorphisms between buildings of rank
>
2
and spherical type associated to algebraic or classical simple groups and, in particular, description of the full automorphism groups of such buildings
(cf. 5.8, 5.9,
5.10, 6.6, 6.1;, 8.6, 9.;, 10.4).
Except for the appendices, the notes are rather strictly oriented toward these goals.
In particular, the general theory of buildings and BN-pairs is de-
veloped only inasmuch as it
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